Interactive Real Analysis - part of

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Example 7.4.10(b): Properties of the Lebesgue Integral

If f is Lebesgue integrable over E and A f(x) B then show that
A m(E) E f(x) dx B m(E)
We should add the condition that m(E) is finite before we start ...

Now define the simple functions

s(x) = A XE(x)
S(x) = B XE(x)
Because f is bounded by A and B we have
s(x) f(x) S(x)
But then the result follows easily from the properties of the Lebesgue integral:
A m(E) = s(x) dx E f(x) dx S(x) dx = B m(E)
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