Definition 7.4.3: Lebesgue Integral for Simple Function
If
s(x) =
an
XAn(x)
is a simple function and m(An) is finite
for all n, then the Lebesgue Integral of s
is defined as

If E is a measurable set, we defines(x) dx =
an m(An)
E s(x) dx =
XE(x) s(x) dx