Expectation of dirac measure of a function

Question

Suppose $\pi(dx)$ is a measure, $\phi$ is a deterministic function and $\delta_{\phi(x)}(A)$ is the dirac measure. The dirac measure can be considered a measurable function (I think? Should be a kernel) for a fixed set $A$. What does the expectation of $\delta_{\phi(\cdot)}(A)$ becomes with respect to $\pi$? $$ \int \delta_{\phi(x)}(A)\,\, \pi(dx) = \int_A\pi(d\phi(x)) \qquad ??? $$

Answer 1

The Dirac measure is : $$\delta_{\phi(x)}(A) = \left\{\begin{array}{cl} 1 & \text{if } \phi(x) \in A \\ 0 & \text{else} \end{array}\right. = \chi_{\phi^{-1}(A)}(x)$$

with $\chi$ the indicator function.

Therefore, $$\int \delta_{\phi(x)}(A) \pi(\text{d}x) = \int \chi_{\phi^{-1}(A)}(x) \pi(\text{d}x) = \pi\circ\phi^{-1}(A)$$