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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 44 #44
8-44 Real Analysis
and
L(P, f α
) L(P, f , α) M . (8.24)
Taking infimum over partitions in upper sums and supremum over
partitions in lower sums in the above inequalities, we have
b
a
fdα
b
a
f α
dx
M and
b
a
fdα
b
a
f α
dx
M.
Since >0 is arbitrary, it follows that
b
a
fdα =
b
a
f α
dx,
b
a
fdα =
b
a
f α
dx
From these equalities, the entire theorem follows using the defini-
tions.
Theorem 8.8.8 (Change of variable for Riemann-Stieltjes integrals)
Let f
R(α) on [a, b] and φ : [A, B]→[a, b] be strictly increasing,
continuous and onto. Define β, gon[A, B] by
β(y) = α(φ(y)), g(y) = f (φ(y))(y ∈[A, B]).
Then g
R(β) on [A, B] and
B
A
gdβ =
b
a
fdα.
Proof Using the bijectivity of
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Publisher Resources

ISBN: 9781299447561Publisher Website