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If we have a probability space $(\Omega,\mathcal{A},P)$ and pairwise disjoint events, say $B_i$ for $1\leq i \leq n \leq \infty$ such that $\bigcup B_i = \Omega$ and if we further set $\mathcal{F}=\sigma(B_i,1\leq i\leq N)$ why is then the conditional expectation constant on $B_i \;\forall i$, $E[X\cdot 1_{B_i}|\mathcal{F}] = b_i$? Because it is $\mathcal{F}$-measurable?

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  • $\begingroup$ Do you mean to say that $E[X\cdot 1_{A}|\mathcal{F}]=b_{i}$ for all $A\subset B_{i},$ for all $i$? This would make the conditional expectation "constant" on $B_{i},$ for each $i.$ Otherwise $E[X|\mathcal{F}]$ does not appear to depend on $i,$ but the rhs clearly does. $\endgroup$ – RideTheWavelet Jul 11 '17 at 2:12
  • $\begingroup$ Yes, thank you. I edited the question. But why is it constant? $\endgroup$ – laguna Jul 11 '17 at 10:35
  • $\begingroup$ Because every $\mathcal F$-measurable random variable is $$\sum_ix_i\mathbf 1_{B_i}$$ for some $x_i$s. $\endgroup$ – Did Jul 11 '17 at 10:37
  • $\begingroup$ This helps. Thanks you. $\endgroup$ – laguna Jul 11 '17 at 10:41

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