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(Properties of Conditional Expectation). (a) Tower Property. Use the definition of conditional expectation to prove the tower property: If ℋ is a subfield of 𝒢, then, E(E(X |G) |ℋ)=E(X |ℋ) (b) Factorization Property. If Y ∈𝒢 ⊂, show that if E(|X|)<∞and Y is bounded then E(X Y |G)=Y E(X |G) A good part of the challenge of problems like these is the introduction of notation that makes clear that there has been no "begging of the question." Without due care, one can easily make accidental use of
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