Definition 7.4.5: Lebesgue Integral for Bounded Functions

Suppose f is a bounded function defined on a measurable set E with finite measure. Define the upper and lower Lebesgue integrals, respectively, as

I^*(f)_L = inf{ s(x) dx: s is simple and s f }
I_*(f)_L = sup{ s(x) dx: s is simple and s f }

If I^*(f)_L = I_*(f)_L the function f is called Lebesgue integrable over E and the Lebesgue integral of f over E is denoted by

_E f(x) dx

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