| Preface | p. v |
| Notations | p. xv |
| Measure Spaces | p. 1 |
| Introduction | p. 1 |
| Measure on a [sigma]-algebra of Sets | p. 3 |
| [sigma]-algebra of Sets | p. 3 |
| Limits of Sequences of Sets | p. 4 |
| Generation of [sigma]-algebras | p. 6 |
| Borel [sigma]-algebras | p. 8 |
| Measure on a [sigma]-algebra | p. 9 |
| Measures of a Sequence of Sets | p. 12 |
| Measurable Space and Measure Space | p. 15 |
| Measurable Mapping | p. 17 |
| Induction of Measure by Measurable Mapping | p. 19 |
| Outer Measures | p. 24 |
| Construction of Measure by Means of Outer Measure | p. 24 |
| Regular Outer Measures | p. 27 |
| Metric Outer Measures | p. 29 |
| Construction of Outer Measures | p. 32 |
| Lebesgue Measure on R | p. 35 |
| Lebesgue Outer Measure on R | p. 35 |
| Some Properties of the Lebesgue Measure Space | p. 39 |
| Existence of Non Lebesgue Measurable Sets | p. 42 |
| Regularity of Lebesgue Outer Measure | p. 44 |
| Lebesgue Inner Measure on R | p. 49 |
| Measurable Functions | p. 61 |
| Measurability of Functions | p. 61 |
| Operations with Measurable Functions | p. 65 |
| Equality Almost Everywhere | p. 68 |
| Sequence of Measurable Functions | p. 69 |
| Continuity and Borel and Lebesgue Measurability of Functions on R | p. 72 |
| Cantor Ternary Set and Cantor-Lebesgue Function | p. 74 |
| Completion of Measure Space | p. 83 |
| Complete Extension and Completion of a Measure Space | p. 83 |
| Completion of the Borel Measure Space to the Lebesgue Measure Space | p. 86 |
| Convergence a.e. and Convergence in measure | p. 88 |
| Convergence a.e. | p. 88 |
| Almost Uniform Convergence | p. 92 |
| Convergence in Measure | p. 94 |
| Cauchy Sequences in Convergence in Measure | p. 100 |
| Approximation by Step Functions and Continuous Functions | p. 102 |
| The Lebesgue Integral | p. 113 |
| Integration of Bounded Functions on Sets of Finite Measures | p. 113 |
| Integration of Simple Functions | p. 113 |
| Integration of Bounded Functions on Sets of Finite Measures | p. 116 |
| Riemann Integrability | p. 124 |
| Integration of Nonnegative Functions | p. 135 |
| Lebesgue Integral of Nonnegative Functions | p. 135 |
| Monotone Convergence Theorem | p. 137 |
| Approximation of the Integral by Truncation | p. 142 |
| Integration of Measurable Functions | p. 147 |
| Lebesgue Integral of Measurable Functions | p. 147 |
| Convergence Theorems | p. 155 |
| Convergence Theorems under Convergence in Measure | p. 158 |
| Approximation of the Integral by Truncation | p. 160 |
| Translation and Linear Transformation of the Lebesgue Integral in R | p. 164 |
| Integration by Image Measure | p. 166 |
| Signed Measures | p. 175 |
| Signed Measure Spaces | p. 175 |
| Decomposition of Signed Measures | p. 179 |
| Integration on a Signed Measure Space | p. 185 |
| Absolute Continuity of a Measure | p. 189 |
| The Radon-Nikodym Derivative | p. 189 |
| Absolute Continuity of a Signed Measure Relative to a Positive Measure | p. 190 |
| Properties of the Radon-Nikodym Derivative | p. 195 |
| Differentiation and Integration | p. 201 |
| Monotone Functions and Functions of Bounded Variation | p. 201 |
| The Derivative | p. 201 |
| Differentiability of Monotone Functions | p. 205 |
| Functions of Bounded Variation | p. 214 |
| Absolutely Continuous Functions | p. 221 |
| Absolute Continuity | p. 221 |
| Banach-Zarecki Criterion for Absolute Continuity | p. 225 |
| Singular Functions | p. 226 |
| Indefinite Integrals | p. 227 |
| Calculation of the Lebesgue Integral by Means of the Derivative | p. 233 |
| Convex Functions | p. 246 |
| Continuity and Differentiability of a Convex Function | p. 246 |
| Monotonicity and Absolute Continuity of a Convex Function | p. 255 |
| Jensen's Inequality | p. 257 |
| The Classical Banach Spaces | p. 261 |
| Normed Linear Spaces | p. 261 |
| Banach Spaces | p. 261 |
| Banach Spaces on R[superscript k] | p. 263 |
| The Space of Continuous Functions C([a, b]) | p. 266 |
| A Criterion for Completeness of a Normed Linear Space | p. 268 |
| Hilbert Spaces | p. 269 |
| The Dual of a Banach Space | p. 271 |
| Semicontinuous Functions | p. 274 |
| The L[superscript p] Spaces | p. 278 |
| The L[superscript p] Spaces for p [set membership] (0, [infinity]) | p. 278 |
| The Linear Spaces L[superscript p] for p [set membership] (1, [infinity]) | p. 281 |
| The L[superscript p] Spaces for p [set membership] (1, [infinity]) | p. 286 |
| The Space L[superscript infinity] | p. 290 |
| The L[superscript p] Spaces for p [set membership] (0, 1) | p. 294 |
| Relation among the L[superscript p] Spaces | p. 298 |
| The Modified L[superscript p] Norms for L[superscript p] Spaces with p [set membership] [1, infinity] | p. 298 |
| Approximation by Continuous Functions | p. 300 |
| L[superscript p] Spaces with p [set membership] (0, 1) | p. 303 |
| The l[superscript p] Spaces | p. 304 |
| Bounded Linear Functionals on the L[superscript p] Spaces | p. 310 |
| Bounded Linear Functionals Arising from Integration | p. 310 |
| Approximation by Simple Functions | p. 313 |
| A Converse of Holder's Inequality | p. 314 |
| Riesz Representation Theorem on the L[superscript p] Spaces | p. 317 |
| Integration on Locally Compact Hausdorff Space | p. 325 |
| Continuous Functions on a Locally Compact Hausdorff Space | p. 325 |
| Borel and Radon Measures | p. 329 |
| Positive Linear Functionals on C[subscript c](X) | p. 331 |
| Approximation by Continuous Functions | p. 338 |
| Extension of Additive Set Functions to Measures | p. 343 |
| Extension of Additive Set Functions on an Algebra | p. 343 |
| Additive Set Function on an Algebra | p. 343 |
| Extension of an Additive Set Function on an Algebra to a Measure | p. 348 |
| Regularity of an Outer Measure Derived from a Countably Additive Set Function on an Algebra | p. 348 |
| Uniqueness of Extension of a Countably Additive Set Function on an Algebra to a Measure | p. 351 |
| Approximation to a [sigma]-algebra Generated by an Algebra | p. 353 |
| Outer Measure Based on a Measure | p. 356 |
| Extension of Additive Set Functions on a Semialgebra | p. 358 |
| Semialebras of Sets | p. 358 |
| Additive Set Function on a Semialgebra | p. 359 |
| Outer Measures Based on Additive Set Functions on a Semialgebra | p. 362 |
| Lebesgue-Stieltjes Measure Spaces | p. 365 |
| Lebesgue-Stieltjes Outer Measures | p. 365 |
| Regularity of the Lebesgue-Stieltjes Outer Measures | p. 368 |
| Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure | p. 369 |
| Decomposition of an Increasing Function | p. 378 |
| Product Measure Spaces | p. 385 |
| Existence and Uniqueness of Product Measure Spaces | p. 385 |
| Integration on Product Measure Space | p. 389 |
| Completion of Product Measure Space | p. 400 |
| Convolution of Functions | p. 404 |
| Some Related Theorems | p. 411 |
| Measure and Integration on the Euclidean Space | p. 417 |
| Lebesgue Measure Space on the Euclidean Space | p. 417 |
| Lebesgue Outer Measure on the Euclidean Space | p. 417 |
| Regularity Properties of Lebesgue Measure Space on R[superscript n] | p. 422 |
| Approximation by Continuous Functions | p. 424 |
| Lebesgue Measure Space on R[superscript n] as the Completion of a Product Measure Space | p. 427 |
| Translation of the Lebesgue Integral on R[superscript n] | p. 429 |
| Linear Transformation of the Lebesgue Integral on R[superscript n] | p. 430 |
| Differentiation on the Euclidean Space | p. 438 |
| The Lebesgue Differentiation Theorem on R[superscript n] | p. 438 |
| Differentiation of Set Functions with Respect to the Lebesgue Measure | p. 448 |
| Signed Borel Measures on R[superscript n] | p. 448 |
| Differentiation of Borel Measures with Respect to the Lebesgue Measure | p. 450 |
| Differentiation of the Indefinite Integral | p. 454 |
| Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure | p. 455 |
| Change of Variable of Integration on the Euclidean Space | p. 460 |
| Change of Variable of Integration by Differentiable Transformations | p. 460 |
| Spherical Coordinates in R[superscript n] | p. 471 |
| Integration by Image Measure on Spherical Surfaces | p. 478 |
| Hausdorff Measures on the Euclidean Space | p. 485 |
| Hausdorff Measures | p. 485 |
| Hausdorff Measures on R[superscript n] | p. 485 |
| Equivalent Definitions of Hausdorff Measure | p. 489 |
| Regularity of Hausdorff Measure | p. 495 |
| Hausdorff Dimension | p. 498 |
| Transformations of Hausdorff Measures | p. 503 |
| Hausdorff Measure of Transformed Sets | p. 503 |
| 1-dimensional Hausdorff Measure | p. 508 |
| Hausdorff Measure of Jordan Curves | p. 509 |
| Hausdorff Measures of Integral and Fractional Dimensions | p. 514 |
| Hausdorff Measure of Integral Dimension and Lebesgue Measure | p. 514 |
| Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in R[superscript n] | p. 516 |
| Transformation of Hausdorff Measure of Integral Dimension | p. 522 |
| Hausdorff Measure of Fractional Dimension | p. 527 |
| Bibliography | p. 537 |
| Index | p. 539 |
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