A Refinement of Ball's Theorem on Young Measures

摘要

For a sequence uj:Ω⊂ Rn→ Rmgenerating the Young measure νx, x∈Ω, Ball's Theorem asserts that a tightness condition preventing mass in the target from escaping to infinity implies that νx is a probability measure and thatf(uk)⇀νx,f in L1 provided the sequence is equiintegrable. Here we show that Ball's tightness condition is necessary for the conclusions to hold and that in fact all three, the tightnesscondition, the assertion ∥νx∥=1, and the convergenceconclusion, are equivalent. We give some simple applications of thisobservation.