Markov inequalities on partially ordered spaces

Let X([omega]) be a random element taking values in a linear space 4 endowed with the partial order <=; let 10 be the class of nonnegative order-preserving functions on 4 such that, for each g[set membership, variant]10, E[g(X)] is defined; and let 11n10 be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X >= x) <= inf10E[g(X)]/g(x). Moreover, if E(X) = [xi] is defined and if Jensen's inequality applies, we have a further inequality P(X>=x) <= inf11E[g(X)]/g(x) <= inf11g([xi])/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability.