Past exam of the real and harmonic analysis course of the University of Oxford / 2008 / 3 / e / Solution

Read the coefficient inside the limit as $\widehat{{\mu }_k}(n)$; the printed omission of its index would otherwise make that condition independent of the sequence. Weak-star convergence immediately implies convergence of every coefficient by testing $e^{-inx}$. Conversely coefficient convergence gives convergence against every trigonometric polynomial. Since $\Vert {\mu }_k\Vert =1$, the dense-test argument yields weak-star convergence to $\mu$.

The limit is necessarily a probability. Testing the constant function gives $\mu (\mathbb{T})=1$. For every nonnegative continuous $f$, $\int fd\mu ={\lim }_k\int fd{\mu }_k\ge 0$. A Radon measure whose integrals against all such functions are nonnegative is positive, by the positive case of the Riesz representation theorem (or continuous approximation of Borel indicators). Thus $\mu$ is nonnegative and has total mass one.