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Someone please explain this: Presume that for each $x \in \mathbb{R}$ and $A \subseteq \mathbb{R}$, that $x + A = \big\{ x + a \mid a \in A \big\}$. Here, A and x + A are Borel sets for all $x \in \mathbb{R}$.

Then, if $\mu$ is a σ-finite measure on B (so that λ($\mathbb{R}$) = ∞) and λ is a translation invariant, how can it be shown that there exists a real number c > 0 such that $\mu$ = c λ? I know that $\mu$(A) = c λ(A) for all Borel sets A. I also know that the Lebesgue measure is the unique translation invariant measure on B, up to a multiplicative constant.

Sarah Donovan
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  • Maybe this helps: http://math.stackexchange.com/questions/147089/translation-invariant-measures-on-mathbb-r – Adam Mar 23 '16 at 11:51
  • Could you please explain? – Sarah Donovan Mar 23 '16 at 17:36
  • What is actually your question? If $\mu$ is an arbitrary measure, then your statement is wrong, consider for example $\mu=0$. I thought you maybe ask: If $\mu$ is $\sigma$-finite on the Borle set $\mathcal{B}(\mathbb{R})$ and translation invariant, does then a constant $c > 0$ exist with $\mu = c \lambda$. But this is exactly the answer of the link I posted in my first comment. – Adam Mar 23 '16 at 19:44
  • I just noticed that you wrote: I know that $\mu(A)=c \lambda(A)$ for all Borel-sets. But this is the definition of $\mu = c \lambda$. What are you asking for? – Adam Mar 23 '16 at 19:45

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