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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_13” 2011/5/22 23:35 page 37 #37
Lebesgue Measure and Integration 13-37
13.5 INTEGRATION OF REAL-VALUED FUNCTIONS
In this section, we shall define the integral of a real-valued measur-
able functions (not necessarily non-negative) defined on a measurable
subset of
R and study its properties.
Definition 13.5.1 Let f : E
R be a measurable function where E
is a measurable subset of
R. We define
E
f (x)dm(x) =
E
f
+
(x)dm(x)
E
f
(x)dm(x)
where f
+
(x) and f
(x) are the positive and negative variations of f ,
respectively.
This integral exists as an extended real number only if either
E
f
+
(x)dm(x)< or
E
f
(x)dm(x)<.
One sufficient condition under which
E
f (x)dm(x) exists as a real num-
ber is to assume that
E
f
+
(x)dm(x)< and
E
f
(x)dm(x)<.I ...
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Publisher Resources

ISBN: 9781299447561Publisher Website