Following is an excerpt from Lecture 7 of these notes, about the independence of random variables. I have some very basic questions about what's going on. Please scroll down to read them.
Independence can always be viewed in a canonical way. Let $(\Omega,\mu)$ be a product space $(\Omega_1\times\Omega_2, \mu_1\otimes \mu_2)$ where $\mu_1$ and $\mu_2$. Suppose $X$ and $Y$ are random variables on $\Omega$ for which the value $X(\omega_1,\omega_2)$ depends only upon $\omega_1$, while $Y(\omega_1,\omega_2)$ depends only upon $\omega_2$. Then any integral (that converges appropriately) can be written as the product of integrals by Fubini's theorem. $$\mathbb E[f(X)g(Y)] = \int f(X(s)) g(Y(t))\ \mathrm d\mu_1\otimes\mu_2 (s,t)$$ $$= \int f(X(s))\ \mathrm d \mu_1(s) \int g(Y(t))\ \mathrm d\mu_2(t)= \mathbb E[f(X)] \mathbb E[g(Y)]$$
- What do we mean by "canonical" here? I didn't get much on searching on Google.
- What is $\mu_1\otimes\mu_2$? I have never seen this notation before. It'd be helpful if someone could explain the product space (or point me to some references) to me.
- Where is Fubini's theorem being used here? From what I know, Fubini's theorem is a prescription for swapping integrals.
- Inside the integrals, I see that the author chose to write $\mathrm d\mu_1(s)$ instead of just $\mathrm d\mu_1$ (similarly in two other places). Is there a specific reason for this? I haven't seen such notation being used before. When integrating some function $f$ with respect to a measure $\mu$, we typically just write $\int_X f\ \mathrm d \mu$.