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For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?

More precisely, do we have equality for separable spaces? and for separable Hilbert spaces ?

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    $\begingroup$ In a separable normed space, every open set is a countable union of closed balls, so the closed balls and hence the closed convex sets will generate the Borel $\sigma$-algebra. $\endgroup$
    – Michael Greinecker
    Commented Apr 19, 2017 at 13:15
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    $\begingroup$ You should get the same for any separable locally convex space induced by a countable family of seminorms. I wonder whether it is true for $L^p$, $p < 1$. $\endgroup$
    – Nate Eldredge
    Commented Apr 19, 2017 at 17:35

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