The Haar measure of a profinite $n$-ary group

TL;DR

This paper establishes the existence and uniqueness of Haar measures for profinite n-ary groups and relates them to measures on associated binary groups and Post covers.

Contribution

It introduces a Haar measure for profinite n-ary groups and connects it to classical Haar measures on related structures.

Findings

Unique Haar measure exists for every profinite n-ary group.

The Haar measure relates to measures on the binary group and Post cover.

Explicit measure formula for measurable subsets in profinite n-ary groups.

Abstract

We prove that every profinite $n$-ary group $(G, f)=\Gf$ has a unique Haar measure $m_p$ and further for every measurable subset $A\subseteq G$, we have $$ m_p(A)=m(A)=(n-1)m^{\ast}(A) $$ where $m$ and $m^{\ast}$ are the normalized Haar measures of the profinite groups $(G, \bullet)$ and the Post cover $G^{\ast}$, respectively.