Characteristic function criterion for weak convergence

If ${\mu }_n\Rightarrow \mu$, the forward continuity theorem gives the right-hand side at every frequency, including zero, where all values are one.

Conversely suppose the right-hand side. The specified target $\mu$ has a characteristic function continuous at zero. Apply the converse theorem with $\psi ={\phi }_{\mu }$: it gives a unique law $\nu$ with this characteristic function and ${\mu }_n\Rightarrow \nu$. The law $\mu$ itself satisfies the characterizing property, so that uniqueness gives $\nu =\mu$. AC is inherited through the converse theorem's Prokhorov, Fourier-uniqueness and integration-bridge uses. Point masses and constant sequences are included, and no nonzero variance, moment, or density is required.