Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and $\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be extended to a measure on $\Sigma_{1}?$
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and $\Sigma_{2}$ the sigma algebra generated by balls (open and closed). If $\mu$ is a probability measure on $\Sigma_{2}$ can it be extended to a measure on $\Sigma_{1}?$ 


Take a set $X$ of power $\aleph_1$, with the discrete metric where two distinct points have distance $1$. The balls are singletons and the whole space. The ball sigmaalgebra is the countable and cocountable sets. Let countable sets have measure zero, cocountable sets have measure 1. Now all subsets are open, so the Borel sigmaalgebra is the power set. There is no extension of this measure! 

