Equipartitioning Common Domains of Non-Atomic Measures

Let h(a1 , •.. , an) denote the harmonic mean n(a1 1 + ... +a; 1) -1 of positive num­ bers a1 , ••. , all' and extend this definition (by continuity) to the non-negative case by setting h(a l' ... , an) = 0 if ai = 0 for some i ~ n. Recall that a measure J1. is non-atomic if every measurable set A of strictly positive measure has a measur­ able subset B ~ A satisfying J1.(A) > J1.(B) > O. A measurable partition of a measure space (X, gF) is a collection of disjoint subsets At, ... , An of X satisfying A;EgF n for all i, and UAi=X. Throughout this paper, "measure" means "countably i= 1 additive, non-negative measure", and 11J1.11 is the total mass (or total variation) of the measure J1.; the main purpose is to prove the following theorem.