Taking the expectation of a probability density function

Question

I am trying to understand what it means to have an expectation of a probability density function taken over a different, independent random variable.

Let's say I have a random variable $X$ with density $f_X$ and support $\mathcal{X}$ and an independent random variable $Y$ with density $g_Y$ and support $\mathcal{Y}$. Then what does $$\mathbb{E}_{X}[g_Y(x)]$$ mean?

Does this mean $$\int_{\mathcal{X}} g_Y(x) f_X(x) dx$$ And if so, can the integrand be simplified since $X$ and $Y$ are independent?

Answer 1

Theorem Let $u$ be a measurable function and $x$ a random variable with distribution $F.$ Then $\mathbf{E}(u(x)) = \int\limits_\mathbf{R} u(t) dF(t);$ if $F$ has a density $f,$ so that "$dF(t) = f(t) dt$", then $\mathbf{E}(u(x)) = \int\limits_\mathbf{R} u(t) f(t) dt.$

You now apply this theorem when $u = g_Y.$ Whether $u$ has additional meaning is irrelevant.