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Given a probability measure space $(\Omega, \mathcal{F}, P)$, if a random variable X and a sub $\sigma$-algebra $\mathcal{A}$ are independent, I was wondering why:

  1. $E (X|\mathcal{A}) = (EX)I_Ω;$
  2. $E(I_A \times X) = P (A)EX, \, \forall A \in \mathcal{A}.$

Thanks and regards!

1 Answers 1

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Hint for 1.: Recall the definition of the random variable $E(X|\mathcal{A})$. Hint for 2.: Recall the definition of $X$ being independent from $\mathcal{A}$. (And add the condition that $X$ is integrable.)

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    It is true. For a simple construction and for the answers to many of your other questions, see the small book *Probability with martingales* by David Williams http://www.goodreads.com/book/show/615497.Probability_with_Martingales.2011-05-07