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Cover
MEASURE THEORY AND INTEGRATION

Page s1
PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J.Taft
Rutgers University
New Brunswick, New Jersey
Zuhair Nashed
University of Central Florida
Orlando, Florida
EDITORIAL BOARD
M.S.Baouendi
University of California,
San Diego
Jane Cronin
Rutgers University
Jack K.Hale
Georgia Institute of Technology
S.Kobayashi
Berkeley
Marvin Marcus
Santa Barbara
W.S.Massey
Yale University
Anil Nerode
Cornell University
Donald Passman
University of Wisconsin,
Madison
Fred S.Roberts
Rutgers University
David L.Russell
Virginia Polytechnic Institute and State University
Walter Schempp
Universität Siegen
Mark Teply
University of Wisconsin,
Milwaukee

Page s2
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED
MATHEMATICS
1. K.Yano, Integral Formulas in Riemannian Geometry (1970)
2. S.Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970)
3. V.S.Vladimirov, Equations of Mathematical Physics (A.Jeffrey, ed.; A.Littlewood, trans.) (1970)
4. B.N.Pshenichnyi, Necessary Conditions for an Extremum (L.Neustadt, translation ed.; K.Makowski, trans.) (1971)
5. L.Narici et al., Functional Analysis and Valuation Theory (1971)
6. S.S.Passman, Infinite Group Rings (1971)
7. L.Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971, 1972)
8. W.Boothby and G.L.Weiss, eds., Symmetric Spaces (1972)
9. Y.Matsushima, Differentiable Manifolds (E.T.Kobayashi, trans.) (1972)
10. L.E.Ward, Jr., Topology (1972)
11. A.Babakhanian, Cohomological Methods in Group Theory (1972)
12. R.Gilmer, Multiplicative Ideal Theory (1972)
13. J.Yeh, Stochastic Processes and the Wiener Integral (1973)
14. J.BarrosNeto, Introduction to the Theory of Distributions (1973)
15. R.Larsen, Functional Analysis (1973)
16. K.Yano and S.Ishihara, Tangent and Cotangent Bundles (1973)
17. C.Procesi, Rings with Polynomial Identities (1973)
18. R.Hermann, Geometry, Physics, and Systems (1973)
19. N.R.Wallach, Harmonic Analysis on Homogeneous Spaces (1973)
20. J.Dieudonné, Introduction to the Theory of Formal Groups (1973)
21. I.Vaisman, Cohomology and Differential Forms (1973)
22. B.Y.Chen, Geometry of Submanifolds (1973)
23. M.Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975)
24. R.Larsen, Banach Algebras (1973)
25. R.O.Kujala and A.L.Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973)
26. K.B.Stolarsky, Algebraic Numbers and Diophantine Approximation (1974)
27. A.R.Magid, The Separable Galois Theory of Commutative Rings (1974)
28. B.R.McDonald, Finite Rings with Identity (1974)
29. J.Satake, Linear Algebra (S.Koh et al., trans.) (1975)
30. J.S.Golan, Localization of Noncommutative Rings (1975)
31. G.Klambauer, Mathematical Analysis (1975)
32. M.K.Agoston, Algebraic Topology (1976)
33. K.R.Goodearl, Ring Theory (1976)
34. L.E.Mansfield, Linear Algebra with Geometric Applications (1976)
35. N.J.Pullman, Matrix Theory and Its Applications (1976)
36. B.R.McDonald, Geometric Algebra Over Local Rings (1976)
37. C.W.Groetsch, Generalized Inverses of Linear Operators (1977)
38. J.E.Kuczkowski and J.L.Gersting, Abstract Algebra (1977)
39. C.O.Christenson and W.L.Voxman, Aspects of Topology (1977)
40. M.Nagata, Field Theory (1977)
41. R.L.Long, Algebraic Number Theory (1977)
42. W.F.Pfeffer, Integrals and Measures (1977)
43. R.L.Wheeden and A.Zygmund, Measure and Integral (1977)
44. J.H.Curtiss, Introduction to Functions of a Complex Variable (1978)
45. K.Hrbacek and T.Jech, Introduction to Set Theory (1978)
46. W.S.Massey, Homology and Cohomology Theory (1978)
47. M.Marcus, Introduction to Modern Algebra (1978)
48. E.C.Young, Vector and Tensor Analysis (1978)
49. S.B.Nadler, Jr., Hyperspaces of Sets (1978)
50. S.K.Segal, Topics in Group Kings (1978)
51. A.C.M.van Rooij, NonArchimedean Functional Analysis (1978)
52. L.Corwin and R.Szczarba, Calculus in Vector Spaces (1979)
53. C.Sadosky, Interpolation of Operators and Singular Integrals (1979)
54. J.Cronin, Differential Equations (1980)
55. C.W.Groetsch, Elements of Applicable Functional Analysis (1980)

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56. I.Vaisman, Foundations of ThreeDimensional Euclidean Geometry (1980)
57. H.I.Freedan, Deterministic Mathematical Models in Population Ecology (1980)
58. S.B.Chae, Lebesgue Integration (1980)
59. C.S.Rees et al., Theory and Applications of Fourier Analysis (1981)
60. L.Nachbin, Introduction to Functional Analysis (R.M.Aron, trans.) (1981)
61. G.Orzech and M.Orzech, Plane Algebraic Curves (1981)
62. R.Johnsonbaugh and W.E.Pfaffenberger, Foundations of Mathematical Analysis (1981)
63. W.L.Voxman and R.H.Goetschel, Advanced Calculus (1981)
64. L.J.Corwin and R.H.Szczarba, Multivariable Calculus (1982)
65. V.I.Istrătescu, Introduction to Linear Operator Theory (1981)
66. R.D.Järvinen, Finite and Infinite Dimensional Linear Spaces (1981)
67. J.K.Beem and P.E.Ehrlich, Global Lorentzian Geometry (1981)
68. D.L.Armacost, The Structure of Locally Compact Abelian Groups (1981)
69. J.W.Brewer and M.K.Smith, eds., Emmy Noether: A Tribute (1981)
70. K.H.Kim, Boolean Matrix Theory and Applications (1982)
71. T.W.Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982)
72. D.B.Gauld, Differential Topology (1982)
73. R.L.Faber, Foundations of Euclidean and NonEuclidean Geometry (1983)
74. M.Carmeli, Statistical Theory and Random Matrices (1983)
75. J.H.Carruth et al., The Theory of Topological Semigroups (1983)
76. R.L.Faber, Differential Geometry and Relativity Theory (1983)
77. S.Barnett, Polynomials and Linear Control Systems (1983)
78. G.Karpilovsky, Commutative Group Algebras (1983)
79. F.Van Oystaeyen and A.Verschoren, Relative Invariants of Rings (1983)
80. I.Vaisman, A First Course in Differential Geometry (1984)
81. G.W.Swan, Applications of Optimal Control Theory in Biomedicine (1984)
82. T.Petrie and J.D.Randall, Transformation Groups on Manifolds (1984)
83. K.Goebel and S.Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984)
84. T.Albu and C.Năstăsescu, Relative Finiteness in Module Theory (1984)
85. K.Hrbacek and T.Jech, Introduction to Set Theory: Second Edition (1984)
86. F.Van Oystaeyen and A.Verschoren, Relative Invariants of Rings (1984)
87. B.R.McDonald, Linear Algebra Over Commutative Rings (1984)
88. M.Namba, Geometry of Projective Algebraic Curves (1984)
89. G.F.Webb, Theory of Nonlinear AgeDependent Population Dynamics (1985)
90. M.R.Bremner et al., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985)
91. A.E.Fekete, Real Linear Algebra (1985)
92. S.B.Chae, Holomorphy and Calculus in Normed Spaces (1985)
93. A.J.Jerri, Introduction to Integral Equations with Applications (1985)
94. G.Karpilovsky, Projective Representations of Finite Groups (1985)
95. L.Narici and E.Beckenstein, Topological Vector Spaces (1985)
96. J.Weeks, The Shape of Space (1985)
97. P.R.Gribik and K.O.Kortanek, Extremal Methods of Operations Research (1985)
98. J.A.Chao and W.A.Woyczynski, eds., Probability Theory and Harmonic Analysis (1986)
99. G.D.Crown et al., Abstract Algebra (1986)
100. J.H.Carruth et al., The Theory of Topological Semigroups, Volume 2 (1986)
101. R.S.Doran and V.A.Belfi, Characterizations of C*Algebras (1986)
102. M.W.Jeter, Mathematical Programming (1986)
103. M.Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986)
104. A.Verschoren, Relative Invariants of Sheaves (1987)
105. R.A.Usmani, Applied Linear Algebra (1987)
106. P.Blass and J.Lang, Zariski Surfaces and Differential Equations in Characteristic p> 0 (1987)
107. J.A.Reneke et al., Structured Hereditary Systems (1987)
108. H.Busemann and B.B.Phadke, Spaces with Distinguished Geodesics (1987)
109. R.Harte, Invertibility and Singularity for Bounded Linear Operators (1988)
110. G.S.Ladde et al., Oscillation Theory of Differential Equations with Deviating Arguments (1987)
111. L.Dudkin et al., Iterative Aggregation Theory (1987)
112. T.Okubo, Differential Geometry (1987)

Page s4
113. D.L.Stancl and M.L.Stancl, Real Analysis with PointSet Topology (1987)
114. T.C.Gard, Introduction to Stochastic Differential Equations (1988)
115. S.S.Abhyankar, Enumerative Combinatorics of Young Tableaux (1988)
116. H.Strade and R.Farnsteiner, Modular Lie Algebras and Their Representations (1988)
117. J.A.Huckaba, Commutative Rings with Zero Divisors (1988)
118. W.D.Wallis, Combinatorial Designs (1988)
119. W.Więsław, Topological Fields (1988)
120. G.Karpilovsky, Field Theory (1988)
121. S.Caenepeel and F.Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989)
122. W.Kozlowski, Modular Function Spaces (1988)
123. E.LowenColebunders, Function Classes of Cauchy Continuous Maps (1989)
124. M.Pavel, Fundamentals of Pattern Recognition (1989)
125. V.Lakshmikantham et al., Stability Analysis of Nonlinear Systems (1989)
126. R.Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989)
127. N.A.Watson, Parabolic Equations on an Infinite Strip (1989)
128. K.J.Hastings, Introduction to the Mathematics of Operations Research (1989)
129. B.Fine, Algebraic Theory of the Bianchi Groups (1989)
130. D.N.Dikranjan et al., Topological Groups (1989)
131. J.C.Morgan II, Point Set Theory (1990)
132. P.Biler and A.Witkowski, Problems in Mathematical Analysis (1990)
133. H.J.Sussmann, Nonlinear Controllability and Optimal Control (1990)
134. J.P.Florens et al., Elements of Bayesian Statistics (1990)
135. N.Shell, Topological Fields and Near Valuations (1990)
136. B.F.Doolin and C.F.Martin, Introduction to Differential Geometry for Engineers (1990)
137. S.S.Holland, Jr., Applied Analysis by the Hilbert Space Method (1990)
138. J.Oknínski, Semigroup Algebras (1990)
139. K.Zhu, Operator Theory in Function Spaces (1990)
140. G.B.Price, An Introduction to Multicomplex Spaces and Functions (1991)
141. R.B.Darst, Introduction to Linear Programming (1991)
142. P.L.Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991)
143. T.Husain, Orthogonal Schauder Bases (1991)
144. J.Foran, Fundamentals of Real Analysis (1991)
145. W.C.Brown, Matrices and Vector Spaces (1991)
146. M.M.Rao and Z.D.Ren, Theory of Orlicz Spaces (1991)
147. J.S.Golan and T.Head, Modules and the Structures of Rings (1991)
148. C.Small, Arithmetic of Finite Fields (1991)
149. K.Yang, Complex Algebraic Geometry (1991)
150. D.G.Hoffman et al., Coding Theory (1991)
151. M.O.González, Classical Complex Analysis (1992)
152. M.O.González, Complex Analysis (1992)
153. L.W.Baggett, Functional Analysis (1992)
154. M.Sniedovich, Dynamic Programming (1992)
155. R.P.Agarwal, Difference Equations and Inequalities (1992)
156. C.Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992)
157. C.Swartz, An Introduction to Functional Analysis (1992)
158. S.B.Nadler, Jr., Continuum Theory (1992)
159. M.A.AlGwaiz, Theory of Distributions (1992)
160. E.Perry, Geometry: Axiomatic Developments with Problem Solving (1992)
161. E.Castillo and M.R.RuizCobo, Functional Equations and Modelling in Science and Engineering (1992)
162. A.J.Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992)
163. A.Charlier et al., Tensors and the Clifford Algebra (1992)
164. P.Biler and T.Nadzieja, Problems and Examples in Differential Equations (1992)
165. E.Hansen, Global Optimization Using Interval Analysis (1992)
166. S.GuerreDelabrière, Classical Sequences in Banach Spaces (1992)
167. Y.C.Wong, Introductory Theory of Topological Vector Spaces (1992)
168. S.H.Kulkarni and B.V.Limaye, Real Function Algebras (1992)
169. W.C.Brown, Matrices Over Commutative Rings (1993)
170. J.Loustau and M.Dillon, Linear Geometry with Computer Graphics (1993)
171. W.V.Petryshyn, ApproximationSolvability of Nonlinear Functional and Differential Equations (1993)

Page s5
172. E.C.Young, Vector and Tensor Analysis: Second Edition (1993)
173. T.A.Bick, Elementary Boundary Value Problems (1993)
174. M.Pavel, Fundamentals of Pattern Recognition: Second Edition (1993)
175. S.A.Albeverio et al., Noncommutative Distributions (1993)
176. W.Fulks, Complex Variables (1993)
177. M.M.Rao, Conditional Measures and Applications (1993)
178. A.Janicki and A.Weron, Simulation and Chaotic Behavior of αStable Stochastic Processes (1994)
179. P.Neittaanmäki and D.Tiba,
Optimal Control of Nonlinear Parabolic Systems (1994)
180. J.Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994)
181. S.Heikkilä and V.Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)
182. X.Mao, Exponential Stability of Stochastic Differential Equations (1994)
183. B.S.Thomson, Symmetric Properties of Real Functions (1994)
184. J.E.Rubio, Optimization and Nonstandard Analysis (1994)
185. J.L.Bueso et al., Compatibility, Stability, and Sheaves (1995)
186. A.N.Michel and K.Wang, Qualitative Theory of Dynamical Systems (1995)
187. M.R.Darnel, Theory of LatticeOrdered Groups (1995)
188. Z.Naniewicz and P.D.Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995)
189. L.J.Corwin and R.H.Szczarba, Calculus in Vector Spaces: Second Edition (1995)
190. L.H.Erbe et al., Oscillation Theory for Functional Differential Equations (1995)
191. S.Agaian et al., Binary Polynomial Transforms and Nonlinear Digital Filters (1995)
192. M.I.Gil’, Norm Estimations for OperationValued Functions and Applications (1995)
193. P.A.Grillet, Semigroups: An Introduction to the Structure Theory (1995)
194. S.Kichenassamy, Nonlinear Wave Equations (1996)
195. V.F.Krotov, Global Methods in Optimal Control Theory (1996)
196. K.I.Beidar et al., Rings with Generalized Identities (1996)
197. V.I.Amautov et al., Introduction to the Theory of Topological Rings and Modules (1996)
198. G.Sierksma, Linear and Integer Programming (1996)
199. R.Lasser, Introduction to Fourier Series (1996)
200. V.Sima, Algorithms for LinearQuadratic Optimization (1996)
201. D.Redmond, Number Theory (1996)
202. J.K.Beem et al., Global Lorentzian Geometry: Second Edition (1996)
203. M.Fontana et al., Prüfer Domains (1997)
204. H.Tanabe, Functional Analytic Methods for Partial Differential Equations (1997)
205. C.Q.Zhang, Integer Flows and Cycle Covers of Graphs (1997)
206. E.Spiegel and C.J.O’Donnell, Incidence Algebras (1997)
207. B.Jakubczyk and W.Respondek, Geometry of Feedback and Optimal Control (1998)
208. T.W.Haynes et al., Fundamentals of Domination in Graphs (1998)
209. T.W.Haynes et al., eds., Domination in Graphs: Advanced Topics (1998)
210. L.A.D’Alotto et al., A Unified Signal Algebra Approach to TwoDimensional Parallel Digital Signal Processing (1998)
211. F.HalterKoch, Ideal Systems (1998)
212. N.K.Govil et al., eds., Approximation Theory (1998)
213. R.Cross, Multivalued Linear Operators (1998)
214. A.A.Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications (1998)
215. A.Favini and A.Yagi, Degenerate Differential Equations in Banach Spaces (1999)
216. A.Illanes and S.Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999)
217. G.Kato and D.Struppa, Fundamentals of Algebraic Microlocal Analysis (1999)
218. G.X.Z.Yuan, KKM Theory and Applications in Nonlinear Analysis (1999)
219. D.Motreanu and N.H.Pavel, Tangency, Flow Invariance for Differential Equations, and Optimization Problems (1999)
220. K.Hrbacek and T.Jech, Introduction to Set Theory, Third Edition (1999)
221. G.E.Kolosov, Optimal Design of Control Systems (1999)
222. N.L.Johnson, Subplane Covered Nets (2000)
223. B.Fine and G.Rosenberger, Algebraic Generalizations of Discrete Groups (1999)
224. M.Väth, Volterra and Integral Equations of Vector Functions (2000)
225. S.S.Miller and P.T.Mocanu, Differential Subordinations (2000)

Page s6
226. R.Li et al., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000)
227. H.Li and F.Van Oystaeyen, A Primer of Algebraic Geometry (2000)
228. R.P.Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition (2000)
229. A.B.Kharazishvili, Strange Functions in Real Analysis (2000)
230. J.M.Appell et al., Partial Integral Operators and IntegroDifferential Equations (2000)
231. A.I.Prilepko et al., Methods for Solving Inverse Problems in Mathematical Physics (2000)
232. F.Van Oystaeyen, Algebraic Geometry for Associative Algebras (2000)
233. D.L.Jagerman, Difference Equations with Applications to Queues (2000)
234. D.R.Hankerson et al., Coding Theory and Cryptography: The Essentials, Second Edition, Revised and Expanded (2000)
235. S.Dăscălescu et al., Hopf Algebras: An Introduction (2001)
236. R.Hagen et al., C*Algebras and Numerical Analysis (2001)
237. Y.Talpaert, Differential Geometry: With Applications to Mechanics and Physics (2001)
238. R.H.Villarreal, Monomial Algebras (2001)
239. A.N.Michel et al., Qualitative Theory of Dynamical Systems: Second Edition (2001)
240. A.A.Samarskii, The Theory of Difference Schemes (2001)
241. J.Knopfmacher and W.B.Zhang, Number Theory Arising from Finite Fields (2001)
242. S.Leader, The KurzweilHenstock Integral and Its Differentials (2001)
243. M.Biliotti et al., Foundations of Translation Planes (2001)
244. A.N.Kochubei, PseudoDifferential Equations and Stochastics over NonArchimedean Fields (2001)
245. G.Sierksma, Linear and Integer Programming: Second Edition (2002)
246. A.A.Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions (2002)
247. B.G.Pachpatte, Inequalities for Finite Difference Equations (2002)
248. A.N.Michel and D.Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002)
249. J.R.Weeks, The Shape of Space: Second Edition (2002)
250. M.M.Rao and Z.D.Ren, Applications of Orlicz Spaces (2002)
251. V.Lakshmikantham and D.Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Second Edition (2002)
252. T.Albu, Cogalois Theory (2003)
253. A.Bezdek, Discrete Geometry (2003)
254. M.J.Corless and A.E.Frazho, Linear Systems and Control: An Operator Perspective (2003)
255. I.Graham and G.Kohr, Geometric Function Theory in One and Higher Dimensions (2003)
256. G.V.Demidenko and S.V.Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the HighestOrder Derivative (2003)
257. A.Kelarev, Graph Algebras and Automata (2003)
258. A.H.Siddiqi, Applied Functional Analysis: Numerical Methods, Wavelet Methods, and Image Processing (2004)
259. F.W.Steutel and K.van Harn, Infinite Divisibility of Probability Distributions on the Real Line (2004)
260. G.S.Ladde and M.Sambandham, Stochastic Versus Deterministic Systems of Differential Equations (2004)
261. B.J.Gardner and R.Wiegandt, Radical Theory of Rings (2004)
262. J.Haluška, The Mathematical Theory of Tone Systems (2004)
263. C.Menini and F.Van Oystaeyen, Abstract Algebra: A Comprehensive Treatment (2004)
264. E.Hansen and G.W.Walster, Global Optimization Using Interval Analysis: Second Edition, Revised and Expanded (2004)
265. M.M.Rao, Measure Theory and Integration, Second Edition, Revised and Expanded
Additional Volumes in Preparation

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Page i
MEASURE THEORY AND INTEGRATION
Second Edition, Revised and Expanded
M.M.RAO
University of California, Riverside
Riverside, California, U.S.A.
MARCEL DEKKER INC.
NEW YORK • BASEL

Page ii
The first edition was published by John Wiley & Sons (1987).
This edition published in the Taylor & Francis eLibrary, 2006.
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Current printing (last digit):
10 9 8 7 6 5 4 3 2 1

Page iii
To the memory of
my brother
MUKUNDA RAO
and my friend
P.R.KRISHNAIAH
whose lives were cut short
so suddenly

Page iv

Page v
Preface to the Second Edition
It is over sixteen years since the original edition of this book was published, and it has been out of print since 1992. A paperback copy of it was printed by World
Publishing Corporation in Beijing in 1990, exclusively “for sale in P.R.China.” The current edition contains all the original material and many new topics including an
additional chapter (a considerably long one) as well as new sections and exercises; both elementary and advanced. I shall elaborate on these points and discuss their
significance at the same time.
Since the first six chapters and the first two sections of Chapter 7 can be used, even with selected omissions, as a text for a standard “Real Analysis” course, as
originally intended, that structure is maintained. Thus the point of view explained in the preface of the first edition (appended here) is still valid, and the added material
has the following special features. Several elementary problems are included for practice by students in a real analysis course, and they are usually numbered with
Greek letters, α, β, γ, δ in most of the sections. Also I have rewritten some passages for greater readability. Following some user suggestions, a new section on the
“four basic theorems of Banach spaces” is included whereas only the uniform boundedness principle was discussed in the earlier edition. This is Section 4.6 and along
with these results there is also a substantial discussion on their role in both abstract and concrete applications. Moreover the presentation illuminates the fact that
integration has two closely related aspects, namely the classical (or absolute or Lebesgue) type and the generalized (or nonabsolute or Riemann) type. These are so
labeled as Chapters 4 and 7 respectively, the latter being the new one, and its significance will now be explained.
Although the Denjoy/Perron integral is treated in Saks’ ((1937); corrected reprint (1964) as second edition) classical monograph, and its abstract extensions by
Romanovskii (1941) and Solomon (1969) among others are available, they are relatively complicated to employ in applications. This has changed with a new approach
(independently) discovered by Henstock and Kurzweil around 1960, and a

Page vi
modified version by McShane (1969) (see also his book (1983) and one by McLeod (1980)) who has, moreover, generalized it to vector integration—all keeping the
nonabsolute feature in view. While teaching a “Real Analysis” class recently, I used Bartle’s (2001) nice exposition of this new version of generalized integration, and
also followed Pfeffer’s (1993) RiemannStieltjes treatment. These authors have restricted their presentations to the level of advanced calculus or just the beginning
graduate classes, avoiding measurability concepts as far as possible. This is clear from Bartle’s book which contains a sketch of the latter in the last chapters. However,
to appreciate the new approach fully, it is necessary to discuss the analog of the classical inverse Hölder inequality, namely to answer the following question: If f and f g
are integrable in the (generalized) sense already defined, for all such f, what must g be? This is important because it leads directly to the introduction of a (norm)
topology in the space of HenstockKurzweil (HK), or equivalently, DenjoyPerron integrable functions analogous to the Lebesgue case. A solution is already available
in Sargent (1948) and the work depends on the structure of Lebesgue measurable functions, and thus one has to consider measurability (as, for instance, treated in
Chapter 3). It leads naturally to an analysis of the adjoint space of this class, similar to the classical Lpspaces that are so important in applications. This was not
included in any of the above books, and not even in the encyclopedic work of Schector’s (1997) who also treated the nonabsolute case. I have briefly discussed these
ideas in my class, and decided to fill the gap in the above works, with this revised version of my volume to exhibit at the same time the interplay between the absolute
and the nonabsolute integration processes since this is important in contemporary applications.
The material in the new Chapter 7 is concerned with the HKprocedures and some variations. It is seen that the basic geometric structure of the generalized
RiemannStieltjes integral is essential to the theory here, and even motivates an extended later study of the volume Geometric Integration, by Whitney (1957) as well as
the monograph Geometric Measure Theory, by Federer (1969). Further, it leads to vector integrals of various types, e.g., the Pettis, McShane, Bochner, Wiener, Itô,
Stratonovich, and Feynman. I have indicated how these arise as a flow of ideas of nonabsolute integration, and explained the existence of a deep relationship, through a
boundedness principle for

Page vii
mulated by Bochner (1956), connecting it with absolute integration locally. Some highlights of this work will be mentioned here.
Section 7.1 contains an essentially complete treatment of the HKprocedure for the Denjoy integral on the line. This long section can be directly covered (omitting a
few statements on LebesgueStieltjes measures) for those who do not need the properties of measurable functions and the LSintegrals. This may be augmented by
Section 7.2 which is concerned with product (Denjoy) integrals and explains some of the inherent problems with this generalization especially for the Fubini theorem.
Here the method of Perron, as modified by Leader, is discussed in some detail. The remaining sections of this chapter treat vector integrals, and their products, for
more advanced readers. (This is somewhat similar to the work of Sections 6.3–6.5.) They are as follows.
The material in Sections 7.3–7.5 depends on (and uses) the contents of the first six chapters, which anyway is normally included in a standard graduate real analysis
course. Thus Section 7.3 treats integrals of vectorvalued functions relative to a scalar measure, including those of Pettis, McShane and Bochner classes as well as their
interrelations. Here McShane’s approach bridges the other two and illuminates the structure of all these integrals and this is explained. For this the early part of Chapter
4 is required. On the other hand integrating scalar functions relative to vector measures uses the ideas of the HKprocedure and the work of Section 7.1 is useful. This
is the content of Section 7.4. It is noted here that the resulting nonabsolute vector integral, for which the dominated convergence statement is valid, obeys an extended
form of the Bochner boundedness principle. As a consequence, one finds that these nonabsolute integrals are bounded locally by absolute integrals relative to some
regular σfinite measure on a class of basic objects containing the compactly supported continuous functions. There are standard techniques related to truncation (or
“stopping times” procedures) to extend the local theory to a global version. That is indicated here. [This is usually employed, almost routinely, in areas such as
stochastic integration.] Finally the last parts of Sections 7.4 and 7.5 contain several important complements on characterizing functions that are Fourier transforms of
(possibly vectorvalued) measures and integration on general (not necessarily locally compact) topological spaces.
To keep the exposition within bounds, and to cover the main is

Page viii
sues, a number of results of interest are outlined in exercises with extended sketches in each of these sections. They cover Fourier transforms on Hilbert (and certain
other vector) spaces including some “matrix spaces” (or their direct sums), as well as the Wiener and Feynman integrals. It is an interesting fact that the finitely additive
HKintegrals play a key role in defining the Feynman integral as shown by Henstock (1973) and elaborated by Muldowney (1987) who detailed the existence proof
via Fresnel’s (nonabsolute) integrals. Other methods for the same integral use a sequence of steps (with Lebesgue type integrals) as was done, for instance, by
R.H.Cameron and his associates. These take considerably longer arguments. The direct approach thus seems to justify a study of the HKmethods of nonabsolute
finitely additive integration. [For the Wiener as well as the “white noise” integrals, one can use the projective limit procedure of Section 6.4, as discussed in an exercise
here, but this is not directly applicable for the Feynman integral!]
Another class of vector integrals related to “stable processes” admits extensions to “formally stable classes” introduced and detailed by Bochner (1975). Those are
briefly discussed. Such results, intended for advanced readers to pursue the subject further, are discussed in a sequence of graded exercises with detailed hints in
Chapter 7. I would like to urge young researchers to study them closely since they supplement the subject of integration in many ways and present interesting topics for
possible further investigation. In fact, here and throughout the book, I have presented several problems, with copious hints, to enhance the value of the text without
lengthening the volume too much. There are over 490 problems spread over various sections of the book constituting an increase of nearly 100 new ones in this edition,
many with several parts.
The work in Chapter 7 also shows that local absolute integration plays a vital role in the study of nonabsolute integrals, and exemplifies Professor McShane’s
description (1983) of Lebesgue’s integral as “the workhorse of contemporary mathematical analysis.” Thus a further treatment of the latter integrals and sharper
properties of measures are again the main items of the last four chapters. Their contents are adequately described already in the Preface of the original edition. Some
additions are made to the chapters also. Thus a classification of measures on locally compact and general topological spaces, Daniell’s integral (obtained from
Choquet’s capacity theory) and the lifting the

Page ix
orem are the topics there. The final chapter contains an extended discussion of the Stone isomorphism theorem along with some of its many important applications. As
noted in the Preface to the first edition, to ease cross referencing, I restate certain concepts at various places, at the cost of annoying some readers. I hope this will help
in recalling those concepts immediately.
I hope that the new material, and especially a detailed analysis of nonabsolute integration, makes this edition more useful both as a graduate text (the first half) as well
as for reference purposes and selfstudy by advanced readers (the last half). The revision was accomplished with a UCR Academic Senate Grant. The composition of
the text was ably and expeditiously done by Ms. Ambika Vanchinathan using the word processing, I am indeed very grateful to her for this help, given from a
distance (Chennai, India) using the email service. Finally I shall fondly hope that the material covered here stimulates the interest of graduate students as well as young
researchers in abstract analysis and its applications.
M.M.Rao

Page x

Page xi
Preface to the First Edition
This book presents a detailed exposition of the general theory of measure and integration. It is meant to be a text for a first year graduate course, often given under such
titles as “Measure Theory”, “Integration”, “Real Analysis”, or “Measure and Integral”. The material is unified from various sets of notes, and of experience gained, from
my frequent teaching of such a class since 1960.
Generally the subject is approached from two points of view as evidenced from the standard works. Traditionally one starts with measure, then defines the integral
and develops the subject following Lebesgue’s work. Alternatively one can introduce the integral as a positive linear functional on a vector space of functions and get a
measure from it, following the method of Daniell’s. Both approaches have their advantages, and eventually one needs to learn both methods. As the preponderance of
existing texts indicates, the latter approach does not easily lead to a full appreciation of the distinctions between the (sigma) finite, localizable, and general measures, or
their impact on the subject. On the other hand, too often the former approach appears to have little motivation, rendering the subject somewhat dry. Here I have tried
to remedy this by emphasizing the positive and minimizing the negative aspects of these methods, essentially following the natural growth of the subject in its
presentation. This book covers all the standard theory and includes several contemporary results of interest for different applications.
Each topic is introduced with ample motivation. I start with an abstraction of lengths, areas, volumes and other measurements of known geometric figures and
develop the basic ideas of Lebesgue in This is then used as a model and a reference for the general study leading to the Carathéodory process. The measure
approach as a basic step is essentially natural in such areas as functional analysis, probabil

Page xii
ity and statistics, and ergodic theory, whereas reference to Lebesgue’s method keeps in view the applications to differential equations and mathematical physics among
others. I now indicate some features of the present treatment and contrast it with earlier works.
The Carathéodory process, which here takes center stage and helps in an efficient presentation, was effectively used earlier by Dunford and Schwartz (1958), by
Zaanen (1967), and more recently by Sion (1968, 1973). In addition, inner measures have a special role in several types of extension procedures. This is particularly
true in obtaining regular expressions of topological measures. It was indicated by Royden (1968), but the full potential is utilized and emphasized here. In the context of
topological measures, I have presented the Henry extension theorem and used it later in shortening and illuminating the structure of some other results. (See, e.g.,
Theorems 6.4.7–8 for novel applications.)
Inclusion of image measures and vague convergence is discussed for sequences. For instance, Skorokhod’s representation (cf. Theorem 3.3.5) in this context is of
interest in probability and Fourier analysis. A few results given in Section 4.3, on integration of not necessarily measurable functions, exhibit the power of
Carathéodory’s process and also help in simplifying some arguments for product integrals in Chapter 6, while enlarging the scope of applications of Lebesgue’slimit
theorems. An account of Lpspaces is included in Sections 4.5 and 5.5, illustrating the methods of integration. Then signed measures and the VitaliHahnSaks theorem
find a natural place there. Further a detailed treatment of differentiation of set functions is given. The RadonNikodým theorem is presented with multiple proofs and
shown to imply the JordanHahn decomposition. This exhibits a deeper equivalence between these two theorems since each is also shown to be provable
independently of the other and deducible from one another. The localizability concept introduced earlier is used to establish Segal’s theorem on the equivalence of the
RadonNikodým property for µ, with the dual of L1(μ) as L∞(μ). Also absolutely continuous and completely monotone real functions on the line are treated. Only
Zaanen (1967) had considered an extended discussion of the RadonNikodým theorem. However, localizability is also found to be interesting in product measure
theory. (See, e.g., Exercises 6.2.7 and 6.2.8.)
Infinite product measures are given an extended treatment. I include the KolmogorovBochner, Prokhorov, Tulcea, and FubiniJessen

Page xiii
theorems. Their relation with two martingale convergence results is established. In the earlier work, only Hewitt and Stromberg (1965) have considered an aspect of
this theory. These results find an important place in the current work on stochastic analysis. As useful applications, Bochner’s representation theorem on continuous
positive definite functions on the line, and a realization of an abstract Hilbert space as a subspace of an L2(μ)space are presented (Section 6.5).
A novel treatment is an inclusion of Choquet’s capacity theorem for analytic sets from which one obtains the Daniell integration as a consequence. This approach
was indicated by Meyer (1966), and a comprehensive account is given here. Recently Jacobs (1978) also considered Choquet’s theorem, but my purpose is to obtain
Daniell’s results early and quickly from the former. Next an elementary proof of the lifting theorem, due to T.Traynor (1974), is included. This result vividly shows the
facility and problems created by sets of measure zero in the Lebesgue theory, in addition to its intrinsic importance. Finally the interplay of topology and measure is
expounded in Chapters 9 and 10. Here regular measures on locally compact and some general topological spaces, as well as Pettis’s theorem on extension of a
measure from a lattice (usually of compact sets) to the σalgebras generated by them, the RieszMarkov theorem, and an integral representation of local functionals of
Gel’fandVilenkin on compactly based continuous function spaces are presented. Topologies induced by a measure, the Stone isomorphism theorem of a measure
algebra, and some applications as well as a treatment of the Haar measure find a place here.
I have presented both the classical and some contemporary topics often used in the current mathematical activity. Indeed, almost all the measure and integration
theory needed by probabilists and functional analysts, and in particular most of what is needed for my earlier books (1979, 1981, 1984), is found here. I hope it will be
useful to others in similar applications in which measure and integration play an important role.
The book is primarily intended as text for a year’s or a semester’s course on contemporary real analysis. The following suggestions are offered for this purpose.
Omitting a few special topics, such a standard analysis course is covered by the first six chapters.
A respectable course for a semester (or a two quarter) length course is obtained by the selection: Chapter 1, Sections 2.1–2.3, 2.6, 3.1, 3.2,

Page xiv
4.1, 4.3, 4.5, 5.1; the first two results of Section 5.2; Section 5.3; the first half of Section 5.5 and Sections 6.1 and 6.2. If any time is left one can cover Chapter 7 for
either of the above two classes. However, Chapter 9 can be studied immediately after the first four chapters, with only a reference to the RadonNikodým theorem, or
by omitting Theorem 9.3.5. For a year’s course, it is possible to cover all the first seven chapters. Chapters 8, 9 are essentially independent and can be taken up in any
order (after Chapter 4) and then Chapter 10 may be appended.
There is more than enough material for a year’s course, even with selected omission of certain sections, according to one’s tastes. However, the treatment
throughout is considerably detailed with alternative arguments (including some repetitions of notation and definitions to ease a search by the reader), keeping the
student’s needs in mind. Therefore, the book is also suitable for selfstudy.
A prerequisite for this text is a knowledge of advanced calculus such as that found in Bartle (1976) or Rudin (1976). Essentially everything else is detailed here. A
short appendix presents some results from topology and set theory with references. I have included many exercises (over 400) of varying difficulty at the end of each
section and those which are less simple are provided with hints. As the study progresses, the reader is expected to gain sophistication, and in any case, some of the
more advanced topics can be skipped in a first reading.
The numbering system is standard: m.n.p denotes the chapter (m), the section (n), and the proposition, definition, or exercise, etc. (p). In a given chapter, m is
omitted, and in a section, m.n is also omitted.
The material is influenced by the many texts used before, but I should especially like to acknowledge that my point of view has shifted from the traditional one with
the appearance of Dunford and Schwartz (1958) at the beginning of my career. This and that of Sion’s books (1968, 1973) have strengthened my belief in the efficacy
of the Carathéodory process even for pedagogical purposes. Also, the reactions of my audiences have encouraged me in this approach.
The preparation of the manuscript over the past two years has been facilitated by a year’s UCR sabbatical leave, spent at the Institute for Advanced Study during
1984–1985, partially supported by an ONR contract. Typing of my handwritten and difficult manuscript, and its revision, was patiently carried out by Mrs. Eva
Stewart. This preparation was helped by a UCR Academic Senate grant. Joseph Sroka

Page xv
and Derek Chang assisted me in proofreading and preparation of indexes. To all these people and institutions I wish to express my deep appreciation.
Riverside, California
May 1987
M.M.Rao

Page xvi

Page xvii
Contents
Preface to the Second Edition v
Preface to the First Edition xi
1. Introduction and Preliminaries
1
1.1. Motivation and Outlook, 1
1.2. The Space as a Model, 4
1.3. Abstraction of the Salient Features, 14
2. Measurability and Measures
21
2.1. Measurability and Class Properties, 21
2.2. The Lebesgue Outer Measure and the Carathéodory Process, 30
2.3. Extensions of Measures to Larger Classes, 67
2.4. Distinction between Finite and Infinite Measures, 86
2.5. Metric Outer Measures, 92
2.6. LebesgueStieltjes Measures, 99
3. Measurable Functions
110
3.1. Definition and Basic Properties, 110
3.2. Measurability with Measures and Convergence, 120
3.3. Image Measures and Vague Convergence, 136
4. Classical Integration 147
4.1. The Abstract Lebesgue Integral, 147
4.2. Integration of Nonmeasurable Functions, 163
4.3. The Lebesgue Limit Theorems, 171

Another Random Document on
Scribd Without Any Related Topics

Bjarne Grimolfsson, Wineland voyager, i. 319, 320, 326, 329, 330; ii.
20
Bjarne Herjulfsson, traditional discoverer of Wineland, i. 314, 317,
334; ii. 21
Bjarneyjar (Bear-islands), Greenland, i. 301, 302, 304, 321, 322,
323, 335, 336
Björn Breidvikingekjæmpe, i. 360; ii. 49-50, 53, 54, 56
Björn Einarsson Jorsalafarer, ii. 82, 106, 112, 113
Björn Jónsson of Skardsá (Annals of Greenland), i. 263, 282-3, 288,
292, 295, 299, 301, 308, 309, 321, 377; ii. 35, 37, 82, 83, 239
Björn Thorleifsson, shipwrecked in Greenland, ii. 82
Björnbo, Dr. A. A., i. 200, 201, 202, 297; ii. 2, 31, 32, 116, 123, 127,
132, 147, 154, 193, 220, 221, 223, 224, 225, 226, 233, 234, 240,
249, 250, 253, 261, 262, 264, 273, 277, 278, 281, 283, 284, 287,
289, 332, 353, 368, 369, 370, 374, 375
Björnbo and Petersen, i. 226; ii. 85, 123, 124, 127, 219, 231, 234,
249, 250, 252, 253, 254, 255, 256, 258, 262, 263, 267, 273, 275,
277, 377
Bláserkr (Greenland), i. 267, 291-6
Blom, O., ii. 8
Boas, F., ii. 69, 70
Boats of hides (coracles, &c.), in the Œstrymnides, i. 38, 39;
Scythians, Saxons, &c., i. 154, 242;

Greenlanders’, i. 305;
Irish, ii. 92;
Skrælings’, in Wineland, i. 327; ii. 10, 19;
in Trondhjem cathedral, ii. 85, 89, 117, 269, 270;
in Irish tales, i. 336; ii. 20;
in Newfoundland (?), ii. 367;
Eskimo, see Kayaks and Women’s Boats
Bobé, Louis, ii. 126
Borderie, A. de la, i. 234
Borgia mappamundi, ii. 284-5
Bornholm, i. 169, 180; ii. 204, 265
Bothnia, Gulf of, i. 169, 187; ii. 269;
in mediæval cartography, ii. 219
“Boti,” i. 87
Bran, Voyage of, i. 198, 354, 356, 365, 370; ii. 56
Brandan, Legend of, i. 281-2, 334, 337, 344, 345, 358-364, 366,
376; ii. 9, 10, 13, 18, 19, 43-5, 50, 51, 61, 64, 75, 151, 206, 214,
228-9, 234
Brattalid, in Greenland, i. 268, 270, 271, 275, 317, 319, 320, 331
Brauns, D., i. 377; ii. 56
“Brazil,” Isle of (Hy Breasail, O’Brazil, &c.), i. 3, 357, 379; ii. 30, 228-
30, 279, 294-5, 318;
expeditions to find, ii. 294-5, 301, 325
Breda, O. J., ii. 31

Brenner, O., i. 58
Brinck (Descriptio Loufodiæ), i. 378
Bristol, trade with Iceland, ii. 119, 279, 293;
Norwegians living at, ii. 119, 180;
expeditions sent out from, ii. 294-5, 298, 301, 304, 325, 326, 327,
330, 331
Britain, i. 193, 234, 240, 241;
visited by Pytheas, i. 49, 50-3;
Cæsar on, i. 79-80;
Mela on, i. 97;
Pliny on, ii. 106;
Ptolemy on, i. 117;
in mediæval cartography, ii. 220, 227
Brittany, cromlechs in, i. 22;
tin in, i. 23, 26, 27, 29-31, 38-42
Broch, Prof. Olaf, ii. 142, 175, 176
Brögger, A. W., i. 14
Brönlund, Jörgen, i. 2-3
Bruun, D., i. 164, 270, 271, 274, 275
Bugge, Prof. A., i. 136, 137, 138, 146, 163, 164, 166, 170, 173, 234,
245, 246, 258, 297, 304; ii. 7, 55, 80, 168, 201
Bugge, Sophus, i. 93, 94, 103, 132, 134, 135, 136, 138, 146, 148,
207, 273; ii. 27, 28, 175
Bulgarians of the Volga, ii. 142-5, 195, 200, 210

Bunbury, E. H., i. 30, 107
“Burgundians” (== Bornholmers ?), i. 169, 180
Burrough, Stephen, ii. 173
Cabot, John, i. 3, 115, 312; ii. 130, 295-330, 333, 343, 374, 377;
settles at Bristol, ii. 297;
voyage of 1496, ii. 299-301;
voyage of 1497, ii. 301-23;
voyage of 1498, ii. 311, 324-8, 349;
his discovery premature, ii. 343
Cabot, Sebastian, ii. 129, 130, 295-6, 299, 301-2, 308, 319, 326,
329, 330, 332, 333, 336-43;
reported voyage of 1508-9, ii. 336-40;
doubtful voyage of 1516 or 1517, ii. 340-2;
his credibility, ii. 296, 298, 303, 329, 338-40;
map of 1544, attributed to, ii. 303, 309, 310, 314-5, 319-20
Cæsar, C. Julius, i. 39, 40, 79-80, 92, 242
Callegari, G. V., i. 43, 58, 59
Callimachus, i. 375
Callisthenes (Pseudo-), ii. 213, 234
Calypso, i. 347, 355, 370; ii. 43
“Cananei,” i. 154-5
Canary Isles, i. 117, 348-50, 362, 376; ii. 2

Canerio map (1502-07), ii. 368
Cannibalism, among the Irish, Scythians, Celts, Iberians, i. 81;
Issedonians, i. 81;
Massagetæ, i. 81, 148;
in Scandinavia, i. 149
Cantino, Alberto, his map of 1502, ii. 316, 350-1, 355, 361, 362,
364, 365, 368-74;
his letter of Oct. 1501, ii. 349-52, 360, 361, 362, 363, 367, 372
Canto, Ernesto do, ii. 331
Cape Breton, i. 324, 329, 335; ii. 309, 312, 314, 315, 316, 317, 319,
321, 322;
John Cabot’s probable landfall in 1497, ii. 314-15
Capella, Marcianus, i. 123, 126, 184, 188, 195, 197, 334
Carignano, Giovanni da, compass-chart by, ii. 220-2, 227, 235
“Carte Pisane,” ii. 220
Carthage, Sea-power of, i. 45, 75
Caspian Sea, i. 10, 74, 76, 122; ii. 142, 183, 195, 197, 213
Cassiodorus, i. 120, 128-30, 132, 137, 138, 142, 154, 155, 203
Cassiterides, i. 23, 24, 25, 27-9, 89; ii. 47, 48
Catalan Atlas, mappamundi of 1375, ii. 233, 266, 292
Catalan compass-chart at Florence, ii. 231, 232-3, 235
Catalan compass-chart (15th century) at Milan, ii. 279, 280

Catalan sailors and cartographers (see Compass-charts), ii. 217
Catapult, used by the Skrælings, i. 327; ii. 6-8, 92
Cattegat, The, i. 93, 100, 101, 102, 105, 169, 180
“Cauo de Ynglaterra” on La Cosa’s map, ii. 314-5, 317, 321-2;
probably Cape Breton, ii. 314;
or Cape Race (?), ii. 321-2
Celts, i. 19, 41, 42, 68, 81, 208;
early Celtic settlement of the Faroes, i. 162-4;
of Iceland, i. 167, 258;
possible Celtic population in Scandinavia, i. 210;
mythology of the, i. 379
Chaldeans, i. 8, 47
Chancellor, Richard, ii. 135
Chinese myths of fortunate isles, i. 377; ii. 213
Christ, The White, ii. 44, 45, 46
Christ, Wilhelm, i. 14, 37
Christianity introduced in Iceland, i. 260, 332;
introduced in Greenland, i. 270, 272, 357, 332, 380;
decline of, in Greenland, ii. 38, 100-2, 106, 113, 121
Christian IV. of Denmark, ii. 124, 178
Christiern I. of Denmark, ii. 119, 125, 127, 128, 132, 133, 134, 345
Chukches, i. 212

Church, ii. 301
Cimbri, i. 14, 21, 82, 85, 91, 94, 99, 100, 101, 118, 145
Cimmerians, i. 13, 14, 21, 79, 145
Circumnavigation, Idea of, i. 77, 79; ii. 271, 291-3, 296-7
Clavering, ii. 73
Clavus, Claudius, i. 226, 303; ii. 11, 17, 85, 86, 89, 117, 248-76,
284;
his Nancy map and text, ii. 249, 250, 253, 255-69;
his later map and Vienna text, ii. 250, 251, 252-3, 254, 265-76;
his methods, ii. 252-3, 259-61;
his influence on cartography, ii. 276-9, 335, 368, 369, 370, 371
Cleomedes, i. 44, 52, 53, 55, 57, 134
Codanovia, island, i. 91, 93-4, 103
Codanus, bay, i. 90-5, 101, 102, 103, 105, 118
Collett, Prof. R., i. 345; ii. 91
Collinson, R., ii. 129
Columbus, i. 3, 77, 79, 115, 116, 312, 376; ii. 291, 292, 293, 294,
295, 296, 297, 300, 307, 310, 325
Compass, Introduction of, i. 248; ii. 169, 214, 215-6;
variation of, ii. 217, 307-8, 370-1
Compass-charts, ii. 215-36, 265, 279, 280, 282, 308, 313;
development of, ii. 215-8;

limits of, ii. 218
Congealed or curdled sea, beyond Thule, i. 65-9, 70, 100, 106, 121,
165, 181, 195, 363, 376; ii. 149, 200, 231
Connla the Fair, Tale of, i. 371
Contarini, G., ii. 303, 336, 337, 338, 342, 343
Converse, Harriet Maxwell, i. 377
Cornwall, Tin in, i. 23, 29, 31
Corte-Real, Gaspar, ii. 130, 328, 330, 331, 332, 347-53, 354, 357,
358-66, 373;
letters patent to (1500), ii. 347;
voyage of 1500, ii. 360;
voyage of 1501, ii. 347-53, 360-75;
his fate, ii. 353, 375;
his discoveries, ii. 354-5, 362, 364
Corte-Real, João Vaz, unhistorical expedition attributed to, ii. 359
Corte-Real, Miguel, ii. 353, 360, 361;
letters patent to, ii. 353, 355, 376;
voyage of 1502 or 1503, ii. 353, 376;
probably reached Newfoundland, ii. 376;
his fate, ii. 376
Corte-Real, Vasqueanes, refused leave to search for his brothers, ii.
377
Corte-Real, Vasqueanes IV., reported expedition of, in 1574, ii. 378
Cosa, Juan de la, map by, ii. 302, 309-18, 321, 374;
represents Cabot’s discoveries of 1497, ii. 311-2

Cosmas Indicopleustes, i. 126, 127, 128; ii. 183
Costa, B. T. de, ii. 129, 214
“Cottoniana” mappamundi, i. 180, 182, 183; ii. 192-3, 208, 220, 284
Cottonian Chronicle, ii. 303, 324, 326
Crassus, Publius, visits the Cassiterides, i. 27
Crates of Mallus, i. 44, 78-9
Croker, T. Crofton, i. 379
Cromlechs, Distribution of, i. 22, 239
Cronium, Mare, i. 65, 100, 106, 121, 182, 363, 376
Crops, in Thule, i. 63;
in Britain, i. 63;
in Greenland, i. 277
Cuno, J. G., i. 59
Cwên-sæ̂ , i. 169
Cyclopes, i. 189, 196; ii. 10, 147, 148, 238
Cylipenus, i. 101, 104, 105
Cynocephali, i. 154-5, 159, 187, 189, 198, 383
Cystophora cristata (bladder-nose seal), i. 276, 286

Daae, L., i. 226; ii. 125, 129
Dalorto (or Dulcert), Angellino, ii. 226-30;
his map of 1325, ii. 177, 219, 226, 229, 235, 236;
his map of 1339 (Dulcert), ii. 229, 230, 235, 265, 266
Damastes of Sigeum, i. 16
Danes, i. 94, 121, 136, 139, 142, 143, 145, 146, 153, 167, 169, 180,
188, 245; ii. 115, 161
Darkness, Sea of, i. 40-1, 192, 195, 199, 363, 382; ii. 149, 204, 206,
212
Dauciones, i. 120, 121
Davis Strait, i. 269
Dawson, S. E., ii. 295, 307, 319, 321
Debes, Lucas, i. 375
Delisle, L., ii. 161
Delos, i. 375
Delphi, i. 18, 19
Democritus, i. 127
Denmark, i. 82, 94, 180, 185, 234; ii. 179, 201, 204, 205, 208, 237;
called “Dacia” on mediæval maps, ii. 188, 190, 222, 225;
representation of, in mediæval cartography, ii. 219, 225, 235, 250,
286
Denys, Nicolas, ii. 3

Desimoni, C., ii. 325
Deslien’s map of 1541, ii. 322
Detlefsen, D., i. 43, 70, 71, 72, 83, 84, 85, 93, 97, 99, 102, 119
Dicæarchus, i. 44, 73
Dicuil, i. 58, 160, 162-7, 252, 362; ii. 43, 51, 229
Dihya, Ibn, ii. 200-1, 209
Dimashqî, ii. 212-3
Diodorus Siculus, i. 23, 29-30, 44, 50, 51, 52, 58, 63, 71, 80, 87, 90,
346; ii. 48
Dionysius Periegetes, i. 114-5, 123, 356; ii. 47, 48, 192
Dipylon vases, i. 236-7
Disappearing (fairy) islands, i. 370, 378-9; ii. 213
Disc, Doctrine of the earth as a, i. 8, 12, 126, 127, 153, 198; ii. 182
Disco Bay, Greenland, i. 298, 300, 301, 302, 306, 307; ii. 72
“Dœgr” (== half a 24 hours’ day), used as a measure of distance, i.
287, 310, 322, 335; ii. 166, 169, 170, 171
Dogs as draught-animals, ii. 69, 72, 145, 146
Down Islands (Duneyiar), i. 285, 286
Dozy, R., ii. 55, 200, 201

Dozy and de Goeje, ii. 51, 204
Drapers’ Company, Protest of, against Sebastian Cabot, ii. 302, 330,
338, 342
Draumkvæde, i. 367, 381
Driftwood, in Greenland, i. 299, 305, 307, 308; ii. 37, 96
Drusus (The elder Germanicus), i. 83
“Dumna,” island, i. 106, 117; ii. 257
Dumont d’Urville, i. 376
Dvina, river, i. 173, 174, 222; ii. 135, 136, 137, 142, 146, 164, 176
Eastern Settlement of Greenland, i. 263, 265, 267, 271, 272, 274,
275, 276, 296, 301, 302, 307, 310, 311, 321; ii. 71, 82, 90, 107,
108, 112, 116;
decline of, ii. 95-100, 102
Ebstorf map, i. 102, 191; ii. 187
Edda, The older (poetic), i. 273
Edda, the younger (Snorra-Edda), i. 273, 298, 304, 342, 364
Eden, Richard, ii. 341
Edrisi, i. 182, 382; ii. 51-53, 202-8, 209, 210, 216;
his map, ii. 192, 203, 208, 220, 284
Egede, Hans, ii. 40, 41, 74, 101, 104, 105, 106

Egil Skallagrimsson’s Saga, i. 175, 218
Egyptian myths, i. 347
Einar Sokkason, i. 283, 294
Einar Thorgeirsson, lost in Greenland, i. 284
Einhard, i. 167, 179, 180, 185
Elk (achlis), i. 105, 191
Elymus arenarius (lyme-grass), ii. 5
Elysian Fields, i. 347, 349, 351
Empedocles, i. 12, 127
England (see Britain), Arab geographers on, ii. 204, 211;
maritime enterprise of, ii. 180, 294-5, 343;
English State document (1575) on North-West Passage, ii. 129-30,
132
“Engronelant,” ii. 277, 279, 373
d’Enjoy, Paul, i. 377
Eratosthenes of Cyrene, i. 20, 29, 44, 47, 52, 55, 61, 73, 75-7, 78,
82, 115; ii. 292
Eric Blood-Axe, ii. 136
Eric of Pomerania, ii. 118, 119

Eric the Red, i. 252, 256, 259, 262, 280, 288, 293, 318-21, 324, 330,
337, 344, 368; ii. 22, 77, 88;
discovers Greenland, i. 260, 263, 266-70
Eric the Red, Saga of, i. 260, 266, 273, 291, 292, 293, 296, 310,
313, 314, 318, 322, 331, 332-5, 337, 338, 342, 343, 367, 382; ii. 4,
6, 8, 10, 11, 14, 15, 22, 23, 24, 42, 43, 50, 59, 61, 89, 91, 206;
its value as a historical document, ii. 62
Eric’s fjord (Greenland), i. 267, 268, 271, 275, 317, 318, 319, 321; ii.
112
Eric Upsi, bishop of Greenland, ii. 29-31
Eridanus, river, i. 31, 32, 34, 42
Eruli, i. 21, 94, 136, 137-8, 139-49, 153, 235, 245
Erythea, i. 9
Erythræan Sea, i. 10
Eskimo, i. 19, 51, 150, 212, 215, 216, 223, 231-2, 260, 298, 306,
307, 308, 309, 310, 368; ii. 10, 12, 16, 17, 19, 66-94, 102-6, 107,
111-2, 113-6, 333, 366-7;
fairy-tales and legends of, ii. 8, 105, 115;
ball-game among, ii. 40-1;
distribution of, ii. 66-74;
racial characteristics of, ii. 67-8;
their culture, ii. 68-9, 91-2;
Norse settlers absorbed by, ii. 100, 102-105, 106, 107-11, 117;
unwarlike nature of, ii. 114, 115-6
Esthonians (Æstii, Osti), Esthonia, i. 69, 72, 104, 109, 131, 167, 169,
170, 181, 186; ii. 205

“Estotiland,” fictitious northern country, ii. 131
Eudoxus, i. 46
Eyrbyggja-saga, i. 313, 376; ii. 42, 46, 48, 50
Fabricius, A., ii. 55
Fabyan, Robert, Chronicle (quoted by Hakluyt), ii. 303, 324, 326,
333
Fadhlân, Ibn, ii. 143
Fairies, Names for, i. 372-3
Fairylands, Irish, i. 357, 370-1, 379; ii. 60;
Norwegian, i. 369-70, 378; ii. 60, 213;
laudatory names for, i. 374;
characteristics of, i. 375-9; ii. 213-4
Faqîh, Ibn al-, ii. 197
Farewell, Cape, i. 261, 267, 280, 282, 284, 288, 291, 295, 307, 316;
ii. 73
Faroes, The, i. 254, 255, 257, 316, 324, 362; ii. 51, 229, 262;
discovered by the Irish, i. 162-4, 233;
Irish monks expelled from, i. 252, 253;
early Celtic population in, i. 164, 253
Felix, The monk, in mediæval legend, i. 381
Fenni (Finns), i. 109, 112, 113, 114, 120, 149, 203
Ferdinand and Isabella of Spain, letter from, ii. 300

Fernald, M. L., ii. 3, 5-6
Fernandez, João (called “Lavorador”), ii. 331-2, 356;
letters patent to (1499), ii. 346, 356;
probably sighted Greenland (1500), ii. 356, 357, 375;
took part in Bristol expedition (1501), ii. 331, 356, 357;
Greenland (Labrador) named after him, ii. 358
Filastre, Cardinal, ii. 249-50, 278
Finland (see Kvænland), i. 206, 209, 210, 214;
the name confused with Vinland, i. 198, 382; ii. 31, 191;
and with Finmark, i. 382; ii. 191, 205;
Finmark, i. 61, 173, 175, 177, 191, 198, 204, 210, 213, 220, 222,
225; ii. 86, 141, 163, 164, 172, 178, 179, 205, 211, 237;
the name confused with Finland, i. 382; ii. 32, 191, 205;
“Finn,” The name, i. 198, 205-7, 210
“Finnaithæ” (Finnédi, Finvedi) (see Finns), i. 135, 137, 189, 198,
203, 204, 206, 382
Finn mac Cumhaill, i. 363; ii. 45
Finns, i. 109, 112, 113, 114, 120, 135, 136, 137, 149, 171, 173-8,
189, 198, 203-32, 382; ii. 68, 143;
Horned Finns, ii. 167
“Finns,” in southern Scandinavia, i. 103, 203, 205, 206-11; ii. 159
Finn’s booths (Finnsbuðir), in Greenland, i. 283, 296, 305

“Finnur hinn Friði,” Faroese lay of, ii. 33-4
Fisher, J., ii. 33, 121, 229, 249, 276, 277, 278, 279, 281
Fischer, M. P., ii. 161
Fischer, Theobald, ii. 216, 220, 230, 234
Fishing Lapps, i. 204, 205, 207, 218, 221, 223-32
Flateyjarbók, i. 254, 283, 313, 304, 317, 318, 324, 329, 331, 334,
338, 340, 343, 344, 359, 360; ii. 4, 14, 15, 18, 21, 22, 23, 25, 59, 61
Fletcher, Giles, i. 226
Floamanna-Saga, i. 280, 281; ii. 46, 81
Floating islands, Legends of, i. 375-7; ii. 213-4
Floki Vilgerdarson, sails to Iceland, i. 255, 257, 269
Florus, L. Annæus, i. 350
Forbiger, A., i. 58, 102
Forster, i. 179
Fortunate Isles (Insulæ Fortunatæ), i. 117, 198, 334, 345-53, 367,
370, 372, 373, 382-4; ii. 1-6, 24, 31, 42, 55, 59-61, 64, 191, 228,
280, 304
Fortunate Lake, Irish myth of, ii. 229-30
Foster-Brothers’ Saga, i. 276, 320; ii. 9, 18
Frähn, C. M., ii. 143, 145

Franks Casket, The, i. 176
Freydis, daughter of Eric the Red, i. 320, 328, 332, 333; ii. 11, 51
Friesland, Frisians, i. 95, 153, 205
Friis, J. A., i. 372
Friis, Peder Claussön, i. 224, 227-9, 232, 369; ii. 153, 158, 178, 268
Frisian noblemen’s polar expedition, i. 195-6, 200, 383; ii. 147-8
Frisius, Gemma, ii. 129, 132
Frisland, fabulous island south of Iceland, i. 377; ii. 131
Fritzner, ii. 9
Furðustrandir, i. 273, 312, 313, 322, 323, 324, 325, 326, 334, 336,
337, 339, 357; ii. 24, 36
Fyldeholm (island of drinking), i. 352
Gadir (Gadeira, Gades, Cadiz), i. 24, 27, 28, 30, 36, 37, 66, 79
Galvano, Antonio, ii. 336, 337, 338, 354, 364, 376
Gandvik (the White Sea), i. 218-9, 228; ii. 136-8, 164, 223, 237, 239
Gardar, discoverer of Iceland, i. 255-7, 263
Garðar, Greenland, i. 272, 273, 275, 311; ii. 106, 107, 108, 121, 122
“Gautigoth” (see Goths), i. 135

Gautrek’s Saga, i. 18-9
Geelmuyden, Prof. H., i. 52, 54, 311; ii. 23
Geijer, E. G., i. 60, 102, 111, 131, 205, 207
Gellir Thorkelsson, i. 366
Genoese mappamundi (1447 or 1457), ii. 278, 286, 287
Geminus of Rhodes, i. 43, 44, 53, 54, 57, 63, 64
Geographia Universalis, i. 382; ii. 32, 177, 188-91, 220, 227, 339
Gepidæ, i. 139, 142, 153
Gerfalcons, Island or land of, ii. 208, 227, 266, 289
Germania, i. 69, 71, 73, 87, 90, 95, 101, 108-14, 154, 169;
Roman campaigns in, i. 81, 83, 85, 97
Germanicus, The younger, i. 83
Germanus, Nicolaus, ii. 251, 276-9, 288, 290, 373
Germany, coast of, in mediæval cartography, ii. 219, 257
Gesta Francorum, i. 234
Gilbert, Sir Humphrey, ii. 340
Gildas, i. 234, 364
Ginnungagap, i. 12, 84, 158; ii. 35, 150, 154, 239-41

Giraldus Cambrensis, i. 379; ii. 151, 220, 245
Gisle Oddsson’s Annals, ii. 82, 100-2, 109
Gissur Einarsson, Bishop, i. 285
Gjessing, H., ii. 31
Glæsaria, island, i. 101, 106
Glastonbury, Legend of sow at, i. 378-9
“Gli,” mythical island, i. 364
Globes, used by the Greeks, i. 78;
introduced by Toscanelli, ii. 287;
Behaim’s, ii. 287-9;
Laon globe, ii. 290;
used by Columbus, ii. 287;
and Cabot, ii. 304, 306
Gnomon, The, i. 11, 45-6
Godthaab, Greenland, i. 271, 304, 307, 321; ii. 73, 74
Goe, month of, i. 264, 265
Goeje, M. de, i. 344, 362; ii. 51, 194, 197, 198
Goes, Damiam de, ii. 354, 366, 376, 377
Gokstad ship, i. 246
Gomara, Francesco Lopez de, ii. 129, 130, 131, 336, 337, 354, 364
Gongu-Rólv’s kvæði, i. 356

Göta river, i. 131; ii. 190, 205
Göter (Gauter), i. 120, 135, 141, 144, 147; ii. 190
Goths (Gytoni, Gythones, Getæ), i. 14, 21, 71, 120, 129, 130, 135,
137, 139, 145, 147, 153; ii. 143, 190
Gotland, i. 121, 180, 378; ii. 125, 237;
in mediæval cartography, ii. 219, 221, 224, 233, 265
Gourmont, Hieronymus, map of Iceland, ii. 122-3, 127
Graah, Captain, i. 297; ii. 104
Grail, Legends of the, i. 382
Grampus, i. 50-1
Granii, i. 136
Grape Island (Insula Uvarum), i. 358, 361, 363, 365, 366
Greenland, i. 184, 192, 194, 197, 199, 200, 201, 215, 223, 252, 315-
21, 322; ii. 1, 5, 12, 25, 36, 38, 40-2, 66-94, 95-134, 167, 169, 177,
244, 345, 366;
Eskimo of, ii. 71-5;
discovered and settled by Norwegians, i. 258-78;
estimated population of settlements, i. 272;
conditions of life in i. 274-8, 319; ii. 96-7;
voyages along the coasts of, i. 279-311;
glaciers (inland ice) of, i. 288-95, 301, 308; ii. 246-7;
decline of Norse settlements in, ii. 90, 95-100;
last voyage to (from Norway), ii. 117;
last ship from, ii. 118;
geographical ideas of, ii. 237-40, 246-8, 254-5, 259-62, 270-6,

278, 279, 280;
east coast of, i. 271-2, 279-96, 308; ii. 168, 170, 171, 238;
uninhabited parts (ubygder) of, i. 279-311, 320, 321; ii. 28, 166,
172;
sixteenth-century discovery of, ii. 315, 332, 335, 352, 363, 364,
375;
called Labrador, ii. 129, 132, 133, 315, 335, 353;
in sixteenth-century maps, ii. 368-75
Gregory of Tours i. 234
“Greipar,” in Greenland, i. 298, 299, 300-1, 304
Grettis-saga, i. 313, 367
Griffins, i. 19, 254; ii. 263
Grim Kamban, i. 253
Grimm, J., i. 18, 94, 95, 355, 372; ii. 45, 56
Grimm, W., i. 373
Grip, Carsten, letter to Christiern III., ii. 126-8
Gripla, i. 288; ii. 35-6, 237, 239, 241
Gröndal, B., i. 371, 375
Grönlands historiske Mindesmærker, i. 262, 263, 271, 281, 282, 283,
284, 285, 288, 292, 294, 295, 296, 297, 298, 299, 300, 301, 302,
304, 305, 311, 333, 359, 377; ii. 1, 9, 14, 17, 22, 25, 31, 35, 46, 79,
82, 86, 100, 102, 106, 108, 112, 113, 117, 119, 120, 125, 127, 172,
237, 278
Grönlendinga-þáttr (see Flateyjarbók)

Groth, Th., ii. 103
Grottasongr, i. 159
Gudleif’ Gudlaugsson, story of his voyage, ii. 49-50, 53-4;
compared with Leif Ericson, ii. 50-1
Gudmund Arason’s Saga, i. 284
Gudmundsson, Jón, map by, ii. 34, 241
Gudmundsson, V., ii. 25
Gudrid, wife of Karlsevne, i. 318, 319, 320, 321, 329, 330, 333; ii.
14-5, 51
Guichot y Sierra, A., i. 376
Gulathings Law, ii. 140
Gulf Stream, i. 251; ii. 54
Gunnbjörnskerries, i. 256, 261-4, 267, 280; ii. 276
Gunnbjörn Ulfsson, i. 256, 261-4, 267, 280, 296
Gustafson, Prof. G., i. 237, 240
Gutæ, i. 120
Guta-saga, i. 378
Gutones (see Goths), i. 70, 71, 72, 72, 93
Gytoni (see Goths), i. 71

Hægstad, Prof. M., ii. 242
Hægstad and Torp (Gamal-norsk Ordbog), ii. 9
Hæmodæ (“Acmodæ,” “Hæcmodæ”), i. 90, 106
“Hafsbotn” (the Polar Sea), i. 283, 303; ii. 137, 151, 165, 166, 167,
168, 171, 172, 237, 240
Hakluyt. R., i. 226; ii. 129, 132, 152, 261, 319, 321, 326, 333
Håkon Håkonsson’s Saga, i. 299; ii. 139, 141
Halichoerus grypus (grey seal), i. 217; ii. 91, 155
Halli Geit, Tale of, ii. 239
Hallinger, i. 104, 247
Hallstatt, i. 24, 36
Hâlogaland (Hålogaland, Hâlogi, Halgoland, Halagland, Halogia,
Helgeland), i. 61, 62, 64, 132, 135, 138, 175, 179, 194, 197, 200,
231, 247, 264, 381, 383; ii. 64, 137, 139, 140, 142, 165, 168, 172;
in mediæval cartography, ii. 227, 236
Halsingia, or Alsingia, i. 104
Hamberg, Axel, ii. 69
Hammershaimb, V. U., i. 356, 375; ii. 33
Hamy, ii. 220, 223, 229, 230, 234

Hanno, i. 37, 88, 350; ii. 45
Hans (John), king of Denmark, ii. 125, 128
Hanseatic League, ii. 99, 119, 125, 179, 218
Hansen, Dr. A. M., i. 149, 192, 206, 207, 208, 218, 221, 222, 228,
229, 230, 236-7, 239
Harold Fairhair, i. 253-4, 255, 258
Harold Gråfeld, ii. 136, 153, 154
Harold Hardråde, i. 185, 195, 201, 283, 383; ii. 147, 199;
his voyage in the Polar Sea, i. 195; ii. 148-54
Harpoons, i. 214-7, 277; ii. 145-6, 156-63
Harrisse, Henry, ii. 132, 230, 293, 294, 295, 296, 297, 300, 302,
303, 304, 305, 309, 314, 315, 319, 320, 326, 327, 329, 331, 332,
333, 334, 336, 341, 347, 348, 349, 353, 358, 359, 360, 365, 374
Harudes (Charydes, Charudes, Horder), i. 85, 118, 136, 143, 148,
246
Hauksbók, i. 188, 251, 256, 257, 261, 262, 264, 268, 286, 291, 293,
308, 309, 322, 327, 331, 333, 353, 367, 369; ii. 10, 11, 166, 169,
172, 216, 261
Hebrides (Ebudes, Hebudes), i. 57, 90, 106, 117, 123, 158, 159, 160,
161, 234, 273, 316; ii. 151, 200
Hecatæus of Abdera, i. 8, 9, 10, 15, 16, 98
Heffermehl, A. V., ii. 242

Heiberg, Prof. J., i. 219, 220
Heimskringla, i. 270, 313, 331; ii. 59, 137, 171, 239
Heiner, i. 138
Heinrich of Mainz, map by, ii. 185, 187
Helge Bograngsson, killed in Bjarmeland, ii. 139-40
Heligoland, i. 197
Helland, A., i. 226, 231, 369, 372, 373, 378, 381; ii. 46, 152, 177,
228
Helluland, i. 312, 313, 322, 323, 334, 336, 357; ii. 1, 23, 35-6, 61,
237
Helm, O., i. 14
Helsingland, Helsingers, i. 189; ii. 237
Henry V. of England, ii. 119
Henry VI. of England, ii. 119
Henry VII. of England, ii. 130, 298, 299, 302, 303, 322, 324, 326,
327, 331, 332, 333, 334, 337, 338, 340
Henry VIII. of England, ii. 319, 330, 334, 338, 341, 342, 343
Heraclitus, i. 12
“Herbrestr” (war-crash), ii. 8-9
Hereford map, i. 91, 92, 102, 154, 157, 190; ii. 186, 187

Hergt, G., i. 43, 51, 60, 65, 66, 67, 71, 72
Herla, mythical king of Britain, ii. 76
Hermiones, i. 91, 104
Hermits, in Irish legends, ii. 19, 43-6, 50
Herodotus, i. 9, 12, 20, 23, 24, 27, 31-2, 46, 76, 78, 81, 88, 114,
148, 155, 156, 161, 187
Hertzberg, Ebbe, ii. 38, 39, 40, 61, 93
Hesiod, i. 9, 11, 18, 42, 84, 348
Hesperides, i. 9, 161, 334, 345, 376; ii. 2, 61
Heyman, i. 342; ii. 8
Hielmqvist, Th., i. 381
Hieronymus, i. 151, 154
Higden, Ranulph (Polychronicon), i. 346, 382; ii. 31-2, 288-92, 220;
his mappamundi, ii. 188, 189, 192
Hilleviones, i. 101, 104, 121
Himilco’s voyage, i. 29, 36-41, 68, 83
Himinrað (Hunenrioth, &c.), mountain in Greenland, i. 302-4; ii. 108
Hipparchus, i. 44, 47, 52, 56, 57, 73, 77-8, 87, 116; ii. 197
Hippocrates, i. 13, 88

Hippopods, i. 91
Hirri, i. 101
Historia Norwegiæ, i. 204, 229, 252, 255, 256, 257, 298; ii. 1, 2, 17,
29, 61, 79, 87, 88, 135, 151, 167, 168, 172, 222, 227, 235, 239,
240, 280
Hjorleif, settles in Iceland with Ingolf, i. 166, 252, 254, 255
Hoegh, K., ii. 31
Hoffmann, W. J., ii. 39, 40
Hofmann, C., i. 59
Holand, H. R., ii. 31
Holberg, Ludvig, ii. 118
Holm, G. F., i. 271, 274
Holz, G., i. 85, 102
Homer, i. 8, 10-11, 13, 14, 25, 33, 77, 78, 196, 347, 348, 371; ii. 53,
54, 160
Homeyer, C. G., i. 214
Hönen, Ringerike, Runic stone from, ii. 27-9, 58
Honorius Augustodunensis, i. 375
Honorius, Julius, i. 123; ii. 183

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