Let be continuous functions. We call the *-th moment* of . Prove that if and have identical -th moments for all , then .

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# One moment please

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Let be continuous functions. We call the *-th moment* of . Prove that if and have identical -th moments for all , then .

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We will prove a more general result. Suppose are in and they have identical -th moments for all . Let . By hypothesis, for every polynomial . Thus, by the Weierstrass Approximation Theorem, it follows that for every continuous function . Consider the continuous function for and such that . Then, the convolution , as , for almost every . Thus, a.e. and, if they are continuous, everywhere.

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