Covariance of Square-Integrable Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $X$ and $Y$ be square-integrable random variables on it.

The covariance of $X$ and $Y$ is

This is defined: square-integrable random variables are integrable by Square-Integrable Random Variables and the Mean-Square Inner Product, so $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ are real numbers; and the centered random variables $X-\mathbb{E}[X]$ and $Y-\mathbb{E}[Y]$ are square-integrable with integrable product, by the closure properties of the same definition (constant random variables are square-integrable).

The random variables $X$ and $Y$ are called uncorrelated if $\operatorname{Cov}(X,Y)=0$.

The covariance of $X$ with itself, $\operatorname{Cov}(X,X)=\mathbb{E}\bigl[(X-\mathbb{E}[X])^{2}\bigr]$, is the variance $\operatorname{Var}(X)$.