Say we have some function $f$ on the reals and a Borel set $\sigma$ from the $\sigma$-algebra of $\mathbb{R}$. My understanding from https://en.wikipedia.org/wiki/Dirac_measure is that the Dirac measure $\delta_x(\sigma) = \begin{cases} 1 & x\in \sigma \\ 0 & x\not\in \sigma \\ \end{cases}$ satisfies the following properties
- $f(x) = \int_{y\in\mathbb{R}} f(y) d\delta_{x}(y)$
- $\delta_{f(x)}(\sigma) = \int_{y\in\sigma} d\delta_{f(x)}(y)$
Is it the case that: \begin{align*} \delta_{f(x)}(\sigma) = \int_{y\in\sigma} d\delta_{f(x)}(y) = \int_{z \in \mathbb{R}} \int_{y \in \sigma} d\delta_{f(z)}(y) d\delta_{x}(z) \end{align*}
If not, why?