Journal of Convex Analysis Volume 4 (1997), No. 1, 129–147 Non-Uniform Integrability and Generalized Young Measures J.J. Alibert, G. Bouchitt´ e Laboratoire ANLA, UFR Science, Universit´ e de Toulon et du Var, BP 132, 83957 La Garde Cedex, France. Received September 6, 1995 Revised manuscript received February 27, 1996 Given a bounded sequence (u n ) in L 1 (Ω,μ; IR d ), we describe the weak limits in the sense of measures of f (x, u n ) μ for a class of continuous integrands with linear growth at infinity. The defect of uniform integrability of the sequence f (x, u n ) 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. 1. Introduction The oscillatory properties of a weakly convergent sequence (u n ) of functions in L 1 (Ω, µ, IR d ) can be very well described by the parametrized measure (or Young measure) it generates. This parametrized measure is a family of probabilities (ν x ) x∈Ω on IR d such that: f (x, u n (x)) ⇀  IR d f (x, z)ν x (dz) for every Caratheodory function f :Ω × IR d → IR such that f (x, u n (x)) is weakly con- vergent in L 1 (see [1,2,3]). In practice this representation formula is applied in case (u n ) is bounded in some L p and f satisfies a suitable growth condition: |f (x, z)|≤ C (1 + |z| q ) (1 ≤ q<p). In many situations f (x, u n (x)) represents a density of energy and is bounded in L 1 (Ω) but non uniformly integrable (problems in Plasticity and fracture mechanics). A way to overcome this difficulty already used in Calculus of Variations was to replace the weak convergence in L 1 by the convergence in the biting sense introduced by Gaposhkin and Chacon (see for example [4]). In this case, one can construct also a parametrized measure denoted “biting Young measure” by Kinderlehrer and Pedregal [5] (see also [2]). In fact in this approach, we have no information on the singular part of the weak limit of energies f (x, u n (x)) µ in the sense of Radon measures. Note also that the regular part of this weak limit do not coincide in general with the biting limit (see Ball and Murat [3] or Example 3.2 in this paper). An important precursor to the present paper is the work by Di Perna and Majda [6] where an extension of the concept of Young measure was introduced in ISSN 0944-6532 / $ 2.50 c  Heldermann Verlag