Consider a fixed density function $f$ on $\mathbb{R}$, and for each $y$, define $\mu_y$ as the probability measure over $\mathbb{R}$ induced by the random variable $y+\epsilon$, where $\epsilon$ is drawn from the density $f$. In other words, $\mu_y(A)=\int_A f(x-y)dx$. If necessary, $f$ can be assumed to be continuous or continuously differentiable.
Let $g(x)$ be any bounded, measurable function. Is it true that the function $y\mapsto \int_{-\infty}^{\infty} g(x)d\mu_y(x)$ is uniformly continuous in $y$?
