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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 21 #21
Lebesgue Measure and Integration 13-21
Proof Take g : [−∞, ∞] [−∞, ∞] as g(x ) =|x|, which is
continuous and apply Theorem 13.3.7.
The converse of the above corollary is not true. This is given in the
following example
Example 13.3.9 Let E
R be a non-Lebesgue measurable set. Define
f (x) =
1ifx E
1ifx E
c
Since |f (x)|=1, |f (x)| is a measurable function on R but f (x) is not
(note that otherwise f
1
({1}) = E will be measurable).
For a sequence of functions {f
n
} defined on a common domain, we
can always define the following new functions:
sup
1jn
f
j
(x), inf
1jn
f
j
(x), sup
j1
f
j
(x), inf
j1
f
j
(x), lim sup
j→∞
f
j
(x)
and lim inf
j→∞
f
j
(x)
For example, lim sup
j→∞
f
j
(x) is defined as the function h(x) whose ...
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Publisher Resources

ISBN: 9781299447561Publisher Website