Intersection of minimal sigma algebras of two independent random variables Consider a probability space $(\mathcal{X},\mathcal{M},P).$ Let $Y$ and $Z$ be two independent random variables. What are the contents of $\sigma(Y)\cap\sigma(Z)$? It is trivial that $\{\varnothing,\mathcal{X}\}\subset\sigma(Y)\cap\sigma(Z).$ I am not able to prove if there exist any more sets (possibly of measure zero or one). Can some one please help? 

All sets in the intersection have a measure of $0$ or $1$.
Note that if $A \in \sigma(Y) \cap \sigma(Z)$ we have by independence $P(A) = P(A \cap A) = P(A) \cdot P(A)$, which can only hold true for $P(A) \in \{0, 1\}$.
I don't think you can say much more about the contents, take for example two constant random variables.
