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:
- $E (X|\mathcal{A}) = (EX)I_Ω;$
- $E(I_A \times X) = P (A)EX, \, \forall A \in \mathcal{A}.$
Thanks and regards!