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