Ramseys Theorem

Given a rational set of Preferences there exists a real valued function $U$ (Utility function) such that

U(A)\geq U(B)

iff $A\succeq B$ and

U\left(\left[p_1, S_1 ; \ldots ; p_n, S_n\right]\right)=\sum_i p_i U\left(S_i\right)

The utility of a lottery is the Expected Utility of all of the individual states with their probabilities.

This function is not unique. For example positive linear transformations will not change the function in a meaningful way.