Measure and integral : an introduction to real analysis

524 Pages·2015·4.148 MB·English

by  Wheeden, Richard L., Zygmund, Antoni

Mathematics
Measure and Integral
M
Now considered a classic text on the topic, Measure and Integral: An
e
Introduction to Real Analysis provides an introduction to real analysis by
a
An Introduction to Real Analysis
first developing the theory of measure and integration in the simple setting s
of Euclidean space, and then presenting a more general treatment based on u
r
abstract notions characterized by axioms and with less geometric content. e Second Edition

Published nearly forty years after the first edition, this long-awaited Second a
n
Edition also:
d
• Studies the Fourier transform of functions in the spaces L1, L2, and Lp,
I
n
1 < p < 2
t
• Shows the Hilbert transform to be a bounded operator on L2, as e
an application of the L2 theory of the Fourier transform in the one- g
r
dimensional case
a
• Covers fractional integration and some topics related to mean l

oscillation properties of functions, such as the classes of Hölder
S
continuous functions and the space of functions of bounded mean e
c
oscillation
o
• Derives a subrepresentation formula, which in higher dimensions plays n
d
a role roughly similar to the one played by the fundamental theorem of
Richard L. Wheeden
E
calculus in one dimension
d
• Extends the subrepresentation formula derived for smooth functions to i
t
functions with a weak gradient io Antoni Zygmund
n
• Applies the norm estimates derived for fractional integral operators

to obtain local and global first-order Poincaré–Sobolev inequalities,
including endpoint cases
• Proves the existence of a tangent plane to the graph of a Lipschitz
function of several variables W
h
• Includes many new exercises not present in the first edition e
e
d
This widely used and highly respected text for upper-division undergraduate e
n
and first-year graduate students of mathematics, statistics, probability, or •
Z
engineering is revised for a new generation of students and instructors. The y
g
m
book also serves as a handy reference for professional mathematicians.
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K24634
www.crcpress.com
K24634_cover.indd 1 3/20/15 10:44 AM
Measure and Integral
An Introduction to Real Analysis
Second Edition
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Measure and Integral
An Introduction to Real Analysis
Second Edition
Richard L. Wheeden
Rutgers University
Antoni Zygmund
Late of the University of Chicago
CRC Press
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Contents
PrefacetotheSecondEdition.................................................... xiii
PrefacetotheFirstEdition.........................................................xv
Authors .............................................................................xvii
1. Preliminaries....................................................................1
1.1 PointsandSetsinRn................................................. 1
1.2 RnasaMetricSpace ................................................. 2
1.3 OpenandClosedSetsinRn,andSpecialSets ................... 5
1.4 CompactSetsandtheHeine–BorelTheorem .................... 9
1.5 Functions .............................................................. 10
1.6 ContinuousFunctionsandTransformations..................... 11
1.7 TheRiemannIntegral................................................ 12
Exercises....................................................................... 13
2. FunctionsofBoundedVariationandtheRiemann–Stieltjes
Integral..........................................................................17
2.1 FunctionsofBoundedVariation ................................... 17
2.2 RectifiableCurves.................................................... 25
2.3 TheRiemann–StieltjesIntegral..................................... 26
2.4 FurtherResultsaboutRiemann–StieltjesIntegrals.............. 34
Exercises....................................................................... 37
3. LebesgueMeasureandOuterMeasure ...................................41
3.1 LebesgueOuterMeasureandtheCantorSet .................... 41
3.2 LebesgueMeasurableSets .......................................... 45
3.3 TwoPropertiesofLebesgueMeasure ............................. 50
3.4 CharacterizationsofMeasurability................................ 52
3.5 LipschitzTransformationsofRn ................................... 54
3.6 ANonmeasurableSet................................................ 57
Exercises....................................................................... 58
4. LebesgueMeasurableFunctions ...........................................63
4.1 ElementaryPropertiesofMeasurableFunctions ................ 64
4.2 SemicontinuousFunctions.......................................... 69
4.3 PropertiesofMeasurableFunctionsandTheoremsof
EgorovandLusin..................................................... 71
4.4 ConvergenceinMeasure............................................ 73
Exercises....................................................................... 76
ix