 Solution

ID: past-exam-of-the-probability-measure-and-martingales-course-of-the-university-of-oxford/2023/2/b/iii/solution

A family

(ξ_{i})_{i \in I}

K \to \infty lim i \in I sup E [∣ ξ_{i} ∣ 1_{{∣ ξ_{i} ∣ > K}}] = 0.

{X < 1}

∣ Z_{n} ∣ = 2^{n} ∣ f (X_{n} + 2^{- n}) - f (X_{n}) ∣ \leq 2^{n} L 2^{- n} = L .

Our chosen zero value on

{X = 1}

satisfies the same bound. Hence the random variables

(Z_{n})

are integrable, and for

K \geq L

every tail expectation in the definition vanishes. Therefore the family is uniformly integrable.

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