[Math] Questions about independence between random variable and $\sigma$-algebra

probability theory

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!

Best Answer

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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