One of my favourite cautionary tales in topology and probability is the fact that there is no such thing as a topology of almost-everywhere convergence; or, to borrow the title of E. T. Ordman’s 1966 note in the American Mathematical Monthly, “Convergence Almost Everywhere is not Topological”. This limitation is a healthy thing to know about, because it’s often tempting to think that once you have specified what a notion of convergence means for sequences (or, more generally, nets), one has specified a topology — you probably want to bring some topological tricks to bear on some other problem that interests you, and this is a necessary first step. Unfortunately, convergence almost everywhere has no associated topology; it’s not even that it has a bad topology, like the trivial one; it has no topology.
A quick reminder: if (Ω, Σ, ℙ) is a probability space and (fα)α∈A is a net (or, if you’re scared by nets, a sequence) of measurable functions (i.e. random variables) from Ω into ℝ, then fα converges almost everywhere to f: Ω → ℝ if
ℙ[limα fα = f] = 1,
i.e. the ℙ-probability measure of the set of those ω ∈ Ω for which the limit limα fα(ω) either doesn’t exist at all or exists but is something other than f(ω) is zero. Closely related to this, we say that fα converges in probability to f: Ω → ℝ if, for every δ >; 0,
limα ℙ[|fα − f| ≥ δ] = 0.
One important fact here is that if fα → f in probability, then for every subsequence of (fα), there exists a subsequence of that subsequence that converges to f almost surely.
Ordman shows that, in general, there can be no topology on the space of measurable functions from Ω into ℝ that corresponds to this notion of convergence. There are a few cases in which Ordman’s argument does not work, but they are rather trivial: finite, or other completely atomic, probability spaces. The general idea is easiest to explain, as Ordman does, with Ω = [0, 1], the unit interval with its usual σ-algebra, and ℙ = the usual uniform measure on [0, 1].
Consider the following sequence of functions (random variables) (fn): first, the indicator of [0, 1]; then the indicator of [0, 1⁄2]; then the indicator of [1⁄2, 1]; then the indicator of [0, 1⁄3]; then [1⁄3, 2⁄3]; then [2⁄3, 1]; and so on. For every x ∈ [0, 1], the sequence (fn(x)) fails to converge, since it equals 0 infinitely often and equals 1 infinitely often. In particular, fn does not converge almost everywhere to the zero function. So. If there were a topology of convergence almost everywhere, then there would exist and open set U about the zero function such that fn ∉ U infinitely often. Hence, we can extract a subsequence fn(k) that lies entirely outside U. On the other hand, the full sequence (fn) does converge to 0 in probability, and so from the subsequence (fn(k)) we can extract a further subsequence (fn(k(i))) that converges almost everywhere to 0, and hence is eventually in U. This is a contradiction, so no such topology of convergence almost everywhere can exist.