Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 1 / c

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- Solution c

Solution

If

G \in G \cap H

, independence makes

G

independent of itself, so

P (G) = P (G)^{2} .

Thus every event in

G \cap H

has probability zero or one. The intersection is trivial modulo null sets, and the independent sigma-algebras have trivial intersection result gives

E [X ∣ G \cap H] = E [X]

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