User:Boris Tsirelson/sandbox1

Every narrow neighborhood of a probability measure $\mu$ is also a wide neighborhood of $\mu$. Here is a proof. Given $\varepsilon$, we take a compactly supported continuous $f:X\to[0,1]$ such that $\int f \rd\mu > 1-\varepsilon$. Now, consider another probability measure $\nu$ widely close to $\mu$, namely, satisfying $|\int f \rd(\mu-\nu)| < \varepsilon$. If such $\nu$ satisfies $|\int fg \rd(\nu-\mu)|<\varepsilon$