I have a Borel non-decreasing measure $\mu$ such that
$$ \int_{-\infty}^{+\infty}\left|\sum_{i=1}^n \xi_i e^{-y_i t}\right|^2 d\mu(t)\geq 0 $$ and finite for every $n\in\mathbb{N}$, every $\{\xi_i\}_{i=1\ldots n}$ complex sequence and every $\{y_i\}_{i=1\ldots n}$ real sequence different from (0,...0).
I can conclude that $\mu(\mathbb{R})>0$, but can I say that $\mu(\mathbb{R})<+\infty$? I would like to have finite $\mu$ in order to prove that $\mu$ is Borel-finite and non-negative.