7.4. 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.
Define the sets
En = { x E: f(x) 1/n }
Z = { x E: f(x) # 0 }
Then En = Z and En E. Using the previous two examples we get:
0 = E f(x) dx En f(x) dx 1/n m(En)
so that m(En) = 0 for all n. But then
m(Z) = m(En) En = 0
which is what we had to prove.
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