Let $\Gamma$ and $\Psi$ be probability measures on $(\mathbb{R}, {\cal B}(\mathbb{R}))$ and construct a product probability space $(\mathbb{R}^2, {\cal B}(\mathbb{R^2}), \Gamma\otimes \Psi)$.
Consider the following two subspaces of $L^2(\mathbb{R}^2, {\cal B}(\mathbb{R^2}), \Gamma\otimes \Psi)$:
A:= \left\{f(x) + g(y): f,g \text{ Borel-m'ble with} \int(f(x) + g(y))^2 d\Gamma(x)d\Psi(y)<\infty \right\}. and $ B:= \left\{f(x) + g(y): f,g \text{ continuous and bounded} \right\}. $
My question: what is the closure of $A$ and $B$, respectively ?