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

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.