$\mathbb{E}(XY)$ is known as a joint moment or mixed moment of $X$ and $Y$.

The equivalent central moment $\mathbb{E}\big((X-\mu_X)(Y-\mu_Y)\big)$ is the covariance.

If the variables were also standardised to have $\sigma_X = \sigma_Y = 1$ then their covariance is also their correlation.

$\mathbb{E}(XY)$ is known as a joint moment or mixed moment of $X$ and $Y$.

The equivalent central moment $\mathbb{E}\big((X-\mu_X)(Y-\mu_Y)\big)$ is the covariance. This will be the same as $\mathbb{E}(XY)$ if the variables have been centered so that $\mu_X = \mu_Y = 0$.

The correlation of two variables is their covariance divided by the product of their standard deviations. If the variables have unit variance, so $\sigma_X = \sigma_Y = 1$, then their covariance is the same as their correlation.

If the variables have been standardised so they have mean zero and standard deviation one, then $\mathbb{E}(XY)$ is a simple formula for the correlation. This is one of the advantages of standardising your variables, if possible, before trying to prove something about their correlation.