mu is a measure on the Borel sets of a separable, complete, metric
space X such that mu(X) = 1

Is it possible to choose a Borel set of a prescribed measure (e.g. For
any c in the unit interval there exists a Borel set E in X such that
mu(E) = c).  If so, please prove.

The question is related to a problem from Halmos' Measure Theory text,
chapter 2 sec 9 #10

Perhaps I've missed a critical part of the problem, but I think there
is a trivial counterexample.  Hint:  Pick point x in X.

regards, mathtalk-ga

Hello Mathtalk,

I appreciate your posting a comment on my question.
I'm not quite sure however, how your hint connects 
to a counterexample.  

In any case, I'm no longer convinced that this result is  
necessary to the problem, so I'm cancelling the question.

Regards
halmosreader1-ga