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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 77 #77
Lebesgue Measure and Integration 13-77
(i) each g
n
is bounded.
(ii) f (x) = lim
n→∞
g
n
(x) pointwise a.e. on X .
Proof Taking = 2
n
successively and applying Lusin’s Theorem, we
get a sequence of functions {g
n
} in C
c
(X ) such that each g
n
is bounded
and if E
n
={x X / g
n
(x) = f (x)}, then
µ(E
n
)<2
n
.
Since
n=1
µ(E
n
) 1, we see that
E ={x X / x lies in infinitely many of the sets E
n
}
satisfies µ(E) = 0 (see Theorem 13.6.32).
For x E
c
, f (x) = g
n
(x) for all sufficiently large n and hence
lim
n→∞
g
n
(x) = f (x).
It follows that lim
n→∞
g
n
(x) = f (x) point wise a.e.onX . This completes
the proof.
SOLVED EXERCISES ...
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Publisher Resources

ISBN: 9781299447561Publisher Website