Laplace transform of measure on complex plane

Question

I construct Laplace transform of measure on complex plane，i.e. $F(t)=\int_\mathbb{C}e^{-\lambda t}d\mu(\lambda), t \in \mathbb{R},\mu(\mathbb{C})=1$ and have bounded support. Suppose that $F_n(t)=\int_\mathbb{C}e^{-\lambda t}d\mu_n(\lambda)$ is well defined, and that $F_n$ converges to $F$ pointwise. I guess that $\mu_n$ converges weakly to $\mu$ and $F(t)=\int_\mathbb{C}e^{-\lambda t}d\mu(\lambda)$, but I don't know how to prove it.

Answer 1

It is not true, since Laplace transform on probability measures on $\mathbf{C}$ (with bounded support) is no more injective. For example, the uniform measure on the unit circle $\mathbf{S}_1$ and the Dirac mass at $0$ have the same Laplace transform since holomorphic like $z \mapsto e^{-tz}$ functions satisfy the mean value property. For every $t \in \mathbf{C}$, one has
$$\frac{1}{2\pi}\int_0^{2\pi} \exp(-te^{i\theta}) \mathrm{d}\theta = \exp(-t \times 0).$$