Interactive Real Analysis7.4. Lebesgue Integral
Example 7.4.11(d): Properties of the Lebesgue Integral
Suppose f is a bounded, non-negative function defined on a measurable
set E with finite measure such that
E f(x) dx = 0.
Show that f must then be equal to zero except on a set of
measure zero.
Context
Define the sets
E f(x) dx = 0.
Show that f must then be equal to zero except on a set of
measure zero.
ContextEn = { xThenE: f(x)
1/n }
Z = { x
En = Z
and
En 
E.
Using the previous two examples we get:
0 =so that m(En) = 0 for all n. But then
m(Z) = m(which is what we had to prove.
En = 0