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.

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