Suppose $\varphi_{n}$ is a sequence of characteristic functions converging pointwise to a function $\varphi$ which is continuous and absolutely integrable. I want to show that $\varphi$ is the Fourier Transform of a probability density. I know that convergence of $\varphi_{n}$ pointwise implies weak* convergence of a sequence of corresponding random-variables to the RV whose chf is $\varphi$ but I'm not sure if that helps.
Help would be appreciated!