This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and
For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral.
The representation theory for locally compact abelian groups is described by Pontryagin duality.
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity.
In a Polish group G, the σ-algebra of Haar null sets satisfies the countable chain condition if and only if G is locally compact.
Clausen (2017) has shown that it measures the difference between the algebraic K-theory of Z and R, the integers and the reals, respectively, in the sense that there is a homotopy fiber sequence