Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The measurable Hall theorem fails for treeings

Gábor Kun

Abstract

We construct, for every $d \geq 3$, a $d$-regular acyclic measurably bipartite graphing that admits no measurable perfect matching, resolving a problem of Kechris and Marks. A dense variant of our construction yields a coupling of two standard Borel probability measure spaces whose support contains no deterministic coupling, though the conditional probabilities of the coupling measure are atomless. This refutes a conjecture of Gurel-Gurevich and Peled.

The measurable Hall theorem fails for treeings

Abstract

We construct, for every , a -regular acyclic measurably bipartite graphing that admits no measurable perfect matching, resolving a problem of Kechris and Marks. A dense variant of our construction yields a coupling of two standard Borel probability measure spaces whose support contains no deterministic coupling, though the conditional probabilities of the coupling measure are atomless. This refutes a conjecture of Gurel-Gurevich and Peled.

Paper Structure

This paper contains 5 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Definitions, notation
  3. Inverse limits
  4. The proofs
  5. Further directions

Key Result

Theorem 1.2

For every $d \geq 2$ there exists a $d$-regular measurably bipartite treeing $T$ such that every absolutely integrable antisymmetric circulation on $T$ is zero almost everywhere. In particular, $T$ admits no measurable perfect matching.

Theorems & Definitions (21)

  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 11 more