How to interpret integrals that have conditions written beside them

Question

sorry if this question has been asked before. I tried finding similar questions but couldn't find any. I have very little background in statistical mechanics, but I have been reading some literature, and came across notation as such -

$$\mu_Y(\mathbf{y}) = \int_{\mathbf{x}|E(\mathbf{x}) = y} \mu(\mathbf{x}) dx$$

I would like to know how one interprets such integrals, i.e. integrals with statements that resemble conditions in the limit - $\mathbf{x}|E(\mathbf{x})$. Any qualitative answer on how to read this equation, or on the interpretability of such notation, would be helpful. Thank you!

Answer 1

Let $\Omega$ be the ambient space you're integrating on. You can typically treat integrals over subsets of $\Omega$ as integrals over all of $\Omega$, with the integrand being multiplied by a characteristic function $\mathbf{1}_A$. In your case, you can write the integral as: $$\mu_Y(\mathbf{y})=\int_\Omega\mu(\mathbf{x})\cdot\mathbf{1}_{\{\mathbf{x}\in\Omega:E(\mathbf{x})=\mathbf{y}\}}\,d\mathbf{x}.$$ Now we've shifted perspectives from having a condition on $\mathbf{x}$ in the set we're integrating over to integrating over the entire ambient space, with the condition being shifted to the integrand itself.

Answer 2

Suppose your $x$'s take values in the measure space $X$. Then, by definition, $$\mu_Y(y) = \int_{x \mid E(x) = y}\mu(x)\,dx = \int_{X}\mu(x)I(E(x) = y)\,dx,$$ where $$I(E(x) = y) = \begin{cases}1 & \text{ if } E(x) = y \\ 0 & \text{ if } E(x) \neq y\end{cases}.$$