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ixed x y r displaystyle x y in mathbb r with x 0 displaystyle x 0 associate with g displaystyle g the operation of function composition displaystyle circ which turns g displaystyle g into a non abelian group g displaystyle g can be identified with the right half plane 0 r x y x y r x 0 displaystyle 0 infty times mathbb r left x y x y in mathbb r x 0 right under which the group operation becomes s t u v s u s v t displaystyle s t circ u v su sv t a left invariant haar measure μ l displaystyle mu _ l respectively a right invariant haar measure μ r displaystyle mu _ r on g 0 r displaystyle g 0 infty times mathbb r is given by μ l s s 1 x 2 d x d y displaystyle mu _ l s int _ s frac 1 x 2 dx dy and μ r s s 1 x d x d y displaystyle mu _ r s int _ s frac 1 x dx dy for any borel subset s displaystyle s of g 0 r displaystyle g 0 infty times mathbb r this is because if s 0 r displaystyle s subseteq 0 infty times mathbb r is an open subset then for s t g displaystyle s t in g fixed integration by substitution gives μ l s t s s t s 1 x 2 d x d y s 1 s u 2 s s 0 0 d u d v μ l s displaystyle mu _ l s t circ s int _ s t circ s frac 1 x 2 dx dy int _ s frac 1 su 2 s s 0 0 du dv mu _ l s while for u v g displaystyle u v in g fixed μ r s u v s u v 1 x d x d y s 1 s u u 1 v 0 d s d t μ r s displaystyle mu _ r s circ u v int _ s circ u v frac 1 x dx dy int _ s frac 1 su u 1 v 0 ds dt mu _ r s on any lie group of dimension d displaystyle d a left haar measure can be associated with any non zero left invariant d displaystyle d form ω displaystyle omega as the lebesgue measure ω displaystyle omega and similarly for right haar measures this means also that the modular function can be computed as the absolute value of the determinant of the adjoint representation shaded area is one square unit a representation of the haar measure of positive real numbers in terms of area under the positive branch of the standard hyperbola xy 1 uses borel sets generated by intervals a b b a 0 for example a 1 and b euler s number e yields and area equal to log e 1 1 then for any positive real number c the area over the interval ca cb equals log b a so the area is invariant under multiplication by positive real numbers note that the area approaches infinity both as a approaches zero and b gets large use of this haar measure to define a logarithm function anchors a at 1 and considers area over an interval in b 1 with 0 b 1 as negative area in this way the logarithm can take any real value even though measure is always positive or zero if g displaystyle g is the group of non zero quaternions then g displaystyle g can be seen as an open subset of r 4 displaystyle mathbb r 4 a haar measure μ displaystyle mu is given by μ s s 1 x 2 y 2 z 2 w 2 2 d x d y d z d w displaystyle mu s int _ s frac 1 x 2 y 2 z 2 w 2 2 dx wedge dy wedge dz wedge dw where d x d y d z d w displaystyle dx wedge dy wedge dz wedge dw denotes the lebesgue measure in r 4 displaystyle mathbb r 4 and s displaystyle s is a borel subset of g displaystyle g if g displaystyle g is the additive group of p displaystyle p adic numbers for a prime p displaystyle p then a haar measure is given by letting a p n o displaystyle a p n o have measure p n displaystyle p n where o displaystyle o is the ring of p displaystyle p adic integers construction of haar measure edit a construction using compact subsets edit the following method of constructing haar measure is essentially the method used by haar and weil for any subsets s t g displaystyle s t subseteq g with s displaystyle s nonempty define t s displaystyle t s to be the smallest number of left translates of s displaystyle s that cover t displaystyle t so this is a non negative integer or infinity this is not additive on compact sets k g displaystyle k subseteq g though it does have the property that k u l u k l u displaystyle k u l u k cup l u for disjoint compact sets k l g displaystyle k l subseteq g provided that u displaystyle u is a sufficiently small open neighborhood of the identity depending on k displaystyle k and l displaystyle l the idea of haar measure is to take a sort of limit of k u displaystyle k u as u displaystyle u becomes smaller to make it additive on all pairs of disjoint compact sets though it first has to be normalized so that the limit is not just infinity so fix a compact set a displaystyle a with non empty interior which exists as the group is locally compact and for a compact set k displaystyle k define μ a k lim u k u a u displaystyle mu _ a k lim _ u frac k u a u where the limit is taken over a suitable directed set of open neighborhoods of the identity eventually contained in any given neighborhood the existence of a directed set such that the limit exists follows using tychonoff s theorem the function μ a displaystyle mu _ a is additive on disjoint compact subsets of g displaystyle g which implies that it is a regular content from a regular content one can construct a measure by first extending μ a displaystyle mu _ a to open sets by inner regularity then to all sets by outer regularity and then restricting it to borel sets even for open sets u displaystyle u the corresponding measure μ a u displaystyle mu _ a u need not be given by the lim sup formula above the problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure so is not an outer measure a construction using compactly supported functions edit cartan introduced another way of constructing haar measure as a radon measure a positive linear functional on compactly supported continuous functions which is similar to the construction above except that a displaystyle a k displaystyle k and u displaystyle u are positive continuous functions of compact support rather than subsets of g displaystyle g in this case we define k u displaystyle k u to be the infimum of numbers c 1 c n displaystyle c_ 1 cdots c_ n such that k g displaystyle k g is less than the linear combination c 1 u g 1 g c n u g n g displaystyle c_ 1 u g_ 1 g cdots c_ n u g_ n g of left translates of u displaystyle u for some g 1 g n g displaystyle g_ 1 ldots g_ n in g as before we define μ a k lim u k u a u displaystyle mu _ a k lim _ u frac k u a u the fact that the limit exists takes some effort to prove though the advantage of doing this is that the proof avoids the use of the axiom of choice and also gives uniqueness of haar measure as a by product the functional μ a displaystyle mu _ a extends to a positive linear functional on compactly supported continuous functions and so gives a haar measure note that even though the limit is linear in k displaystyle k the individual terms k u displaystyle k u are not usually linear in k displaystyle k a construction using mean values of functions edit von neumann gave a method of constructing haar measure using mean values of functions though it only works for compact groups the idea is that given a function f displaystyle f on a compact group one can find a convex combination a i f g i g textstyle sum a_ i f g_ i g where a i 1 textstyle sum a_ i 1 of its left translates that differs from a constant function by at most some small number ϵ displaystyle epsilon then one shows that as ϵ displaystyle epsilon tends to zero the values of these constant functions tend to a limit which is called the mean value or integral of the function f displaystyle f for groups that are locally compact but not compact this construction does not give haar measure as the mean value of compactly supported functions is zero however something like this does work for almost periodic functions on the group which do have a mean value though this is not given with respect to haar measure a construction on lie groups edit on an n dimensional lie group haar measure can be constructed easily as the measure induced by a left invariant n form this was known before haar s theorem the right haar measure edit it can also be proved that there exists a unique up to multiplication by a positive constant right translation invariant borel measure ν displaystyle nu satisfying the above regularity conditions and being finite on compact sets but it need not coincide with the left translation invariant measure μ displaystyle mu the left and right haar measures are the same only for so called unimodular groups see below it is quite simple though to find a relationship between μ displaystyle mu and ν displaystyle nu 𝜇 indeed for a borel set s displaystyle s let us denote by s 1 displaystyle s 1 the set of inverses of elements of s displaystyle s if we define μ 1 s μ s 1 displaystyle mu _ 1 s mu s 1 quad then μ 1 displaystyle mu _ 1 is a right haar measure to show right invariance apply the definition μ 1 s g μ s g 1 μ g 1 s 1 μ s 1 μ 1 s displaystyle mu _ 1 sg mu sg 1 mu g 1 s 1 mu s 1 mu _ 1 s quad because the right measure is unique it follows that μ 1 displaystyle mu _ 1 is a multiple of ν displaystyle nu and so μ s 1 k ν s displaystyle mu s 1 k nu s for all borel sets s displaystyle s where k displaystyle k is some positive constant the modular function edit the left translate of a right haar measure is a right haar measure more precisely if ν displaystyle nu is a right haar measure then for any fixed choice of a group element g s ν g 1 s displaystyle s mapsto nu g 1 s quad is also right invariant thus by uniqueness up to a constant scaling factor of the haar measure there exists a function δ displaystyle delta from the group to the positive reals called the haar modulus modular function or modular character such that for every borel set s displaystyle s ν g 1 s δ g ν s displaystyle nu g 1 s delta g nu s quad since right haar measure is well defined up to a positive scaling factor this equation shows the modular function is independent of the choice of right haar measure in the above equation the modular function is a continuous group homomorphism from g to the multiplicative group of positive real numbers a group is called unimodular if the modular function is identically 1 displaystyle 1 or equivalently if the haar measure is both left and right invariant examples of unimodular groups are abelian groups compact groups discrete groups e g finite groups semisimple lie groups and connected nilpotent lie groups citation needed an example of a non unimodular group is the group of affine transformations x a x b a r 0 b r a b 0 1 displaystyle big x mapsto ax b a in mathbb r setminus 0 b in mathbb r big left begin bmatrix a b 0 1 end bmatrix right on the real line this example shows that a solvable lie group need not be unimodular in this group a left haar measure is given by 1 a 2 d a d b displaystyle frac 1 a 2 da wedge db and a right haar measure by 1 a d a d b displaystyle frac 1 a da wedge db measures on homogeneous spaces edit if the locally compact group g displaystyle g acts transitively on a homogeneous space g h displaystyle g h one can ask if this space has an invariant measure or more generally a semi invariant measure with the property that μ g s χ g μ s displaystyle mu gs chi g mu s for some character χ displaystyle chi of g displaystyle g a necessary and sufficient condition for the existence of such a measure is that the restriction χ h displaystyle chi _ h is equal to δ h δ displaystyle delta _ h delta where δ displaystyle delta and δ displaystyle delta are the modular functions of g displaystyle g and h displaystyle h respectively 8 in particular an invariant measure on g h displaystyle g h exists if and only if the modular function δ displaystyle delta of g displaystyle g restricted to h displaystyle h is the modular function δ displaystyle delta of h displaystyle h example edit if g displaystyle g is the group s l 2 r displaystyle sl_ 2 mathbb r and h displaystyle h is the subgroup of upper triangular matrices then the modular function of h displaystyle h is nontrivial but the modular function of g displaystyle g is trivial the quotient of these cannot be extended to any character of g displaystyle g so the quotient space g h displaystyle g h which can be thought of as 1 dimensional real projective space does not have even a semi invariant measure haar integral edit using the general theory of lebesgue integration one can then define an integral for all borel measurable functions f displaystyle f on g displaystyle g this integral is called the haar integral and is denoted as f x d μ x displaystyle int f x d mu x where μ displaystyle mu is the haar measure one property of a left haar measure μ displaystyle mu is that letting s displaystyle s be an element of g displaystyle g the following is valid g f s x d μ x g f x d μ x displaystyle int _ g f sx d mu x int _ g f x d mu x for any haar integrable function f displaystyle f on g displaystyle g this is immediate for indicator functions 1 a t g d μ 1 t 1 a g d μ μ t 1 a μ a 1 a g d μ displaystyle int mathit 1 _ a tg d mu int mathit 1 _ t 1 a g d mu mu t 1 a mu a int mathit 1 _ a g d mu which is essentially the definition of left invariance uses edit 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 9 unless g displaystyle g is a discrete group it is impossible to define a countably additive left invariant regular measure on all subsets of g displaystyle g assuming the axiom of choice according to the theory of non measurable sets abstract harmonic analysis edit the haar measures are used in harmonic analysis on locally compact groups particularly in the theory of pontryagin duality 10 11 12 to prove the existence of a haar measure on a locally compact group g displaystyle g it suffices to exhibit a left invariant radon measure on g displaystyle g mathematical statistics edit in mathematical statistics haar measures are used for prior measures which are prior probabilities for compact groups of transformations these prior measures are used to construct admissible procedures by appeal to the characterization of admissible procedures as bayesian procedures or limits of bayesian procedures by wald for example a right haar measure for a family of distributions with a location parameter results in the pitman estimator which is best equivariant when left and right haar measures differ the right measure is usually preferred as a prior distribution for the group of affine transformations on the parameter space of the normal distribution the right haar measure is the jeffreys prior measure 13 unfortunately even right haar measures sometimes result in useless priors which cannot be recommended for practical use like other methods of constructing prior measures that avoid subjective information 14 another use of haar measure in statistics is in conditional inference in which the sampling distribution of a statistic is conditioned on another statistic of the data in invariant theoretic conditional inference the sampling dis...