Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 1 / a

A measurable space is a set $E$ together with a sigma-algebra $\mathcal{E}$ of its subsets: it contains $E$ and is closed under complements relative to $E$ and countable unions. A measure is a map $\mu :\mathcal{E}\to [0,\infty ]$ with $\mu (\varnothing )=0$ and countable additivity on disjoint measurable sets:A finite measure additionally satisfies $\mu (E)<\infty$.

For $\epsilon >0$, define the measurable decreasing tail eventsIf $f_n\to f$ almost everywhere, every point outside its null exceptional set eventually leaves each such tail event, so $\mu ({\bigcap }_n{A}_n(\epsilon ))=0$. Continuity from above of a measure applies because $\mu (E)<\infty$, giving $\mu (A_n(\epsilon ))\to 0$.

Conversely, assume these measures tend to zero. For each positive integer $k$, the intersection ${\bigcap }_n{A}_n(1/k)$ has measure zero. Outside the union of these countably many null sets, every $k$ has an index after which $|f_m-f|\le 1/k$ for all later $m$. Hence $f_m\to f$ there. This proves the desired equivalence. The argument uses all positive thresholds; replacing the intersection of tail events by a single strict limsup inequality at the same threshold would mishandle equality cases.