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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), johnartin@gmail.com wrote:
>Hi,
>
>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.
DU.
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