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This question is not asking for a general algorithm. Instead, a very specific, preferably simple, example would suffice. Similar questions have been asked and answered for sigma-algebras that are partition-generated. E.g., How to compute conditional expectations with respect to a sigma field?.

An acceptable answer would also require you to prove that

i. The sigma-algebra is not partition-generated.

ii. It is indeed a conditional expectation in the standard sense.

Riemannstein
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  • Could you explain what a "non partition-generated sigma algebra" is? – whuber Mar 19 '20 at 11:33
  • Of course. I presume you are familiar with the notion of a partition (a collection of non-intersecting subsets whose union equals the sample space $\Omega$) and a sigma-algebra generated by a collection of subsets (the smallest sigma-algebra contains that collection). So a sigma-algebra $\mathcal F$ of the sample space $\Omega$ is non-partition-generated if there does not (exist a partition $\mathcal P$ of $\Omega$ such that $\mathcal F$ is the generated sigma-algebra of $\mathcal P$). – Riemannstein Mar 19 '20 at 12:07
  • Thank you. That sounds like it would be isomorphic to a discrete sigma algebra (namely, the power set of the partition)--and therein lies a key to the solution. – whuber Mar 19 '20 at 12:31

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