Fubini's theorem and symmetry arguments

Question

I'm currently going a textbook in a physics course and in there is the following expression $$\mathbb{E}_{x}\left[\mathbb{E}_{y}\left[f(x,y)\right]\right],$$ where $y \overset{\mathrm{i.i.d}}{\sim} \mathcal{N}(0, 1)$ and $x \overset{\mathrm{i.i.d}}{\sim} P_X(x)$. The author of the notes then transforms this expression to $$\mathbb{E}_{x,y}\left[f(x,y)\right],$$ with the following argument "because the standard normal distribution is symmetric around zero and a change of variable $y\to -y$".

Now I already did some searching and found this very related question: Expectation over 2 random variables, help needed

The answers talk about the so called Fubini theorem so I tried looking it up on wikipedia. The problem is that I have a very hard time understanding what the article says and I especially have a problem connecting it to the symmetry argument the author made...

I wanted to ask if anyone could help and point me out where the connection is such that I understand this step in the notes a little bit better. Thank you!

Answer 1

For any multiple integral one has the problem to define $$\int_{x \in \Omega} f(x) d^nx $$ understood as a Riemann sum over a partition of the region $\Omega$ that converges to a limit if the region is partioned into n-simplices with their maximum side length going to zero.

This problem practically crashes, because summable functions on regions with irregular, non smooth boundaries are not easily to characterize.

But for smooth functions with smooth boundaries its possible to show, that the multiple integral can be reduced to an iterated integral, eg in 2d

$$\int_{a<x<b, c<y<d} f(x,y ) dx \ dy = \int_{a<x<b} \left( \int_{c<y<d} f(x,y ) dx \right) \ dy = \int_{c<y<d} \left( \int_{a<x<b} f(x,y ) dy \right) \ dx$$

The generalized theorem of integration over regions by successive integrations over a single dimension with adapted boundaries is named after Fubini, and it is true in general for measurable functions in regions with smooth boundaries, where the integrals are understood as Lebesgue integrals.

Since expectations are integrals over a measurable density multiplied by polynomials and characteristic functions, the same algebra of summations is true for $E_{x,y} = \int \dots dx dy $, that is not much more than $$\sum_i \sum_k = \sum_{i,k}$$