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Topic: Haar measure on locally compact group
Replies: 3   Last Post: Feb 6, 2013 2:32 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Haar measure on locally compact group
Posted: Feb 6, 2013 10:45 AM
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On Tue, 5 Feb 2013 14:17:15 -0800 (PST), wrote:

>Can somebody help me by showing the outline or key steps to prove that if G is a locally compact group with the Haar
>measure (just assume G is Abelian), then if the Haar measure of G is finite, then G must be compact. I tried many times,
>got some progress but did not get to the final solution yet. Thanks

Must be in the standard books... ok, do this:

Say V is a neighborhood of 0 with compact closure. By continuity of
the group operations there exists a nbd W of 0 with

W - W subset V.

Now let S be a subset of G maximal subject to the condition that

{x + W : w in S}

is pairwise disjoint.

Since W is open, m(W) > 0 and hence S is finite.

Now, for all x in G there exists y in S such that

(x + W) intersect (y + W) is nonempty.

This says that

y - x is in W - W,

hence y - x is in V, so that y is in x + V.

So G is the union of x + V for x in S, hence G is
a finite union of compact sets, hence compact.


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