This chapter will treat some classes of sets satisfying a combinatorial condition. In Chapter 6 it will be shown that under a mild measurability condition to be treated in Chapter 5, these classes have the Donsker property, for all probability measures P on the sample space, and satisfy a law of large numbers (Glivenko–Cantelli property) uniformly in P. Moreover, for either of these limit-theorem properties of a class of sets (without assuming any measurability), the Vapnik–Červonenkis property is necessary (Section 6.4).

The name Červonenkis is sometimes transliterated into English as Chervonenkis. The present chapter will be self-contained, not depending on anything earlier in this book, except in some examples.

Vapnik–Červonenkis Classes of Sets

Let X be any set and C a collection of subsets of X. For A ⊂ X let CA:= C ⊓ A:= A ⊓ C:= {C ⋂ A: C ∈ C}. Let card(A):= |A| denote the cardinality (number of elements) of A and 2A:={B: B ⊂ A}. Let ΔC(A):=|CA|. If A ⊓ C = 2A, then C is said to shatter A. If A is finite, then C shatters A if and only if ΔC(A) = 2|A|.

Universal Lower Bounds

This chapter is primarily about asymptotic lower bounds for ∥Pn − P∥F on certain classes F of functions, as treated in Chapter 8, mainly classes of indicators of sets. Section 11.2 will give some upper bounds which indicate the sharpness of some of the lower bounds. Section 11.4 gives some relatively difficult lower bounds on classes such as the convex sets in ℝ3 and lower layers in ℝ2. In preparation for this, Section 11.3 treats Poissonization and random “stopping sets” analogous to stopping times. The present section gives lower bounds in some cases which hold not only with probability converging to 1, but for all possible Pn. Definitions are as in Sections 3.1 and 8.2, with P:= U(Id) = λd = Lebesgue measure on Id. Specifically, recall the classes G(α, K, d):= Gα,K, d of functions on the unit cube Id ⊂ ℝd with derivatives through αth order bounded by K, and the related families C(α, K, d) of sets (subgraphs of functions in G(α, K, d − 1)), both defined early in Section 8.2.

Theorem 11.1 (Bakhvalov) For P = U(Id), any d = 1, 2, … and α > 0, there is a γ = γ (d, α) > 0 such that for all n = 1, 2,…, and all possible values of Pn, we have ∥Pn − P ∥G(α1,d) ≥ γn−α/d.

Measurable and measure spaces, extended Borel sets

The discussion up to now has been primarily concerned with the construction and properties of measures on σ-rings. There was some advantage (with a little added complication) in preserving the generality of consideration of σ-rings, rather than σ-fields during this construction process (cf. preface). In this chapter we prepare to use the results obtained so far to develop the theory of integration of functions on abstract spaces. From this point it will usually be convenient to assume that the basic σ-ring on which the measure is defined is, in fact, a σ-field. This will avoid a number of rather fussy details, and will involve negligible loss of generality for integration.

The basic framework for integration will be a space X, a σ-field S of subsets of X, and a measure μ on S. The triple (X, S, μ) will be referred to as a measure space. When μ(X)=1, μ will be called a probability measure. Probabilities are studied in depth from Chapter 9, though also occasionally appear earlier as special cases.

In most of this chapter we shall be not concerned at all with the measure μ, but just with properties of functions and transformations defined on X, in relation to S. To emphasize this absence of μ from consideration, the pair (X, S) will be referred to as a measurable space.

Measurability in Cartesian products

Up to this point, our attention has focussed on just one fixed space X. Consider now two (later more than two) such spaces X, Y, and their Cartesian product X × Y, defined to be the set of all ordered pairs (x, y) with x ∈ X, y ∈ Y. The most familiar example is, of course, the Euclidean plane where X and Y are both (copies of) the real line ℝ.

Our main interest will be in defining a natural measure-theoretic structure in X × Y (i.e. a σ-field and a measure) in the case where both X and Y are measure spaces. However, for slightly more generality it is useful to first consider σ-rings S, T in X, Y, respectively and define a natural “product” σ-ring in X ×; Y.

First, a rectangle in X ×; Y (with sides A ⊂ X, B ⊂ Y) is defined to be a set of the form A × B = {(x, y): x ∈ A, y ∈ B}. Rectangles may be regarded as the simplest subsets of X ×; Y and have the following property.