This is solely a reference request. I have heard a few versions of the following theorem:
If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.
And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems?