 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2010/a/1/b/i/solution

The triangle inequality gives the event inclusion

{∣ f_{n} - f_{m} ∣ > ε} \subseteq {∣ f_{n} - f ∣ > ε /2} \cup {∣ f_{m} - f ∣ > ε /2} .

Countable subadditivity of a measure bounds its measure by the sum of the two terms on the right. Both tend to zero as

m, n \to \infty

by convergence in measure. Thus convergence in measure implies the Cauchy property in measure.

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