Borel sets. For example, the product of the unit circle (with its usual topology) and the real line with the discrete topology is a locally compact group with the product topology and a Haar measure on this group is not inner regular for the closed subset . (Compact subsets of this vertical segment are finite sets and points have measure , so the measure of any compact subset of this vertical segment is . But, using outer regularity, one can show the segment has infinite measure.) The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. In particular an invariant measure on exists if and only if the modular function of restricted to is the modular function of . Example If is the group and is the subgroup of upper triangular matrices, then the modular function of is nontrivial but the modular function of is trivial. The quotient of these cannot be extended to any character of , so the quotient space (which can be thought of as 1-dimensional real projective space) does not have even a semi-invariant measure. Haar integral Using the general theory of Lebesgue integration, one can then define an integral for all Borel measurable functions on . This integral is called the Haar integral and is denoted as: where is the Haar measure. One property of a left Haar measure is that, letting be an element of , the following is valid: for any Haar integrable function on . This is immediate for indicator functions: which is essentially the definition of left invariance. Uses In the same issue of Annals of Mathematics and immediately after Haar's paper, the Haar theorem was used to solve Hilbert's fifth problem restricted to compact groups by John von Neumann. Unless is a discrete group, it is impossible to define a countably additive left-invariant regular measure on all subsets of , assuming the axiom of choice, according to the theory of non-measurable sets. Abstract harmonic analysis The Haar measures are used in harmonic analysis on locally compact groups, particularly in the theory of Pontryagin duality. To prove the existence of a Haar measure on a locally compact group it suffices to exhibit a left-invariant Radon measure on . Mathematical statistics In mathematical statistics, Haar measures are used for prior measures, which are prior probabilities for compact groups of transformations. These