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  be a locally compact group. Then a left invariant Haar measure on  is a Borel measure  satisfying the following conditions:

1.  for every  and every measurable .

2.  for every nonempty open set .

3.  for every compact set .

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

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

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.