# alternative definition of Lebesgue integral, an

There is also another way which uses the Riemann integral .

Let $(X,\mathcal{M},\mu)$ be a measure space  . Let $f\colon x\rightarrow\mathbb{R}^{+}\cup\{0\}$ be a nonnegative measurable function. We will define $\int fd\mu$ in $[0,\infty]$ and will call it as the Lebesgue integral of $f$.

If there exists a $t>0$ such that $\mu\left(\left\{x\colon f(x)>t\right\}\right)=\infty$, then we define $\int fd\mu=\infty.$

Otherwise, assume $\mu\left(\left\{x\colon f(x)>t\right\}\right)<\infty$ for all $t\in(0,\infty)$ and let $F_{f}(t)=\mu\left(\left\{x\colon f(x)>t\right\}\right)$. $F_{f}(t)$ is a monotonically non-increasing function on $(0,\infty)$, therefore its Riemann integral is well defined on any interval $[a,b]\subset(0,\infty)$, so it exists as an improper Riemann integral on $(0,\infty)$. We define

 $\int fd\mu=\int_{0}^{\infty}F_{f}(t)dt.$

The definition can be extended first to real-valued functions, then complex valued functions as usual.

## References

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Title alternative definition of Lebesgue integral, an AlternativeDefinitionOfLebesgueIntegralAn 2013-03-22 17:32:46 2013-03-22 17:32:46 Gorkem (3644) Gorkem (3644) 6 Gorkem (3644) Definition msc 26A42 msc 28A25