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In Durrett's book "Probability: Theory and Examples" (pg. 32), he states the following change of variables formula:

Let $X$ be a random element of $(S,\mathcal{S})$ with distribution $\mu$, i.e., $\mu(A) = P(X\in A)$. If $f$ is a measurable function from $(S,\mathcal{S})$ to $(\textbf{R},\mathcal{R})$ so that $f\geq 0$ or $E|f(X)|<\infty$, then $$Ef(X) = \int_S f(y)\mu(dy)$$

As far as I can tell, he never defines the notation $\mu(dy)$. What does this notation mean?

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    $\begingroup$ $\int_S f(y)\mu(dy) = \int_S f(y) d\mu(y)$ (maybe you saw that notation in the second form). It just means the integral with respect to $\mu$ measure. $\endgroup$
    – Dominik Kutek
    Jun 22, 2020 at 23:01

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$$\int_S f(y) \mu(dy)$$ is simply the integral of $f$ against the measure $\mu$ with respect to the variable $y$.

You may also see the other notations $\int_S f d\mu$ or $\int_S f(y) d\mu(y)$.

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