# Extension of measures from the ball sigma-algebra to the borel sigma-algebra

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 $\Sigma_{1}?$

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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.
Very nice answer. But let me add: The nonextension of the measure here amounts to the fact that $\aleph_1$ is not a real-valued measurable cardinal. This is a theorem of ZFC, but we should not expect to prove this without the axiom of choice. Indeed, it is consistent (relative to large cardinals) that one has ZF+DC plus $\aleph_1$ is a measurable cardinal (with a two-valued measure). In such a set-theoretic context, the measure actually would extend. In particular, this situation is a consequence of the axiom of determinacy, which implies that the club filter on $\omega_1$ is a measure. – Joel David Hamkins Feb 28 '13 at 23:07