STOCHASTIC

S. L. BRUMELLE AND R. G. VICKSON
By hypothesis, v ^ h on F. Thus, by the Hahn-Banach theorem, v can be
extended to a linear functional v on E, with v ^ E on E. Clearly, the functional
v may be written as v(fuf2, ...,/„) = Z?=i A^O/i). Now v g h on E =>
v;(/) ^ /;,•(/) for all /E C. Since v = v on F one has v(/) = £*=1 p;Vf(/).
Returning to the proof of Theorem 3.1, define for each /e C the upper
envelope /:
The following properties are evident.
(1) /g/^sup{/(x)|xe/'}.
(2) /e^ for all/e C.
(3) (f+g)^f+g (f.geC).
(4) (c/) = c-/ (/eC;c£0).
f(x) = M{g(x)\ge^,g^f}.
The upper envelope operation thus produces a sublinear functional
ht(f) =f(Xj) for each *; e CI. Define
/>(/)= EM(/) = ipifixdi=l
i=l
From properties (1) and (2) of the upper envelope, and from hypothesis (a)
of Theorem 3.1, it follows that
v(f) ^ v(/) ^ n(J) = £ ^./(x,.) = /,(/)
i=l
for all/e C. By Lemma 1, v = £*=1 p;V;, with v,^ h;. One has
(1') V; is a nonnegative linear functional. For/^ 0 =>/^= 0 => v;(/) ^
A, (/)=/(* v,(/) = 0, U i ^ .
i=l i=l
Define rfs, = Vi({xj}) = P[Yt = *,•], where Yt is the random variable corresponding
to V;. Then
110 PART II QUALITATIVE ECONOMIC RESULTS