Given a sequence of functions bounded in L1([0, 1]), is it possible to extract a subsequence that is pointwise bounded almost everywhere? The main objective of this note is to present an example showing that this is not possible in general. We will also prove a pair of positive results. We show that if the sequence of functions consists of multiples of characteristic functions of measurable sets, the answer is yes. We also show that it is always possible to extract a subsequence that is pointwise bounded on a countable, dense set of points. © 2001 Academic Press.2632447454Billingsley, P., (1986) "Probability and Measure", , 2nd ed., Wiley, New YorkDelort, J.-M., Existence de nappes de tourbillon en dimension deux (1991) J. Amer. Math. Soc, 4, pp. 553-586Falconer, K., (1990) "Fractal Geometry: Mathematical Foundations and Applications", , Wiley, ChichesterLopes Filho, M.C., Nussenzveig Lopes, H.J., Xin, Z., Existence of vortex sheets with reflection symmetry in two space dimensions (2001) Arch. Rat. Mech. Anal, 158, pp. 235-257. , IMECC/UNICAMP, submitted for publicationRoyden, H.L., (1968) "Real Analysis", , 2nd ed., Macmillan, New YorkRudin, W., (1974) "Real and Complex Analysis", , 2nd ed., Tata McGraw-Hill, New DelhiSchochet, S., The weak vorticity formulation of the 2D Euler equations and concentration-cancellation (1995) Comm. Partial Differential Equations, 20, pp. 1077-1104