borel-finite and non-negative measure

Question

Let $\mu_y$ be a finite and non-negative measure on $\mathbb{R}$, and $\mu$ a non-decreasing measure on $\mathbb{R}$, such that for every fixed $y\in (a,b)$ with $(a,b)$ not containing the origin $$ \int_{-\infty}^{+\infty}d\mu_y(t)=\int_{-\infty}^{+\infty} e^{-yt}d\mu(t) $$ Under these assumptions is it possible to conclude that $\mu$ is borel-finite and non-negative? Thank you!