Let
$G$
be a compact topological group and let
$f:\,G\,\to \,G$
be a continuous transformation of
$G$
. Define
${{f}^{*}}:G\to G$
by
${{f}^{*}}\left( x \right)\,=\,f\left( {{x}^{-1}} \right)x$
and let
$\mu \,=\,{{\mu }_{G}}$
be Haar measure on
$G$
. Assume that
$H\,=\,\text{IM}\,{{f}^{*}}$
is a subgroup of
$G$
and for every measurable
$C\,\subseteq \,H,\,{{\mu }_{G}}{{\left( \left( {{f}^{*}} \right) \right)}^{-1}}\left( \left( C \right) \right)\,=\,\mu H\left( C \right)$
. Then for every measurable
$C\,\subseteq \,G$
, there exist
$S\subseteq C$
and
$g\,\in \,G$
such that
$f\left( S{{g}^{-1}} \right)\,\subseteq \,C{{g}^{-1}}$
and
$\mu \left( S \right)\,\ge \,{{\left( \mu \left( C \right) \right)}^{2}}$
.