Quote:
Originally Posted by mSed84
Let X be a partially ordered set and P and Q be probability measures on X. Suppose that for each principle upset U of X, we have that P(U)<=Q(U). Can you give an example where P is not stochastically smaller than Q?
I don't recall having seen these concepts of "upsets" and "stochastically smaller" before. Looking at Wiki I see "stochastically smaller" defined for random varibles. ie. If A and B are random variables then A is smaller than B means that for every real x, P(A > x) <= P(B > x). Conceptually, A has more probability weight on smaller real numbers relative to B.
However, in your question you don't have any random variables. You have probablity measures on X but no function from X to R defining random variables. So I assume you mean to have a function, say A, from X to R defining two random variables via P and Q, say Ap and Aq. And you then want Ap not smaller than Aq.
It seems to me that you would only expect Ap to be smaller than Aq if the partial ordering on X somehow agreed with the natural ording on R under the map A. So look for a counter example where the map A does not respect the ordering on X. How about letting X = R with the natural ordering, assume you have your P striclty smaller than Q, and let the map A:X --> R be A(x) = -x.
Or assume you have a Q and P on R with Q strictly smaller than P, then let X = R with reverse ordering, and A the identity map.
Or am I missing something?
PairTheBoard