I just want to make sure I understand a detail in the proof of the following result in my book:
Proposition. Let $X_1,\dots,X_n$ be independent real random variables. Then
$$E\Big(\prod_{i=1}^{n} X_i\Big)=\prod_{i=1}^{n}E(X_i)$$
if all $X_i$ are non-negative or if each is integrable. In the second case the product $\prod_{i=1}^{n} X_i$ is also integrable.
Proof. Let $Q:=\bigotimes_{i=1}^n P_{X_i}$ denote the joint distribution of $X_1,\dots,X_n$. Then by the change-of-variable formula and Tonelli's theorem
$$E\Big(\Big\lvert \prod_{i=1}^{n} X_i \Big\rvert\Big)=\int | x_1\cdot ...\cdot x_n| Q(dx) =\int\dots\int |x_1|\cdot ...\cdot|x_n| P_{X_1}(dx_1)...P_{X_n}(dx_n)$$
$$=\prod_{i=1}^{n}\int |x_i|P_{X_i}(dx_i)=\prod_{i=1}^{n}E(|X_i|)$$ Hence the assertion in the case that all $X_i\geq 0$, as well as the integrability of $\prod_{i=1}^{n} X_i$ in the case that each $X_i$ is integrable. In the latter case by Fubini's theorem the computations above remain valid if we remove all the absolute value signs.
Question:
Usually when applying Fubini's theorem the iterated integrals are only defined a.e. with respect to the product measure of the coordinates in which integration has not yet occured, which could invalidate the third equality above when absolute values are removed. However I think in this case this does not happen because each $X_i$ is assumed to be integrable, and hence the iterated integrals are can be defined on the whole product space of the coordinates in which integration has not yet occured, and measurable with respect to the product $\sigma$-algebra of that space.
For example, on the first iteration Fubini's theorem tells us that $$\int x_1\cdot ...\cdot x_n dP_{X_1}= x_2\cdot ...\cdot x_n \int x_1 dP_{X_1}$$ is defined $\otimes_{i=2}^n \mu_i$ a.e. and also $\otimes_{i=2}^n \mu_i$ integrable when extending it (on a null set) to a measurable function on $\times_{i=2}^n \Omega_i$. Hence we are free to define $$\int x_1\cdot ...\cdot x_n dP_{X_1}dx_1:= x_2\cdot ...\cdot x_n \int x_1 dP_{X_1}$$ on such a null set to integrate. Repeated application of this reasoning with respect to each coordinate then gives us the product of integrals.
Is this right way to think about this? Thanks a lot for your help.