Measurable rectangles inside a non-null set

measure-theory

The existence of a counterexample is implied by Theorem 2.1 of Kendall and Montana's paper "Small sets and Markov transition densities". Below is an excerpt from pages 180-182, including the beginning of the section for context.

This question is answered with a different approach in Halmos's Problems for mathematicians, young and old. Problem, Hint, and Solution 14.E are given below the Kendall and Montana excerpt.

enter image description here

enter image description here

enter image description here

enter image description here

And here's Halmos:

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

Related

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus [duplicate]
Prove the open mapping theorem by using maximum modulus principle
Maximal logarithmic likelihood function of the Student-t distribution
how to make everything open on external monitor
Mac Mini won't open Startup Manager
VoiceOver in macOS Big Sur echos zoom level
Apple Pencil life span
Remove file association for folders
tmutil doesn't create a real snapshot (since Catalina?)
Safari File Dialog Fails to Show
How do I add a new google contact on an iPhone?
Automatically apply tag to folder based on content