Sigma fields generated by random variables

Question

If $\sigma(X) = \sigma(Y)$, then is it true that $\sigma(X,Z) = \sigma(Y,Z)$, where $X$, $Y$, and $Z$ are real random variables, and where, for example, the notation, $\sigma(X)$, denotes the sigma algebra generated by $X$ .

Answer 1

Yes.

Let $X,Y,Z$ be random variables $\Omega\to \mathbb R$ where $\mathcal B$ denotes the Borel $\sigma$-algebra.

Then actually e.g. $\sigma(X,Z)$ is the smallest $\sigma$-algebra on $\Omega$ such that $X$ and $Z$ are measurable if $\Omega$ is equipped with it. Further it can be proved that e.g. $X^{-1}(\mathcal B):=\{X^{-1}(B)\mid B\in\mathcal B\}$ is a $\sigma$-algebra itself, which leads to the conclusion that $\sigma(X)=X^{-1}(\mathcal B)$.

Then $X^{-1}(\mathcal B)=\sigma(X)=\sigma(Y)\subseteq\sigma(Y,Z)$ and $Z^{-1}(\mathcal B)=\sigma(Z)\subseteq\sigma(Y,Z)$

So $X$ and $Z$ are both measurable if we are dealing with measurable space $(\Omega,\sigma(Y,Z))$.

This allows the conclusion that $\sigma(X,Z)\subseteq\sigma(Y,Z)$.

With symmetry we find that also $\sigma(Y,Z)\subseteq\sigma(X,Z)$ so we conclude that: $$\sigma(X,Z)=\sigma(Y,Z)$$