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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