Let A be the set of all possible empirical distribution functions F1 for one observation x ∈ [0, 1], namely F1(t) = 0 for t < x and F1{t) = 1 for t ≥ x. We noted previously that A in the supremum norm is nonseparable: it is an uncountable set, in which any two points are at a distance 1 apart. Thus A and all its subsets are closed. If x ≔ X1 has a continuous distribution such as the uniform distribution U[0, 1] on [0, 1], then x → (t → 1t≥x) takes [0, 1] onto A, but it is not continuous for the supremum norm. Also, it is not measurable for the Borel σ-algebra on the range space. So, in Chapter 3, functions f* and upper expectations E* were used to get around measurability problems.

Here is a different kind of example. It is related to the basic “ordinal triangle” counterexample in integration theory, showing why measurability is needed in the Tonelli-Fubini theorem on Cartesian product integrals. Let (Ω, ≤) be an uncountable well-ordered set such that for each x ∈ Ω, the initial segment Ix ≔ {y : y ≤ x} is countable. (In terms of ordinals, Ω is, or is orderisomorphic to, the least uncountable ordinal.) Let S be the σ-algebra of subsets of Ω consisting of sets that are countable or have countable complement. Let P be the probability measure on S which is 0 on countable sets and 1 on sets with countable complement.