I was reading the definition of Markov's Inequality on Wikipedia and it says
If $X$ is any nonnegative integrable random variable and $a > 0$, then
$\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}$. So I was just wondering what it means to be an integrable random variable. I thought integrable means that the integral exists and in this case that is clear since $X$ is nonnegative and we cannot get the integral equal $\infty - \infty$. Does it mean that the integral is finite? If that's the case why does it matter for the result?
A random variable $X$ is integrable if and only if $$\mathbb E[|X|] = \int_\Omega |X(\omega)|\mathsf d\mathbb P(\omega) < \infty. $$ If $X\geqslant 0$ almost surely, i.e. $\mathbb P(X<0) = 0$, then this is equivalent to $\mathbb E[X]<\infty$. So in this case, "integrable" just means that it has a finite mean. However, the inequality would still hold even if $\mathbb E[X]=\infty$, since clearly $\mathbb P(X\leqslant a)<\infty$.