= Integrable random variable
{title2=$\mathbb E|X|<\infty$}
A real or complex <random variable> $X$ is integrable when $\mathbb E|X|<\infty$. This is <integrability> with respect to its <probability measure>, and ensures that its <expected value> is finite. A bounded <random variable> is always an <integrable random variable>; so is the example taking $-1,2,8$, for which $\mathbb E|X|=2.7$.
For a real <random variable>, put $X_+=\max(X,0)$ and $X_-=\max(-X,0)$, its <positive part> and <negative part>. Each has a nonnegative <Lebesgue integral>, possibly infinite. The expression $\mathbb E[X]=\mathbb E[X_+]-\mathbb E[X_-]$ is defined when at least one of these two <expected values> is finite; if both are infinite, the <expected value> is undefined. <Integrability> makes both finite. A nonnegative <random variable> can have <expected value> $+\infty$, even though it is not an <integrable random variable>.
The standard <Cauchy distribution> is a useful warning. With its <probability density function> $f(x)=1/[\pi(1+x^2)]$, the positive contribution is $\int_0^Rxf(x)\,dx=\log(1+R^2)/(2\pi)$, which tends to infinity; the negative contribution has the same magnitude. Integrating symmetrically over $[-R,R]$ gives zero for each $R$, but this cancellation does not define an <expected value>. The <Cauchy sample mean has an unchanged distribution>, showing why the <integrability> hypothesis in the <strong law of large numbers> matters.
For an <integrable random variable>, <conditional expectation> preserves <integrability>, and <linearity of expectation> can safely combine finite averages without forming $\infty-\infty$.
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