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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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    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.2012-05-03

0 Answers 0