TITLE: Extension of measures from the ball sigma-algebra to the borel sigma-algebra
QUESTION [8 upvotes]: 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 $\Sigma_{1}?$

REPLY [11 votes]: 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!