Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

Measure Thoery: Chp 9. Haar Measure

Topological Groups

A topological group is a set that has the structure of a group (say with group operation ) and of a topological space and is such that the operations and are continuous.

A locally compact topological group, or simply a locally compact group, is a topological group whose topology is locally compact and Hausdorff.

A compact group is a topological group whose topology is compact and Hausdorff.

The Existence and Uniqueness of Haar Measure

Let be a locally compact group, and let be a nonzero regular Borel measure on . Then is a left Haar measure (or simply a Haar measure) if it is invariant under left translations (or simply translation invariant), in the sense that holds for each in and each in .

Theorem 9.2.2. Let be a locally compact group. Then there is a left Haar measure on .

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }