Let $(\Omega, \mathcal{A}, P)$ be a probability space, and let $\mathcal{F}$ and $\mathcal{G}$ be two $\sigma$-algbras on $\Omega$. Suppose $\mathcal{F} \subset \mathcal{A}$ and $\mathcal{G} \subset \mathcal{A}$. The $\sigma$-algbras $\mathcal{F}$ and $\mathcal{G}$ are independent if for any $A \in \mathcal{F}$, any $B \in \mathcal{G}$, $P(A \cap B) = P(A)P(B)$. Suppose $\mathcal{F}$ and $\mathcal{G}$ are independent $\sigma$-algbras, and a r.v. $X$ is measurable from both $(\Omega, \mathcal{F})$ to $(R, \mathcal{B})$ and from $(\Omega, \mathcal{G})$ to $(R, \mathcal{B} )$ (where $R$ is the set of all real numbers and $\mathcal{B}$ is the Borel $\sigma$-algebra on $R$ ). Show that $X$ is almost surely a constant. i.e. $P(X=c)=1$ for some constant c.