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Haar Measure


Any locally compact Hausdorff topological group has a unique (up to scalars) nonzero left invariant measure which is finite on compact sets. If the group is Abelian or compact, then this measure is also right invariant and is known as the Haar measure.

More formally, let G be a locally compact group. Then a left invariant Haar measure on G is a Borel measure mu satisfying the following conditions:

1. mu(xE)=mu(E) for every x in G and every measurable E subset= G.

2. mu(U)>0 for every nonempty open set U subset= G.

3. mu(K)<infty for every compact set K subset= G.

For example, the Lebesgue measure is an invariant Haar measure on real numbers.

In addition, if G is an (algebraic) group, then G with the discrete topology is a locally compact group. A left invariant Haar measure on G is the counting measure on G.


See also

Lebesgue Measure

Portions of this entry contributed by Mohammad Sal Moslehian

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References

Conway, J. A Course in Functional Analysis. New York: Springer-Verlag, 1990.Feldman M. and Gilles, C. "An Expository Note on Individual Risk Without Aggregate Uncertainty." J. Econ. Theory 35, 26-32, 1985.

Referenced on Wolfram|Alpha

Haar Measure

Cite this as:

Moslehian, Mohammad Sal and Weisstein, Eric W. "Haar Measure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HaarMeasure.html

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