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 (*f*_{α})_{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 *f*_{α} **converges almost everywhere** to *f*: Ω → ℝ if

ℙ[lim_{α} *f*_{α} = *f*] = 1,

i.e. the ℙ-probability measure of the set of those ω ∈ Ω for which the limit lim_{α} *f*_{α}(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 *f*_{α} **converges in probability** to *f*: Ω → ℝ if, for every *δ* >; 0,

lim_{α} ℙ[|*f*_{α} − *f*| ≥ *δ*] = 0.

One important fact here is that if *f*_{α} → *f* in probability, then for every subsequence of (*f*_{α}), 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) (*f*_{n}): 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 (*f*_{n}(*x*)) fails to converge, since it equals 0 infinitely often and equals 1 infinitely often. In particular, *f*_{n} 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 *f*_{n} ∉ *U* infinitely often. Hence, we can extract a subsequence *f*_{n(k)} that lies entirely outside *U*. On the other hand, the full sequence (*f*_{n}) does converge to 0 in probability, and so from the subsequence (*f*_{n(k)}) we can extract a further subsequence (*f*_{n(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.