
“real: chapter_13” — 2011/5/22 — 23:35 — page 11 — #11
Lebesgue Measure and Integration 13-11
and call ‘m’ as the Lebesgue measure on R. Thus the Lebesgue measure
is associated with the pair (
R, M). Further m satisfies the following
properties:
(i) ∅,
R ∈ M and m(∅) = 0, m(R) =∞. For E ∈ M, m(E) ≥ 0
and
(ii) m is countably additive on
M.
For these reasons, we say that the triplet (
R, M, m) is a measure space
and m is the Lebesgue measure on
R.
Theorem 13.2.19 For any subset A of
R there exists a measurable
set E (i.e. E ∈
M) such that A ⊂ E and m
∗
(A) = m(E).
Proof Given =
1
n
> 0 using Corollary 13.2.7 we have a sequence of
open sets {U
n
} such that U
n
⊃ A, m
∗