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Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.

Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.uchicago.edu/~henderson/additive.pdf (Wayback Machine).

Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \mapsto G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.

Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.

Consider a locally compact group $G$. Consider a map $f:G \mapsto G$, which is measurable and has an inverse, which is also measurable with respect to the completed Borel structure on $G$. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.

Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \mapsto G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?

What if $G$ is abelian? If not, what are necessary conditions on $G$, such that this is the case.

For $\mathbb{R}$, it seems to be true: see http://math.stanford.edu/~chris/additive.pdf.