“real: chapter_13” — 2011/5/22 — 23:35 — page 31 — #31
Lebesgue Measure and Integration 13-31
Further if s(x) varies over all non-negative simple measurable
functions less than or equal to f (x), then χ
A
(x)s(x) varies over all non-
negative simple measurable functions less than or equal to χ
A
(x)f (x)
(note that if 0 ≤ s(x) ≤ f (x), then 0 ≤ χ
A
(x)s(x) ≤ χ
A
(x)f (x) and if
s(x) ≤ χ
A
(x)s(x), then s(x) = 0onA
c
and hence s(x) = χ
A
(x)s(x)).
The required result now follows by taking supremum over all simple
measurable functions s(x) with 0 ≤ s(x) ≤ f (x).
(iv) Using (iii) and (i) we have
A
f (x)dm(x) =
D
χ
A
(x)f (x)dm(x)
≤
D
χ
B
(x)f (x)dm(x) =
B
f (x)dm(x).
(Note that A ⊂ B implies χ
A
(x) ≤ χ
B
(x)).
(v) This result is known for simple functions (see Theorem 13.4.4).