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

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- Solution f

Solution

The equation fails for general nonnested sigma-algebras. On the four-point space

Ω = {1, 2, 3, 4}

with uniform probability, let

A = {1, 2}, B = {1, 2, 3}, G = σ (A), H = σ (B),

and take

X = 1_{A}

. The intersection

G \cap H

is trivial, so

E [X ∣ G \cap H] = \frac{1}{2} .

But

X

is

G

-measurable and

E [E [X ∣ G] ∣ H] = E [X ∣ H] = \frac{2}{3} 1_{B},

which is zero on

B^{c}

and is not almost surely

1/2

. This exhibits the failure of iterated conditional expectation over nonnested sigma-algebras.

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