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

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