# Fubini's theorem and symmetry arguments

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!

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