The Riemann integral is defined only for functions on a closed interval. For example, Riemann integral over $\mathbb{R}$ is not defined, and the improper integral $\int_{-\infty}^\infty f$ is defined as a limit of Riemann integral over finite closed interval. How about Lebesgue integrals? Can a reasonably well-behaved nonnegative function $f:\mathbb{R}\to\mathbb{R}$ Lebesgue-integrated? (Forgive me if this is too an elementary question; I know very little of Lebesgue measure theory.)
What I'm having in mind is the probability density function of a random variable:
$$P(X\leq x)=\int_{-\infty}^xf,$$
where $\int_{-\infty}^\infty f=1$. Apparently the density function $f$ is required to be Lebesgue integrable, but I'm wondering if improper integrals like $\int_{-\infty}^x f$ and $\int_{-\infty}^\infty f$ are well-defined.
Best Answer
Lebesgue integration is defined by Lebesgue measure. Specifically you can well define the integral of a positive "measurable" function over any "measurable" subset of $\Bbb R$, and a PDF function like yours and all $\Bbb R$ (or $(-\infty, x] $ for instance) have these properties.