Suppose we have two independent $\sigma$-algebras, $\mathcal{G}$ and $\mathcal{H}$. Let $X$ and $Y$ be two $(\mathcal{G}\cap\mathcal{H})$-measurable random variables. Then $\sigma(X)\subseteq\mathcal{G}\cap\mathcal{H}\subseteq\mathcal{G}$ and similarly $\sigma(Y)\subseteq\mathcal{H}$. Thus for any $A\in\sigma(X)$ and $B\in\sigma(Y)$, we have $A\in\mathcal{G}$ and $B\in\mathcal{H}$, so by the independence of these $\sigma$-algebras, $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$, implying that $X$ and $Y$ are independent random variables.
My question is - is it then necessarily true that any such random variables must be independent, or have I gone wrong somewhere above in my reasoning? It doesn't feel quite right that any such random variables must be independent - if $X$ is $\mathcal{G}\cap\mathcal{H}$-measurable, surely we can define $Y=X+1$ as a contradiction? But I can't see the problem in my reasoning in the first paragraph. Thanks!