Let's say I have a real valued function $f$ and a subset $S \subset \mathbb{R}$. Let's also define the dirac measure to be $ \delta_{x}(S) = \begin{cases} 1 & x \in S \\ 0 & x \not\in S \end{cases}$
I would expect the following to be true:
$\int_{x' \in \mathbb{R}} \int_{z \in S} d\delta_{f(x')}(z) \ d\delta_{0}(x') = \begin{cases} 1 & f(0) \in S \\ 0 & f(0) \not\in S \end{cases}$
Is this always the case? Does this hold if $f$ is discontinuous almost everywhere?