Sets on which several measures agree

It is known that, given n non-atomic probability measures on the space I = 0, 1], and a number α between 0 and 1, there exists a subset K of I that has measure α in each measure. It is proved here that K may be chosen to be a union of at most n intervals. If the underlying space is the circle S¹ 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 α. However, there remain many rational values of α for which we are unable to determine the minimum number of intervals that will suffice.