If $X$ is a metric space and $0<\mu(X)$, where $\mu$ is a radon measure and $\mu(\{x\})=0$. Can we always split $X = X_1 \sqcup X_2$ into two disjoint sets where $\mu(X_1) = a$, for any $0?
Or more generally does there always exist a set $X_1$ such that $\mu(X_1) = a$