Lebesgue Integrable

A real-valued function $f$ defined on the reals $\Bbb{R}$ is called Lebesgue integrable if there exists a Sequence of Step Functions $\{f_n\}$ such that the following two conditions are satisfied:

1. $\sum_{n=1}^\infty \int\vert f_n\vert<\infty$,

2. $f(x)=\sum_{n=1}^\infty f_n(x)$ for every $x\in\Bbb{R}$ such that $\sum_{n=1}^\infty \int\vert f_n\vert<\infty$.

Here, the above integral denotes the ordinary Riemann Integral. Note that this definition avoids explicit use of the Lebesgue Measure.

See also Integral, Lebesgue Integral, Riemann Integral, Step Function