Locally compact group
This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform andFor compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral.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