Lebesgue integral

Remarks

Example: on $[-1, 1]$ with Lebesgue measure, let $f(x) = x$. Then $f^+ = x \mathbf{1}_{[0, 1]}$ and $f^- = -x \mathbf{1}_{[-1, 0]}$ both have integral $1/2$, so $\int f = 0$.

Non-example: $g(x) = x$ on all of $\R$ is not integrable. Both $\int g^+$ and $\int g^-$ are infinite, so the difference $\infty - \infty$ has no meaning.

The same formula is used when only one of $\int f^+$ and $\int f^-$ is finite, with value $\pm \infty$. The integral is linear and monotone on integrable functions, and it satisfies $\abs{\int_X f \, d\mu} \le \int_X \abs{f} \, d\mu$. On a bounded interval, every Riemann integrable function is Lebesgue integrable and the two integrals agree. Common notations are $\int f \, d\mu$, $\int f(x) \, d\mu(x)$ and $\int f(x) \, \mu(dx)$.