P(X,Y), when integrated along all possible diagonals given by x+y=z where z goes through all real numbers, is what a convolution is (of P(X) and P(Y) generated

P(X,Y), when integrated along all possible diagonals given by x+y=z where z goes through all real numbers, is what a convolution is (of P(X) and P(Y) generated by those variables). This gives you the distribution of the variable Z=X+Y. So, we're talking about the same thing, namely that multiplying two probability density functions pointwise roughly corresponds to a single point where X==Y (the same as X-Y==0), but otherwise doesn't seem terribly useful.