Suppose f is a bounded, non-negative function defined on a measurable set E with finite measure such thatE f(x) dx = 0. Show that f must then be equal to zero except on a set of measure zero.
Context
Example 7.4.10(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 |
En = { 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