What is the expectation of the Dirac delta function of a random variable

Question

I have a Dirac delta function as follows:

$\delta_{\epsilon = y}$ where $\epsilon \sim N(0, \sigma^2)$, and $y \in \mathbb{R}$.

I want to take the expectation of this function, but it appears that it would, by definition, be $\infty * P(\epsilon = y)$, which isn't well defined.

Is that accurate, or am I missing something here.

Answer 1

As it turns out, the result is simply the density of $\epsilon$ at y.