“real: chapter_13” — 2011/5/22 — 23:35 — page 21 — #21
Lebesgue Measure and Integration 13-21
Proof Take g : [−∞, ∞] → [−∞, ∞] as g(x ) =|x|, which is
continuous and apply Theorem 13.3.7.
The converse of the above corollary is not true. This is given in the
following example
Example 13.3.9 Let E ⊂
R be a non-Lebesgue measurable set. Define
f (x) =
1ifx ∈ E
−1ifx ∈ E
c
Since |f (x)|=1, |f (x)| is a measurable function on R but f (x) is not
(note that otherwise f
−1
({1}) = E will be measurable).
For a sequence of functions {f
n
} defined on a common domain, we
can always define the following new functions:
sup
1≤j≤n
f
j
(x), inf
1≤j≤n
f
j
(x), sup
j≥1
f
j
(x), inf
j≥1
f
j
(x), lim sup
j→∞
f
j
(x)
and lim inf
j→∞
f
j
(x)
For example, lim sup
j→∞
f
j
(x) is defined as the function h(x) whose ...