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Let $X$ be an integrable random variable defined on probability space $(\Omega , \mathcal F,P)$, and let $\mathcal F_{n},n\ge0$ , be a filtration on this space.
show that $X_{n}= E[X|\mathcal F_{n}]$ is an $\mathcal F_{n}$-martingale.

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To prove this is a martingale, you must show three things. First, that

  1. $X_n$ is $\mathcal F_{n}$ measurable, which is given by definition.
  2. Next, that $E|X_n| < \infty $. Since $X$ is integrable, we have this property as well.
  3. Finally, show that $E[ X_n | \mathcal F_{n-1}] = X_{n-1} $.

$ E[ X_n | \mathcal F_{n-1}] = E[ E[ X | \mathcal F_{n}] | \mathcal F_{n-1}] = E[ X | \mathcal F_{n-1}] = X_{n-1}$

So $X_n$ is a $\mathcal F_{n}$ martingale.

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