Let $\mu$ be a probability measure on the matrix Lie group $G\le GL_n(\mathbb R)$ whose Lie algebra is $\mathfrak{g}$ (should be a subspace of $M_n(\mathbb R)$). Let $B_\eta$ denote the ball in $\mathfrak{g}$ centered at $0$ of radius $\eta$. Now suppose for any $\eta>0$, there exists a nonzero $\omega\in B_\eta$ such that $\mu$ is invariant under $\exp(\omega)$, then I speculate that there exists a $\omega_0\ne 0$ such that $\mu$ is invariant under the one-parameter subgroup $\exp(\mathbb R \omega_0)$.
It is clear from the assumtion that we can have $\mu$ invariant for some $\exp(\mathbb Z \omega)$. I think by an approximation/passing-to-a -subsequence argument we can find a $\omega_0$ as desired but I don't know how to make the argument rigorous. Thanks in advance!