Let $K$ be some compact set and $\mu$ be a probability measure on $K$. Let $f:K\to\mathbb S^{d-1}$ be a continuous function, where $\mathbb S^{d-1}$ is the unit sphere in $\mathbb R^d$ And $g:K\to\mathbb R^d$ be a measurable function. Assume that $f$ is injective.
Is it true that if, for all $y\in\mathbb R^d$, $$\int_K e^{y^{\top} f(x)}y^\top g(x) d\mu(x)=0,$$ Then $g(x)=0$ $\mu$-almost surely?
When $\mu$ is finitely supported, it is true and quite easy to show, but I am not sure about the general case... Thank you!