Is Lebesgue measure the only $\sigma$-finite invariant Borel measure?

Abstract

R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the measure is only defined for Borel sets then the answer is affirmative. Then we show that if the measure is defined on a $\sigma$-algebra \emph{containing} the Borel sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel sets.