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Let $(\mu_n)_{n\in\mathbb N}$ be a sequence of probability measures on $\mathbb R$ with the usual Borel $\sigma$-algebra $\mathcal B(\mathbb R^p)$. That is, $(\mu_n)_{n\in\mathbb N}$ can be considered as a sequence in the Banach space $\mathcal M(\mathbb R^p)$ of all finite signed Borel measures on $\mathbb R^p$ equipped with the total variation norm $\Vert\cdot\Vert_{\text{TV}}$.

In probability theory, the sequence $(\mu_n)_{n\in\mathbb N}$ is called weakly convergent to some probability measure $\mu$ on $\mathcal B(\mathbb R^p)$ iff $$\left\vert\int f\,\mathrm d\mu_n - \int f\,\mathrm d\mu\right\vert$$ converges to zero for every $f\in\mathcal C_b(\mathbb R^p)$, where $\mathcal C_b(\mathbb R^p)$ denote the space of all real-valued bounded, continuous functions on $\mathbb R^p$.

In functional analysis, a sequence $(\phi_n)_{n\in\mathbb N}$ in $(\mathcal C(\mathbb R^p))^*$ is called weak*ly convergent to some $\phi\in(\mathcal C(\mathbb R^p))^*$ iff $$\left\vert\phi_n(f) - \phi(f)\right\vert$$ converges to zero for each $f\in\mathcal C(\mathbb R^p)$. It is known that $(\mathcal C_b(\mathbb R^p))^*\simeq \mathcal M(\mathbb R^p)$ (c.f. the dual space of C(X) (X is noncompact metric space)). So for each $\nu\in\mathcal M(\mathbb R^p)$ there is a $\varphi\in(\mathcal C_b(\mathbb R^p))^*$ such that $$\varphi(f) = \int f\,\mathrm d\nu$$ for $f\in\mathcal C_b(\mathbb R^p)$.

So the sequence $(\mu_n)_{n\in\mathbb N}$ is weak*ly convergent to some some $\mu\in\mathcal M(\mathbb R^p)$ iff $$\left\vert\int f\,\mathrm d\mu_n - \int f\,\mathrm d\mu\right\vert$$ converges to zero for every $f\in\mathcal C_b(\mathbb R^p)$, and the two definitions coincide.

Note: My original question had an error (I mixed up the duals) which caused confusion. In fact, after resolving the error, the question has been answered.

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    $\begingroup$ You do not describe the weak$^*$-convergence on the spaces of signed measures but the weak convergence. The weak topology is strictly finer than the weak$^*$-topology. $\endgroup$ Commented Oct 8, 2023 at 8:52
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    $\begingroup$ What is $\int \phi d\mu$? Is it $(\phi,\mu)$ in the functional analysis sense? $\endgroup$ Commented Oct 8, 2023 at 9:45
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    $\begingroup$ You got the duals and preduals mixed up: it's really $C(K)^*=\mathcal M$, not the other way around. $\endgroup$ Commented Oct 8, 2023 at 18:07
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    $\begingroup$ And yes, on a compact space, the probabilist's weak convergence is exactly the functional analyst's weak $*$ convergence. $\endgroup$ Commented Oct 8, 2023 at 18:08
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    $\begingroup$ Thank you for the replies. I am very confused now: we are looking at sequences in $\mathcal M$. The dual of $\mathcal M$ is $\mathcal C_b(\mathcal K)$, no? The comment of @ChristianRemling is suggesting that there is a mistake, but what's then the dual of $\mathcal M$? Or did I completely mess up duals and pre-duals? $\endgroup$ Commented Oct 9, 2023 at 2:56

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This is just a comment (but I am not entitled) which might help to illuminate the situation. In the case of a compact space $K$, all is clear--$C(K)$ is a Banach space, its dual is the space of bounded, Radon measures and the weak $\ast $ topology in the functional analytic sense coincides with the standard notion for convergence of measure (and both can be restricted to the probability measures of course). The situation for the non compact case has also been studied in detail, both for a locally compact space $S$ and, more generally, completely regular spaces. The natural replacement for $C(K)$, at first sight, is the Banach space $C^b(S)$ but this was soon recognised to be inadequate (it doesn´t distinguish between $S$ and its Stone-Cech compactification and its dual is too large since it contains measures on $S$ which are only finitely additive).

It was soon realised that this situation could be remedied, if one was prepared to leave the comfort zone of Banach spaces and use more esoteric tools of locally convex theory. One of the most prominent examples is a paper of R.C. Buck (1955) who introduced the so-called strict topology in the case of locally compact $S$. This was defined using weighted seminorms and is a complete lc structure on $C^b(S)$ for which the dual space is precisely the space of bounded Radon measures. This was soon extended to general completely regular spaces where the strict topology is most succinctly described as the finest lc topology which agrees with compact convergence on the unit ball. This allows a natural extension to the non-compact case which shows how the measure theoretical and functional analytic approaches coincide just as in the compact one.

Added after OP´s comment on duality. Thus we have the following situation: a locally convex space $C^b(S)$ of functions on a topological space whose dual is identifiable with a space of measures, the bounded Radon measures. On the latter we have two kinds of convergence, weak convergence (in a dual space) and the probabilist´s concept of convergence and these coincide.

In fact, the above duality is perfectly symmetric--the space of measures is, as described above, the dual of the lc space $C^b(S)$ with the strict topology and $C^b(S)$ can also be regarded as the dual of the space of measures. However, for this to hold we have to use the right choice of structures, in the compact case the symmetric duality between Banach and Waelbroeck spaces, in the non compact one between Saks and CoSaks spaces. These are too complex to describe here but they are discussed in detail in the monographs "Functors and Categories of Banach spaces" and "Saks Spaces and Applications to Functional Analysis".

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