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Let $Y$ denote a separable metric space, and $\mathcal{P}(Y)$ the set of probability (understood as countably additive) measures on the Borel $\sigma$-algebra of $Y$. The weak*-topology says that a sequence $\mu_{n}$ (or more generally a net) in $\mathcal{P}(Y)$ converges to $\mu$ when $\int fd\mu_{n}$ converges to $\int fd\mu$ for each bounded continuous mapping $f:Y \to \mathbb{R}$.

My question is: It is known that the set $\mathcal{P}(Y)$ is not necessarily closed when endowed with such a topology (see one example here where the limit is not countably additive). Is it possible to say more generally that whenever $Y$ is not finite (or perhaps "richer" in some sense) there exists a sequence in $\mathcal{P}(Y)$ weakly* converging to zero? Thanks!

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No, a sequence of probability measures cannot converge weakly-* to 0. Consider the bounded continuous function $f=1$. The weak-* limit must be a measure of total mass 1.

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