Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a joint PDF, i.e., $\|f\|_{L^1}=1$, which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$.
What is a necessary and sufficient condition under which the marginal $$ f^y (y)\equiv\int\limits_{\mathbb{R}} f(x,y)\, dx $$ exists for all $y\in \mathbb{R}$?
Known facts: because of Fubini, I'm pretty sure $f^y$ exists almost-everywhere in $y$. However, I don't think continuity or smoothness of $f$ guarantee existence everywhere.
On a compact domain (e.g., $f:[0,1]^2 \to \mathbb{R}$), continuity of $f$ is sufficient to guarantee existence of the marginals everywhere.