Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.