Ramseys Theorem

Given a rational set of Preferences there exists a real valued function $U$ (Utility function) such that $U(A)\ge U(B)$ iff $A⪰B$ and $U\left(\left[p_1,S_1;\dots ;p_n,S_n\right]\right)={\sum }_i{p}_{i}U\left(S_i\right)$ 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.