To establish that the Lebesgue outer measure of any open set in the Euclidean metric space of is equal to the volume of any partitioning of that set into almost disjoint boxes, lemma 1.2.11 in p. 24 of the book “An introduction to measure theory” by Terence Tao first states that any open set can be expressed as a countable union of almost disjoint boxes (and in fact as a countable union of almost disjoint closed cubes).
Note that this lemma is a generalization of the fact that every open subset of can be expressed as a countable union of disjoint open intervals (see theorem 1.3, p. 6, in the book “Real analysis: measure theory, integration and Hilbert spaces” by Elias M. Stein and Rami Shakarchi and also lemma 2 of this blog post of mine).
The purpose of the present blog post is to elaborate on a detail in the proof of lemma 1.2.11 of Tao’s book. In particular, it will be explained why every closed dyadic cube is contained in exactly one maximal cube , without reproducing the rest of the proof.
Introduce the notation to indicate the dependence of closed dyadic cubes on and . For every for which there exists a closed dyadic cube contained in , choose the biggest closed dyadic cube in , i.e. choose . It is now obvious that by having capped the cubes by a sidelength of 1, there exists a maximum cube among , which is a maximal cube among those contained in .
What is left to show is that every closed dyadic cube can’t be contained in two or more maximal cubes. Assume that is contained in two distinct maximal cubes and , i.e. and with . The dyadic nesting property then leads to the contradiction or for the maximal cubes and , since and excludes the possibility of and being almost disjoint.