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 $\Sigma\_{1}?$



Extension of measures from the ball sigma-algebra to the borel sigma-algebra
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 sigma-algebra is the countable and co-countable sets. Let countable sets have measure zero, co-countable sets have measure 1. 


Now all subsets are open, so the Borel sigma-algebra is the power set. There is no extension of this measure!


