Prove the unicity of Lebesgue measure.

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# Prove the unicity of Lebesgue measure.

Submitted by Onezimo on Sun, 10/12/2008 - 17:43

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Can somebody helps me with this problem?

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## Re: Prove the unicity of Lebesgue measure.

Hi Keenan,

since the guy is Brazilian, I think unicity = uniqueness. I have commited the same mistake.

Greetings,

pedro

## Re: Prove the unicity of Lebesgue measure.

I see. Well, Lebesgue measure isn't quite unique...I mean, as a measure on the $\sigma$-algebra of Borel sets in $\mathbb{R}^n$ it is unique up to positive scalar multiples because it is Haar measure on the locally compact group $(\mathbb{R}^n,+)$...the proof of uniqueness of Haar measures usually involves looking at the ratio of the integrals of a positive compactly supported function on your set with respect to one measure and another. For Lebesgue measure in particular, you should try to prove that it is unique up to scalar multiples on bounded boxes (i.e. products of bounded intervals), then extending to open sets. It then follows for all Borel sets by outer regularity.

## Re: Prove the unicity of Lebesgue measure.

What do you mean by "unicity?"