# The closure of a sum space in $L^2$

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 ?

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How do you check that $A$ is a subspace? –  Martin Argerami Mar 29 '12 at 12:55
I think that Minkowski's inequality shows that $A$ is linear. Am I correct? –  Yamamoto Mar 29 '12 at 13:35
Yes, actually it is even simpler as you can use simply the number inequality $(a+b)^2\leq2(a^2+b^2)$. –  Martin Argerami Mar 29 '12 at 13:44
Am I correct in saying that this subspace $A$ is convex? That could be helpful when considering the closure. –  Jeremy Voltz Mar 29 '12 at 13:57
A is closed, and the closure of B is A. Reduce to everything having mean $0$, which is easy. With the product measure X, Y are independent so if you have a cauchy seq $f_n + g_n(Y), \mathbb E(f_n(X) + g_n(Y)-(f_m(X) + g_m(Y))^2 =\mathbb E(f_n(X) -f_m(X))^2 + \mathbb E( g_n(Y)- g_m(Y))^2. f_n, g_n$ are also cauchy etc. –  mike May 3 '12 at 19:12