Independence and combinations of random variables

Question

Setup:

A set of random variables $X_1, \ldots X_n$ is independent if for all Borel-sets $B_1,\ldots,B_n$ it is the case that $$P(X_1 \in B_1,\ldots,X_n \in B_n) = \prod_{i=1}^n P(X_i \in B_i)$$

An alternative definition is that the generated sigma-algebras $\sigma(X_1),\ldots,\sigma(X_n)$ should be independent.

Question:

Let $X, Y, Z$ be independent random variables. I am pretty sure that this implies that $XY$ and $Z$ are also independent, but I have not been able to prove this.

Is there an elementary way to prove this or is there a trick or a general theorem needed?

Thank you.

Answer 1

Hint:

1. Recall (or check) that $(X,Y)$ and $Z$ are independent.
2. Set $g(x,y) := x \cdot y$, then $X \cdot Y = g(X,Y)$. Thus, $$\mathbb{P}(X \cdot Y \in B_1, Z \in B_2) = \mathbb{P}\big((X,Y) \in g^{-1}(B_1), Z \in B_2 \big)$$ for any two Borel sets $B_1$, $B_2$. (Note that $g^{-1}(B_1)$ is the inverse image of $B_1$ under $g$. No need for an inverse function!) Now use step 1 to write the right-hand side as a product of suitable probabilities.