Non-Uniform Integrability and Generalized Young Measures

Given a bounded sequence (uₙ) in L¹ (Ω, µ;$\mathbb{R}^d$), we describe the weak limits in the sense of measures of f(x, uₙ) µ for a class of continuous integrands with linear growth at infinity. The defect of uniform integrability of the sequence f(x, uₙ) is described by a measure m and a family of probability measures on $S^{d-1}$ whereas the classical Young measure is associated with the biting limits in the sense of Chacon’s lemma. Some consequences of this new approach are given in Calculus of Variations.