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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