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Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have $$\int\limits_{X}f\,d\mu_n\to\int\limits_Xf\,d\mu.$$

My question is why this convergence is referred to as a weak one, but not weak*?

In fact, $\mu_n,\mu \in C_b(X)^{*}$, and weak* is exactly what we need! However, the space $C_b(X)$, consisting of all bounded continuous real functions, is usually non-reflexive (I cannot formulate the precise statement immediately, but it is widely known that $C[0,1]=C_b[0,1]$, for example, is not reflexive).

Are there any historical reasons for such kind of confusion?

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  • $\begingroup$ From Wikipedia: en.wikipedia.org/wiki/… "In mathematics and statistics, weak convergence (also known as narrow convergence or weak-* convergence, which is a more appropriate name from the point of view of functional analysis, but less frequently used) is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion." $\endgroup$ – DisintegratingByParts Apr 19 '16 at 18:54
  • $\begingroup$ I would bet that this is due to historic reasons. Maybe the notion of "weak convergence of measures" is older than the weak-* topology on duals of Banach spaces (this was introduced maybe around 1940). $\endgroup$ – gerw Apr 20 '16 at 8:35

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