I have a marginal probability mass distribution $P_X(x)$ and a density function $f_Y(y)$. The probability mass function is given by the following integral:
$$ P_X(x) = \int_y g(x,y) f_Y(y) dy, $$
for some non-negative function $g(x,y)$. Does this mean that $g(x,y)$ is a conditional distribution function $P_{X|Y}(x | y)$, such that $\sum_X P_{X|Y}(x | y) = 1$? In other words, if a integral is a probability mass function and the integrand decomposes into a density and some other function, is the other function automatically a proper conditional probability mass function?