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What are all the translation invariant measures on $\mathbb{R}$?

Except Lebesgue measure on $\mathbb R$ I didn't find any translation invariant measure. So I put this question?

I know that if $\mu$ is a measure then $c \times \mu$ is again a measure where $c>0$.

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Up to a multiplicative constant, Lebesgue measure is the only translation-invariant measure on the Borel sets that puts positive, finite measure on the unit interval. I don't have a reference at hand, though. –  Michael Greinecker May 19 '12 at 17:00
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This is a special case of Haar's theorem: en.wikipedia.org/wiki/Haar_measure. I suppose it might've been known for $\mathbb{R}$ earlier. –  Kevin May 19 '12 at 17:14
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@Kevin: yes, this was known to Lebesgue already. He also asked explicitly whether it was possible to extend Lebesgue measure to the entire power set of $\mathbb{R}$ and whether such an extension was unique. This became known as Le problème de la mesure and influenced Banach's early work. The Banach-Tarski paradox is the most famous outgrowth of these investigations. –  t.b. May 19 '12 at 19:00

3 Answers 3

up vote 5 down vote accepted

Here is a way to argue out. I will let you fill in the details.

  1. If we let $\mu([0,1))=C$, then $\mu([0,1/n)) = C/n$, where $n \in \mathbb{Z}^+$. This follows from additivity and translation invariance.
  2. Now prove that if $(b-a) \in \mathbb{Q}^+$, then $\mu([a,b)) = C(b-a)$ using translation invariance and what you obtained from the previous result.
  3. Now use the monotonicity of the measure to get lower continuity of the measure for all intervals $[a,b)$.

Hence, $\mu([a,b)) = \mu([0,1]) \times(b-a)$.

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$\mathbb{R}$ is a locally compact group with respect to addition and the translation invariant measures are the Haar measures on this group. A general theorem by Von Neumann states that such a measure is unique up to a multiplicative constant.

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Let $\lambda$ be a translation-invariant measure on the Borel sets that puts positive and finite measure on the right-open unit interval $[0,1)$ then $\lambda$ is a positive multiple of Lebesgue measure. Here is an outline of the proof: Every Borel measure is determined by its behavior on finite intervals. By translation invariance, you know that a right-open interval of length $1/2^n$ has measure $1/2^n \lambda[0,1)$, since $2^n$ such pieces form a disjoint cover over $[0,1)$ and every such piece can be translated into every other other such piece. Now you can approximate every interval by such pieces to pin down the measure of each interval.

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