We prove theorems of the following form: if A ⊆ R is a “big set”, then there exists a “big set” P ⊆ R and a perfect set Q ⊆ R such that P × Q ⊆ A. We discuss cases where “big set” means: set of positive Lebesgue measure, set of full Lebesgue measure, Baire measurable set of second Baire category and comeagre set. In the first case (set of positive measure) we obtain the theorem due to Eggleston...

Introduction. An interesting but apparently neglected generalization due to Doob [l], of the ergodic theorem, is the generalization to transformations which are not necessarily one-to-one. In his paper [l], Doob showed that the transformation of the interval [0, l] into itself given by/(x) =nx (modulo 1), where n — 1 is a positive integer, is ergodic under Lebesgue measure and that the law of l...

We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive 3-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations.

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón’s and Zygmund’s theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

There are several generalizations of the space L1(R) of Lebesgue integrable functions taking values in the real numbers R (and defined on the usual Lebesgue measure space (Ω,Σ, μ) on [0, 1] ) to a space of strongly-measurable “integrable” (suitably formulated) functions taking values in a Banach space X. The most common generalization is the space L1(X) of Bochner-Lebesgue integrable functions....