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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 47 #47
Lebesgue Measure and Integration 13-47
The following example shows that the converse of the above
theorem does not hold.
Example 13.5.15 Let f :
R R defined by
f (x) =
0ifx is rational
1ifx is irrational.
Then f L
1
[0, 1] (indeed, f = 1 a.e.onR and so
1
0
f (x)dm(x) = 1).
However for any partition P of [0, 1], U (P, f ) = 1 and L(P, f ) = 0
and hence f is not Riemann integrable.
Theorem 13.5.16 Suppose f : (a, b)
R (−∞ ≤ a < b ≤∞) is
such that f is Riemann integrable over every closed interval [c, d]⊂
(a, b) and that the improper integral
b
a
|f (t)|dt = lim
ca+
lim
db
d
c
|f (t)|dt
exists. Then the improper ...
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Publisher Resources

ISBN: 9781299447561Publisher Website