An Exact Fatou's Lemma for Gelfand Integrals by Means of Young Measure Theory

We show that an exact version of Fatou's lemma for Gelfand integrable functions can be obtained by combining Young measure techniques and results due to E. J. Balder [New fundamentals of Young measure convergence, in: S. Reich, A. Ioffe and I. Shafrir (eds.), Calculus of Variations and Optimal Control, Chapman and Hall 2000, 24–48; and A Fatou lemma for Gelfand integrals by means of Young measure theory, Positivity 6 (2002) 317–329] with a purification result of M. Greinecker and K. Podczeck [Purification and roulette wheels, Economic Theory 58 (2015) 255–272].