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?