We know that under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this statement to linear transformations if we are to keep the same preference representation.
Now the problem is that I don't know how to give to this concept a mathematical and a economic explanation. I know that Von Neumann - Morgenstern utility function is used in these cases, but what this means? Can anybody help me, maybe give me an exhausting and understandable reference? Thanks in advance!