# Is the limit of a sequence of characteristic functions of probability measures a characteristic function of a measure?

Given a sequence of probability measure $$P_n$$ on $$\mathbb R$$ with the Boreal sigma algebra, define $$f_n = \int e^{ixt}dp_n$$, and suppose that $$f_n \to f$$ for some $$f$$ bounded measurable, is it true that there exists some $$p$$ such that $$f = \int e^{ixt}dp$$?

My intuition tells me that this is false, but I cannot come up with an example. The example that I tried: Let $$p_n$$ be the Gaussian with mean $$0$$, variance $$\frac{1}{n^2}$$, then the characteristic function converges to the constant $$1$$. However, $$1$$ is the characteristic function of some dirac delta function.

Instead of having Gaussian with decreasing variance, let $$p_n$$ be Gaussian with mean 0 and variance $$n$$. This gives the counterexample you are looking for. In general the continuity theorem states that if the limiting function $$f$$ is continuous, then there exists such a $$p$$, but not otherwise.