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?
$$\int_S f(y) \mu(dy)$$ 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 $\int_S f(y) d\mu(y)$.