欲證明:An extended real-valued function f is measurable if and only if
the sets Α={x屬於χ:f(x)= +∞},Β={x屬於χ:f(x)= －∞} belong to Χ,
and the real-valued function f1 defined by
f1(x) = f(x) ,if x not belong to Α∪Β
= 0 ,if x belong to Α∪Β
Proof:
左至右：If f is in M (χ,Χ), it has already been noted that Α and Β
belong to Χ, Let α belong to Ｒ and α ≧ 0, then
{x屬於χ:f1(x)＞α} = {x屬於χ:f(x)＞α}\Α
If α < 0, then
{x屬於χ:f1(x)＞α} = {x屬於χ:f(x)＞α}∪Β
Hence, f1 is measurable.
右至左：if Α and Β belong to Χ, and f1 is measurable, then
when α ≧ 0, {x屬於χ:f(x)＞α} = {x屬於χ:f1(x)＞α}∪Α
when α < 0, {x屬於χ:f(x)＞α} = {x屬於χ:f1(x)＞α}\Β
Hence, f is measurable.
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