“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