 Solution

The random variables are independent random variables if the events determined by one are independent of the events determined by the other. For integer-valued variables, this is equivalent to the factorizationIndeed, independence gives the factorization by taking singleton events. Conversely, summing it over $x\in A$ and $y\in B$, for any subsets $A,B\subseteq \mathbb{Z}$, gives $\mathbb{P}(X\in A,Y\in B)=\mathbb{P}(X\in A)\mathbb{P}(Y\in B)$. Thus the singleton condition implies independence of all events in question.