Given the complex plane $\mathbb{C}$.
Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\operatorname{supp}\mu\subseteq\overline{B_r}$$
Then one has: $$\int\lambda^k\,\mathrm{d}\mu(\lambda)=0\quad(k\in\mathbb{N}_0)\implies\mu=0$$
How can I prove this?
This is not true. Cauchy's Theorem provides abundant counterexamples.
In detail: Suppose $\gamma$ is a smooth closed curve in the plane. Cauchy's Theorem says that if $f$ is entire then $\int_\gamma f(z)\,dz=0$. But it's clear that there exists a complex measure $\mu$ such that $$\int f\,d\mu=\int_\gamma f(z)\,dz.\quad(*)$$
(Readers to whom the existence of $\mu$ is not clear are advised to contemplate the Riesz Representation Theorem, describing the dual of $C(K)$.)