I am trying to prove the following statement. Let $F$ and $G$ be two finite measures on $((-\pi,\pi],\mathcal{B}((-\pi,\pi]))$ such that $$\int_{(-\pi,\pi]}e^{ih\lambda}\,dF(\lambda)=\int_{(-\pi,\pi]}e^{ih\lambda}\,dG(\lambda) \quad \text{for every } h \in \mathbb{Z}$$ Then, $F = G$.
The only trick that I know in showing two measures to be equal is the so-called $\pi$-system argument. I don't see how it applies here. The hint in the question uses heavy functional-analytic arguments and those are a bit beyond my knowledge.
Can someone give a not necessarily complete but a tutorial-like proof of this fact?
The objects on the right- and left-hand sides are simply characteristic functions of random variables with distributions $F$ and $G$ respectively. Since the characteristic function uniquely defines the distribution (e.g. Fourier transform is invertible), distributions of these variables coincide.