random variable can be used instead of a probability function

XaY <-> aPx(A)+(1-a)Py(A) for A subset of X.

For X = all simple random variables on (a,b) 
Use the following axioms:
 * C1: (XaX)=X
 * C2: (XaY)=(Y(1-a)X)
 * C3: ((XaY)b(XgY))=X(ab+(1-b)g)Y

X a random variable lets you define an expected value of functions on X.