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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_08” 2011/5/22 23:34 page 39 #39
Riemann Integration 8-39
may exist. As an example of this case, CPV
1
1
dx
x
= 0, whereas CPV
1
1
1
|x|
dx does not exist.
8.8 GENERALIZATIONS
Generalizing the theory of Riemann integration on [a, b], one can
develop the theory of Riemann-Stieltjes integration with respect to an
increasing function α(x) on [a, b] as follows. Let f : [a, b]→
R be a
bounded function. Let P ={x
0
, x
1
, x
2
, ..., x
n
} be a partition of [a, b]
and
M
i
= sup
x
i1
xx
i
f (x), m
i
= inf
x
i1
xx
i
f (x)(1 i n)
U (P, f , α) =
n
i=1
M
i
α
i
, L(P, f , α) =
n
i=1
m
i
α
i
where α
i
= α(x
i
) α(x
i1
). We define
b
a
fdα = inf
P
U (P, f , α),
b
a
fdα = sup
P
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Publisher Resources

ISBN: 9781299447561Publisher Website