Haar uniqueness through shrinking neighborhoods

Free-source companion by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, 5 October 2026. This file records the source and terms of the Fremlin adaptation now included in the Haar lesson. No human review is claimed.

The complete shrinking-neighborhood argument is proved in Theorem 9.2 of the Haar lesson. It compares the two measures on an open subset of the product, squeezes the ratios of small symmetric neighborhoods, and then extends equality from relatively compact open sets to every Borel set. The main lesson supplies each step of this proof, including the open-set product theorem; this companion does not repeat the proof.

Fremlin works with a broader topological measure convention. Our adaptation assumes the convention already defined and constructed in the course: measures are finite on compact sets, inner regular on open sets and outer regular on Borel sets. Accordingly its last step uses Borel outer regularity directly. It does not assume sigma-finiteness or inner compact regularity on all Borel sets. The comparison with Fremlin’s completed-measure convention is proved separately in Proposition 13.7.

Source and terms

D. H. Fremlin, Measure Theory, Volume 4, Topological Measure Spaces, §442B, supplies the neighborhood-ratio argument. Copyright © 1998 D. H. Fremlin. The supplied editable TeX remains attributed to Fremlin. The adapted proof in Theorem 9.2 and this companion are distributed under the Design Science License, with editable Markdown source.

Changes on 3 October 2026: restriction to the course’s LCH/Borel convention, expansion of compact shrinking, and replacement of the quasi-Radon base-uniqueness step by inner regularity on open sets and outer regularity on Borel sets. Changes on 5 October 2026: consolidation of the full argument into Theorem 9.2 and removal of the duplicate proof from this companion. Adaptation and additions: GPT-6.1 Sol (OpenAI), Ultra.

THE WORK IS PROVIDED "AS IS," AND COMES WITH ABSOLUTELY NO WARRANTY, EXPRESS OR IMPLIED, TO THE EXTENT PERMITTED BY APPLICABLE LAW. The full warranty and liability terms are in the accompanying Design Science License.