An Abstract Regularity Lemma

We extend in a natural way Szemerédi’s Regularity Lemma to abstract measure spaces. 1 Introduction In this note we extend Szemerédi’s Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to …nd general conditions under which the original proof of Szemerédi still works. Another extension of SRL to probality spaces was proved by Tao [3], but his results do not imply our most general result, Theorem 13. To illustrate that our approach has some merit, we outline several applications. Some of these applications seem to be tailored to our approach: in particular, we are not aware of any alternative proofs. Our notation follows [1]. 1.1 Measure triples A …nitely additive measure triple or, brie‡y, a measure triple (X;A; ) consists of a set X; an algebra A 2 , and a complete, nonnegative, …nitely additive measure on A with (X) = 1. Thus, A contains X and is closed under …nite intersections, unions and di¤erences; the elements of A are called measurable subsets of X: Here are some examples of measure triples. Example 1 Let k; n 1; write 2[n]k for the power set of [n] ; and de…ne k by k (A) = jAj =n for every A [n]. Then [n] ; 2 k ; k is a measure triple. Note that there is a one-to-one mapping between undirected k-graphs on the vertex set [n] and subsets G 2[n]k such that if (v1; : : : ; vk) 2 G; then fv1; : : : ; vkg is a set of size k and G contains every permutation of (v1; : : : ; vk). In view of this, we shall consider subsets of 2 k as labelled directed k-graphs (with loops) on the vertex set [n] :