Integrable function

Remarks

Example: on $\R$ with Lebesgue measure, $f(x) = e^{-\abs{x}}$ is measurable since it is continuous. Its integral is $\int_\R e^{-\abs{x}} \, dx = 2 < \infty$, so $f$ is integrable.

Non-example: $g(x) = \sin(x) / x$ on $[1, \infty)$ with Lebesgue measure is measurable but not integrable. On each interval $[k\pi, (k+1)\pi]$ the integral of $\abs{g}$ is at least $2 / ((k+1)\pi)$, and these bounds have an infinite sum. Its improper Riemann integral still converges, so improper integrals and Lebesgue integrals differ here.

A real function $f$ is integrable if and only if both $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ have finite integral. A function with values in $[-\infty, \infty]$ is called integrable under the same condition, and it is then finite almost everywhere. The integrable functions form a vector space, often written $\mathcal{L}^1(\mu)$.