Sets on which several measures agree (Q1088802)

It is known that, given n non-atomic probability measures on the space \(I=[0,1],\) and a number \(\alpha\) between 0 and 1, there exists a subset K of I that has measure \(\alpha\) in each measure. It is proved here the K may be chosen to be a union of at most n intervals. If the underlying space is the circle \(S^ 1\) instead of I, then K may be chosen to be a union of at most \(n-1\) intervals. These results are shown to be best possible for all irrational and many rational values of \(\alpha\). However, there remain many rational values of \(\alpha\) for which we are unable to determine the minimum number of intervals that will suffice.