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The book of Lieb and Loss proves the existence of Lebesgue measure in an unorthodox way as theorem 6.22, using the fact that ''positive distributions are measures''.

My question is, whether it is possible to prove the existence of Haar measure using this method, and if so, it is done in any standard reference? If not in the general situation, at least for some nice topological groups?

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    The construction of a Borel measure starting with a positive linear functional on $\mathcal{D}(G) = C_{c}^\infty(G)$ goes through verbatim for Lie groups. This yields Haar measure on arbitrary Lie groups, starting with the functional on $\mathcal{D}(G)$ given by integrating an invariant volume form on $G$ (whose construction can be done in entirely Riemannian terms). This procedure is pretty standard in Lie theory but I can't point to a reference off-hand (but check Knapp's or Helgason's books for a start if no-one answers). Is this what you're looking for or did you expect to go further?2012-08-31

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