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I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.

Does the above result generalize to product of subsets with the RNP? If $C\subset X$ is a (nonempty) bounded, closed, convex subset, then is it true that $$ C\times Y $$ also has the RNP, provided $Y$ has the RNP?

More precisely, I am interested in the product of the form $C\times \Bbb R$, where $C$ is as above. Any result that includes this special case would suffice for my purpose.

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In Section 2 of Bourgin's book "Geometric aspects of convex sets with the Radon-Nikodym Property," it is shown that for a closed, convex set non-empty set $K$, having RNP is equivalent to having the martingale convergence property. Assume $C_1\subset X_1$, $C_2\subset X_2$ are closed, convex with RNP. Any martingale $(f_n)_{n=1}^\infty$ taking values in a closed, bounded, convex subset of $C_1\times C_2$ can be written as $f_n=(g_n, h_n)$, where $g_n$ takes values in a closed, bounded, convex subset of $C_1$ and $h_n$ takes values in a closed, bounded, convex subset of $C_2$. Then since $C_1, C_2$ have RNP, they have the martingale convergence property, and $g_n$ and $h_n$ are a.s. convergent to some limits $g,h$ in $L_1(X_1)$ and $L_1(X_2)$, respectively, Then $f_n=(g_n, h_n)$ converges a.s. to $(g,h)\in L_1(X_1\times X_2)$. This shows that $C_1\times C_2$ has the martingale convergence property.

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  • $\begingroup$ Seems like an interesting approach, thank you. By the way, do you happen to know any reference that state this kind of theorem/proposition explicitly? All I managed to find are some passing remarks and statements like "it is well-known that..." without any actual source to the claim. $\endgroup$
    – BigbearZzz
    Commented Jul 22, 2019 at 23:29
  • $\begingroup$ I'm not aware of an explicit statement about products. I didn't look exhaustively through Bourgin's book, so something like this could possibly be stated there. As your question reflects, a great deal of attention is paid to spaces with RNP, and not to sets with RNP. Bourgin's book is the only reference I know which deals with the latter. $\endgroup$
    – user-1
    Commented Jul 23, 2019 at 10:44

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