7.4. Lebesgue Integral

Definition 7.4.18: The General Lebesgue Integral

Let f be a measurable function and define the positive and negative parts of f, respectively, as:
f +(x) = max(f(x), 0)
f -(x) = max(-f(x), 0)
so that f = f + - f -. Then f is Lebesgue integral if f + and f - are Lebesgue integrable and
E f(x) dx = E f +(x) dx - E f -(x) dx =
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