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
E f(x) dx = 0.
Show that f must then be equal to zero except on a set of
measure zero.
Define the sets
![](../../symbols/int.gif)
En = { xThenE: f(x)
1/n }
Z = { xE: f(x) # 0 }
![](../../symbols/union.gif)
![](../../symbols/subset.gif)
0 =so that m(En) = 0 for all n. But thenE f(x) dx
![]()
En f(x) dx
1/n m(En)
m(Z) = m(which is what we had to prove.En)
![]()
En = 0