A Refinement of Ball’s Theorem on Young Measures

For a sequence uj : Ω ⊂ Rn → Rm generating 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 that f(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 tightness condition, the assertion ‖νx‖ = 1, and the convergence conclusion, are equivalent. We give some simple applications of this observation.