STOCHASTIC

S. L. BRUMELLE AND R. G. VICKSON
Theorem 3.1 was established for a compact interval ft = [a, b] c R 1 by
Blackwell [2] 1 and for discrete, finite multidimensional A'and Fby Sherman
[12]. The technically complex algebraic arguments of the original proofs were
later supplanted by technically simpler, but somewhat "abstract" functional
analysis arguments [8, 13]. The present treatment is a compromise between
the two approaches. The "modern" proofs are followed for the discrete case,
and standard limiting arguments are then used to obtain the general case.
III.l SOME NOTIONS FROM FUNCTIONAL ANALYSIS
Recall that the set C of continuous functions on fi is a vector space over the
real numbers R:
(i) f,geC => f+geC;
(ii) fe C, c e R => cfe C.
The vector space C becomes a separable metric space when endowed with the
sup norm topology, defined by the metric
d(f,g) = swp\f(x)-g(x)\ = max|/(x) - g(x)\.
jtefi IES!
Thus C has many of the properties of ordinary Euclidean space R m , except
that it has infinite dimension. A linear functional on C is a real-valued linear
function on C, that is, a linear map from the functions fe C into the real
numbers. A random variable X may be considered to be a nonnegative linear
functional on C. For if n is the probability measure corresponding to X
\ji(A) = \A dFx(xy\, the functional n(f) = Ef(X) (fe C) has the following
obvious properties:
(i) M/)eR (feC);
(ii) n(f+g) = n(f)+ii(g) (f,geC);
(hi) »(cf) = c-n(f) (feQceR);
(iv) /z(/)^0 (feCJ^O);
(v) n(e) = 1, where e e C is the unit constant function (e(x) = 1 for all
xeQ).
The converse result is also true: Any linear functional fi on C which satisfies
(i)-(v) corresponds to a random variable X such that n(f) = Ef(X) [3,
Chapter IV, p. 120].
Another important class of functionals on C is the class H of sublinear
functionals he H, satisfying
1 For the case EX = EY, this result was recently rediscovered by Rothschild and Stiglitz
[10], and was applied to the qualitative analysis ef several economic problems in a companion
paper [11].
108 PART II QUALITATIVE ECONOMIC RESULTS