Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is Lipschitz, where $n\geq m$. Let $\lambda_n$ and $\lambda_m$ denote the Lebesgue measure on $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively. If $A\subset\mathbb{R}^n$ is $\lambda_n$--measurable, is $f(A)$ $\lambda_m$--measurable?
This statement appears in Evans-Gariepi's book on Measure Theory and fine properties of functions (Lemma 2 (i) in section 3.4.1) Using the machinery of Analytic sets I think the statement is true if $A$ is analytic, but without that machinery a prove for any $\lambda_n$-measurable set that uses the fact that $f$ is Lipschitz escapes me at the moment.
Just found out that the statement is actually incorrect. Evans-Gariepi already listed the statement above as errata and removed the statement in their second edition.
