Lebesgue integral
Via improper Riemann integral

Assuming that f is measurable and non-negative, the function f ∗ (t) = def μ ({ x ∈ E ∣ f (x) > t }). {\displaystyle f^{*}(t)\ {\stackrel {\text{def}}{=}}\ \mu \left(\{x\in E\mid f(x)>t\}\right).} is monotonically non-increasing. The Lebesgue integral may then be defined as the improper Riemann integral of f: ∫ E f d μ = def ∫ 0 ∞ f ∗ (t) d t. {\displaystyle \int _{E}f\,d\mu \ {\stackrel {\text{def}}{=}}\ \int _{0}^{\infty }f^{*}(t)\,dt.} This integral is improper at the upper limit of ∞, and possibly also at zero. It exists, with the allowance that it may be infinite.

As above, the integral of a Lebesgue integrable (not necessarily non-negative) function is defined by subtracting the integral of its positive and negative parts.
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