every pdf can be regarded as a marginal distribution of a joint pdf suppose we have functions $g\ge0,h\ge0$ that $\int g \, dx=1 , \int h \, dy = 1$. it means $g$ is pdf for random variable $X$ and $h$ is pdf for $Y$.

now how we can prove that there is function $f(x,y)$ that $f$ is joint pdf of $X,Y$ and its marginals are $g,h$?
i know that $f$ is not unique in general,but how can i say there is one $f$?

thanks
Assume $X,Y$ are independent, then what is the joint density and marginals?
Define $F(x,y)=g(x|y)*h(y)$  where we know both functions $g,h$ exist. Then $F(x,y) = f(x,y)$ by Bayes Rule/