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Let $G$ be a locally compact group on which there exists a Haar measure, etc..

Now I am supposed to take such a metrisable $G$, and given the existence of some metric on $G$, prove that there exists a translation-invariant metric, i.e., a metric $d$ such that $d(x,y) = d(gx,gy)$ for all $x,y,g \in G$. How to go about this?

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    I know of an answer to this, but It's 5 pages long... This is Birkhoff-Kakutani, no? http://planetmath.org/encyclopedia/BirkhoffKakutaniTheorem.html2011-09-18
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    Birkhoff-Kakutani is something else, according to http://terrytao.wordpress.com/2011/05/17/the-birkhoff-kakutani-theorem/ .. It addresses the question of metrizability in addition .. All I am asking is, is it possible to prove the existence of invariant metric, given already some other metric?2011-09-18
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    I'm pretty sure that there is quite an easy explicit way but I simply don't remember the trick at the moment. You can extract a construction from Theorem 3 on page 8 in Koszul's *[Lectures on Groups of Transformations](http://www.math.tifr.res.in/~publ/ln/tifr32.pdf)* but that's not something you should be able to come up with yourself, I guess.2011-09-18
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    @kneidell: Yes, that's usually called the Birkhoff-Kakutani theorem. However, it's already implicit in Weil's work on uniform structures, he just didn't spell it out. Weil proved that a countably generated uniform structure is metrizable and the metric you get from his construction applied to the left uniform structure of a first countable group is in fact translation invariant.2011-09-18
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    @t.b.you should write your observation that if the group is metizable, it is first countable, so its left uniform structure is countable generated, so metrizable, and the resulting metric is left invariant (a long observation! :) ) as an answer, so that we can vote it up.2012-01-17

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