Function of trivariate random variable

Question

Let $P(X_1,X_2,Y)$ be a discrete trivariate probability distribution and let $B = f(X_1, X_2, Y)$ be a function that maps $X_1$, $X_2$, and $Y$ into a discrete set of symbols $B$.

I would like to know whether it is always licit to assume that a distribution $P(X_1, X_2, B, Y)$ exists such that $$ P(X_1, X_2, B) = \sum_b p(x_1, x_2, b, y) $$

Answer 1

Actually yes. From $p(b|x_1,x_2,y) = 1$ we have $$ p(x_1,x_2,b,y) = p(b|x_1,x_2,y) \cdot p(x_1,x_2,y) = p(x_1,x_2,y) $$