On the lack of equi-measurability for certain sets of Lebesgue-measurable functions

Abstract

Abstract Let Ω be a Lebesgue-measurable set in ℝ n of finite positive Lebesgue measure. In this note we calculate the lack of equi-measurability of the set Kc (Ω), c > 0, of all Lebesgue-measurable functions f : Ω → ℝ such that 0 ≤ f ≤ c, a.e. on Ω. From our result we repair a gap in the Example 2.3 of the paper [APPELL, J.—DE PASCALE, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515].