The Axiomatic Approach

1. Preliminaries

The symbol [parallel] [parallel] for norm is used in many different senses throughout the book; but it is never used in two different senses on the same space, so no confusion can result. In particular, when x is in a euclidean space of finite dimension (i.e. it is a finite-dimensional vector), then [parallel] x [parallel] will always mean the maximum norm, i.e.

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It is important to distinguish notationally between functions and their values. For example, if μ is a measure, then [parallel] μ [parallel] is its total variation, whereas [parallel] μ(S) [parallel] is simply the absolute value of the real number μ(S).

Occasionally it will be necessary to use more than one norm on a given space; in that case, the norms will be distinguished in various ways, for example, by subscript, as in [parallel] [parallel]1.

Closure will be denoted by a bar; thus [bar.A] is the closure of A.

Composition will usually be denoted by the symbol ο; thus if f is defined on the range of μ, then the function whose value at S is f(μ(S)) will be denoted f ο μ. When no confusion can result, especially in the case of compositions of linear operators, the symbol ο will occasionally be omitted.

The origin of any linear space (including, of course, the real line) will be denoted 0; no confusion can result. In euclidean n-space En, x · y will denote the scalar product of two vectors x and y, ei will denote the i-th unit vector (0, ..., 0, 1, 0, ..., 0), and e will denote the vector (1, ..., 1).

The symbol [subset] will be used for inclusion that is not necessarily strict. Set-theoretic subtraction will be denoted by \, whereas — will be reserved for algebraic subtraction. f|A will mean "f restricted to A." The cardinality of a set A is denoted [absolute value of A].

Closed and open intervals are denoted [a, b] and (a, b) respectively; [a, b) and (a, b] denote half-open intervals.

W.l.o.g. means "without loss of generality." W.r.t. means "with respect to."

A measurable space is a pair (I, C) where I is a set and C is a σ-field of subsets of I; the members of 6 are called measurable sets. When no confusion can result, we shall sometimes denote the measurable space (I, C) simply by I. A function f from one measurable space (I, C) into another one (I, D) is called measurable if T [member of] D implies f-1 (T) [member of] C. Two measurable spaces are called isomorphic if there is a one-one function from one onto the other that is measurable in both directions; the mapping is called an isomorphism. The measurable space consisting of the closed unit interval with its Borel subsets will be denoted ([0,1], (B); Lebesgue measure on this space will be denoted λ.

Proposition 1.1. Any uncountable Borel subset of any euclidean space, and indeed of any complete separable metric space, when considered as a measurable space, is isomorphic to ([0, 1], B).

For a proof, see Mackey (1957), or Parthasarathy (1967), Theorems 2.8 and 2.12, pp. 12 and 14.

Proposition 1.2. Let f be a one-one measurable function from a Borel subset of [0, 1] into the real line, both considered as measurable spaces. Then the range off is a Borel set.

For a proof, see Mackey (1957), p. 139, Theorem 3.2, or Parthasarathy (1967), Theorem 3a, p. 21.

Unless otherwise specified, the word "measure" in this book refers to countably additive totally finite signed scalar measures. Recall that a measure [xi] on a measurable space (I, C) is non-atomic if for all S in C with [xi](S) ≠ 0, there is a T [subset] S with [xi](S) ≠ [xi](T) ≠ 0. A vector measure is an n-tuple μ = (μ1, ..., μn) of measures μι with n finite; if the μi are non-atomic then also μ is called non-atomic. The range of μ is the set μ(S): S [member of] C}; it is a subset of En. The following proposition will be used repeatedly throughout the book:

Proposition 1.3 (Lyapunov's Theorem). The range of a non-atomic vector measure is convex and compact.

Since the original proof of Lyapunov (1940), this theorem has been reproved many times. For an elementary proof, the reader is referred to Halmos (1948), and for a quick, though deeper proof, to Lindenstrauss (1966).

NOTES

1. If (I, C) is isomorphic to ([0, 1], (B), then it may be verified that a measure [xi] is non-atomic if and only if [xi]({s}) = 0 for all s in I.

2. Definitions of Game and Value

At the beginning of the section we shall concentrate on the formal basic definitions of "game" and "value," interpolating only a minimum of discussion and illustration. These basic definitions — which form a self-contained unit — will be slightly indented, in order to set them off from the discussion. In the second part of the section we will interpret and motivate the definitions from the game-theoretic point of view.

Let (I, C) be a measurable space; it will be fixed throughout and will be referred to as the underlying space. The term set function will always mean a real-valued function v on C such that v(Ø) = 0.

In most of this book (until Chapter VIII) we will make the following

(2.1) Standardness Assumption: The underlying space (I, C) is isomorphic to ([0, 1], (B).

Because of Proposition 1.1, this assumption is not as drastic as it seems. We add that for much of the material of this book, (2.1) is not needed; see Section 47.

In the interpretation, a set function is a game, I is the player space, and the members of C are coalitions. The number v(S), for S [member of] C, is interpreted as the total payoff that the coalition S, if it forms, can obtain for its members; it will be called the worth of S. This way of representing a game is an obvious generalization of the standard representation in "characteristic function" (or "coalitional") form of a game with finitely many players (cf. von Neumann and Morgenstern, 1953, or Appendix A, below). Sometimes the phrase "game with a continuum of players" is used to distinguish these games from the finite ones. It should be stressed, however, that our treatment is measure-theoretic; the player space is a measurable, not a...