The Haar measure of a profinite n-ary group

Abstract

We prove that every profinite $n$-ary group $(G,f)={\mathrm{\text{der}}}_{𝜃,b}(G,•)$ 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 $(G,•)$ and the Post cover $G^{\ast }$, respectively.