Hey guys, I got this as a homework problem. The homework isn't due for a while, so I thought I'd post it here while I think about it on my own. Shouldn't be too hard for any of you analyst/physicist types out there.

First, definitions used:

Call u* an outer measure on the space X if
1) u*: P(X) -> [0,infty], where P denotes power set
2) u*(empty set) = 0
3) Sigma-subadditivity, so
u*(union of countable A_i) <= sum u*(A_i)
where the last is an inequality even for disjoint unions.

If (X,d) is then a metric space, u* is a metric outer measure if it is an outer measure satisfying
4) If the distance between two sets is d(A,B) = inf{d(a,b): a in A, b in B}, and d(A,B)>0, then u*(A union B) = u*(A)+u*(B)
i.e. the outer measure is finitely additive on separated sets.

A set A is measurable (u*-measurable) if for all subsets E of X,
u*(E) = u*(EA) + u*(E intersect A)
i.e. A splits every subset nicely with respect to the measure.

Now, for the statement of the problem.
Prove that if u* is an outer measure and every open set (with respect to the metric) is measurable, then u* is a metric outer measure. That is, if every open set splits every subset nicely, then the measure itself must split separated sets nicely.

Have fun!

Edit: Ah screw it, I got it already. It was easier than I thought.