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Probability theory can get pretty weird in the non-separable case. For example, write $B$ for the non-separable Banach space of sequences $a = (a_k)_{k \ge 1}$ such that $$ |a|^2 := \sup_{n \ge 0} 2^{-n} \sum_{k=2^n}^{2^{n+1}} |a_k|^2 < \infty\;. $$ Take now an i.i.d. sequence of Gaussian variables $\xi = (\xi_k)_{k \ge 1}$. Then, one has $|\xi| < \infty$ almost surely but, for any fixed $a \in B$, one also has $P(|\xi-a| \ge 1) = 1$, so the notion of "support" becomes pretty problematic...