As we know that weak limit of a sequence of Borel probability measures on metric space is unique. Do we have this property for general sequence of signed Borel measures on metric space? Thank you.
Uniqueness of Weak Limit
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analysis
measure-theory
limits
convergence-divergence
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0Yes, the measures are finite. – 2012-09-06
1 Answers
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We assume that $\{\mu_n\}$ converges weakly to $\nu_1$ and $\nu_2$. Taking $f=1$, we can see that $\nu_1(X)=\nu_2(X)$ and putting $\nu=\nu_1-\nu_2$, we have $\nu(X)=0\mbox{ and }\quad \forall f\in C_b(X), \int_Xf(x)d\nu(x)=0.$ Take $O$ an open subset of $X$. We can approach pointwise the characteristic function of $O$ by a sequence of continuous bounded functions, which gives, by dominated convergence and decomposition of $\nu$ that $\nu(O)=0$ for each open subset $O$. Since the open subsets forms a collections stable under finite intersections which generates Borel $\sigma$-algebra on $X$, we deduce that $\nu=0$, hence $\mu_1=\mu_2$ and the limit is unique.