# Total expectation under measure theory

Mathematics Asked by Jiexiong687691 on December 6, 2020

I want to show that $$mathbb {E}(mathbb {E}(Xmid mathcal {F}))=mathbb {E}X$$. My thought is that

begin{align*}mathbb {E}(mathbb {E}(Xmid mathcal {F}))=sum_{Omega_{i}} frac {mathbb {E}(X;Omega_{i})}{mathbb {P}(Omega_{i})}mathbb {P}(Omega_{i})=mathbb{E}X, end{align*}
where $$Omega_{i}’s$$ are either finite or infinitely countable partition of $$Omega$$ into disjoint sets, and each one has positive probability. I wonder if it’s always true that there exists such $$Omega_{i}’s$$ such that $$mathcal {F}=sigma(Omega_{1},Omega_{2},ldots)$$ with the desired properties? Thanks.

Not every sigma field is countably generated and not many sigma fields are generated by countable partitions.

$$E(E(X|mathcal F) I_A)=EXI_A$$ for all $$A in mathcal F$$ by the definition of $$E(X|mathcal F)$$. Just taking $$A =Omega$$ gives $$E(E(X|mathcal F)) =EX$$

Correct answer by Kavi Rama Murthy on December 6, 2020

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