Let T be any set and (Ω, A, P) a probability space. Recall that a real-valued stochastic process indexed by T is a function (t, ω) ↦ Xt(ω) from T × Ω into ℝ such that for each t ∈ T, Xt (·) is measurable from Ω into ℝ. A modification of the process is another stochastic process Yt defined for the same T and Ω such that for each t, we have P(Xt = Yt) = 1. A version of the process Xt, is a process Zt, t ∈ T, for the same T but defined on a possibly different probability space (Ω1, B, Q) such that Xt and Zt, have the same laws, that is, for each finite subset F of Clearly, any modification of a process is also a version of the process, but a version, even if on the same probability space, may not be a modification. For example, for an isonormal process L on a Hilbert space H, the process M(x) ≔ L(−x) is a version, but not a modification, of L.

One may take a version or modification of a process in order to get better properties such as continuity. It turns out that for the isonormal process on subsets of Hilbert space, what can be done with a version can also be done by a modification, as follows.

TheoremLet L be an isonormal process restricted to a subset C of Hilbert space. For each of the following two properties, if there exists a version M of L with the property, there also is a modification N with the property.

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.

Let (S, ∥·∥) be a Banach space (in general nonseparable). A subset of the unit ball {f ∈ S′ : ∥f∥′ ≤ 1} is called a norming subset if and only if for all s ∈ S. The whole unit ball in S is always a norming subset by the Hahn-Banach theorem (RAP, Corollary 6.1.5).

Conversely, given any set, let be the set of all bounded real functions on, with the supremum norm

Then the natural map f ↦ (s ↦ s(f)) takes one-to-one onto a norming subset of S′.

So, limit theorems for empirical measures, uniformly over a class of functions, can be viewed as limit theorems in a Banach space S with norm Conversely, limit theorems in a general Banach space S with norm ∥ · ∥ can be viewed as limit theorems for empirical measures on S, uniformly over a class of functions, such as the unit ball of S′, since for f ∈ S′ and x1, …, xn ∈ S,

Suppose that Xj are i.i.d. real random variables with mean 0 and variance 1. Let One form of “invariance principle” will say that on some probability space, there exist such Xj and also i.i.d. N(0, 1) variables Y1, Y2, …, with such that as n → ∞ in probability. Since Tn/n½ also has a N(0, 1) distribution for each n, the invariance principle implies that Sn/n½ is close to Tn/n½, which implies the central limit theorem. Although it is not as obvious, central limit theorems generally imply invariance principles.