STOCHASTIC

A UNIFIED APPROACH TO STOCHASTIC DOMINANCE
(i) h(f)eR (heH.feC);
(ii) h(f+g)^h(f) + h(g) (heH;f,geC);
(iii) h(cf) = ch(f) (heH;feC;c^0).
Note that a sublinear functional is simply a positively homogeneous convex
function on the space C. The relevance of sublinear functional is contained
in the following very important theorem.
Hahn-Banach Theorem Let E be a linear space, F a linear subspace of E.
Let / be a linear functional on F, h a sublinear functional on E. If /(/) ^ h(f)
for all/e F, then / can be extended to a linear functional 7 defined on all of
E, such that 1(f) = /(/) for/e F, and 7(/) g h(f) for all/e £.
The proof of this theorem is not technically difficult, but uses somewhat
abstract arguments involving the axiom of choice. The proof is omitted since
it can be found in most books on analysis (e.g., Munroe [9]).
III.2 THE DISCRETE CASE
To prove Theorem 3.1, it suffices to prove the implication (a) => (b),
since the converse (b) => (a) follows trivially from Jensen's inequality. Let
X and Y be discrete random variables concentrated at the points fi =
{xux2, •••,xk}