Kac’s Lemma and Countable Generators for Actions of Countable Groups

Tom Meyerovitch, Benjamin Weiss

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the allocation-based Kac identity by reducing to a disjointness lemma (Lemma 2.1) and an enumeration lemma (Lemma 2.3), then concluding Proposition 1.3, which states ∫_A f_κ dμ = ∫_X f dμ and, in particular, ∫_A |B_κ(x)| dμ(x) = 1. The candidate model instead gives a direct mass-transport/Change-of-Variables proof using Tonelli and measure-preserving bijections. Both arguments are correct and reach the same result; the proofs are substantially different in technique.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This is a concise and well-motivated generalization of Kac's lemma to actions of countable groups, using the neat concept of allocations, and it yields a short proof of the existence of countable generators. The arguments are correct and well-structured. Minor clarifications (especially around measurability and an optional direct proof of Proposition 1.3) would make the exposition even more accessible.