“real: chapter_13” — 2011/5/22 — 23:35 — page 57 — #57
Lebesgue Measure and Integration 13-57
As in the Lebesgue theory, it is easy to see that a simple function
s(x) =
n
i=1
α
i
χ
A
i
is measurable if and only if A
i
’s are measurable.
The following theorem can be proved on lines similar to its
analogous result in the Lebesgue theory.
Theorem 13.6.9 Let (X ,
M) be a measurable space and f : X →
[0, ∞] be a measurable function. Then there are simple measurable
functions s
n
(x) defined on X (one for each n = 1, 2, ...)such that
(i) 0 ≤ s
1
(x) ≤ s
2
(x) ≤ ... ≤ f (x).
(ii) lim
n→∞
s
n
(x) = f (x)(x ∈ X ).
(iii) s
n
(x) → f (x) uniformly on X as n →∞if f (x) is bounded.
We shall now define an abstract measure on a measurable space and
obtain its properties.
Definition 13.6.10 ...