Assumptions of convolution theorem

Question

Given functions $f, g \in L^1 + L^2 (\mathbb R^d)$ such that the product $fg \in L^1(\mathbb R^d)$, is it true that their Fourier transforms satisfy $$ \mathcal F[fg] = \mathcal F[f] * \mathcal F[g]?$$ I tried to approximate by Schwartz functions then use a continuity argument, but I am having a hard time showing that $\mathcal F[f] * \mathcal F[g]$ is well-defined. I think we will need to use $fg \in L^1(\mathbb R^d)$ somehow.