Haar's measure

In mathematical analysis, the Haar measure is a way of assigning an "invariant volume" to the subsets of locally compact topological groups and to define later an integral for the functions on those groups. This measure was introduced by Alfred Haar, a Hungarian mathematician, around the year 1932. See also Pontryagin duality. De Haar measures are used in many parts of analysis and number theory.

Preliminaries

Let G be a locally compact topological group. In this article, the σ-algebra X generated by all compact subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel (Borelian) set.

If a is an element of G and S is a subset of G, then we define the left shift and to the right of S as follows:

• The left translation:

aS={a⋅ ⋅ s:s한 한 S!.{displaystyle aS={acdot s:sin S}. !

• The right translation:

Sa={s⋅ ⋅ a:s한 한 S!.{displaystyle Sa={scdot a:sin S}. !

Left and right translations translate Borel sets to Borel sets.

A measure μ on Borel subsets of G is called left translation invariant if and only if for all Borel subsets S of G and for every a in G we have

μ μ (aS)=μ μ (S).{displaystyle mu (aS)=mu (S).quad }

A similar definition is made for right translation invariance.

Existence of Haar's left measure

It is verified that there is, except for a multiplicative constant, only one regular measure invariant by left translation on X that is finite on all Borel sets of G such that the μ(U) > 0 for any given non-empty Borel open U. Here, μ is said to be regular if

• μ(K) is finite for each compact set K.

• Each set of Borel E is regular outside:

μ μ (E)=inf{μ μ (U):E U,Uopen and Borel!.{displaystyle mu (E)=inf{mu (U):Esubseteq U,U{mbox{ opened and Borel}}}. !

• Yeah. E It's Borel's, then. E is regular interior:

μ μ (E)=sup{μ μ (K):K E,Kcompact!.{displaystyle mu (E)=sup{mu (K):Ksubseteq E,K{mbox{ compact }}}{. !

Observation. Note that in some pathological cases, a set can be open without being Borel. For this reason, in the property of exterior regularity, the rank of the infimal is established specifically on sets that are open and Borel. These pathologies never occur if G is a locally compact group whose underlying topology is metrizable separable; note that in this case the Borel structure is the one generated by all open sets.

Haar's Right Measure

It can also be proved that there exists an essentially unique right translation-invariant regular ν measure, but it need not coincide with the left translation-invariant regular μ measure. These measures are the same only for the groups called unimodular (see below). It is easy, however, to find a relationship between μ and ν.

Indeed, for a given Borel S, S^{- 1} denotes the set of inverse elements of S. Note that if we define

μ μ − − 1(S)=μ μ (S− − 1){displaystyle mu _{-1}(S)=mu (S^{-1})quad }

so this is a right Haar measure. To prove right invariance, apply the definition:

μ μ − − 1(Sa)=μ μ ((Sa)− − 1)=μ μ (a− − 1S− − 1)=μ μ (S− − 1)=μ μ − − 1(S).{displaystyle mu _{-1}(Sa)=mu (Sa)^{-1})=mu (a^{-1S^{-1})=mu (S^{-1})=mu _{-1}(S).

Because the right measure is unique, it follows that μ_-1 is a multiple of ν and then

μ μ (S− − 1)=k.. (S){displaystyle mu (S^{-1})=knu (S),}

for all S of fixed Borel, where k is some positive constant.

The Haar Integral

Using the general Lebesgue theory of integration, one can then define an integral for all Borel measurable functions f on G. This integral is called the Haar integral. If μ is a left Haar measure, then

∫ ∫ Gf(sx)dμ μ (x)=∫ ∫ Gf(x)dμ μ (x){displaystyle int _{G}f(sx) dmu (x)=int _{Gf}(x) dmu (x)}

for any integrable function f. This is immediate for step functions which essentially give the definition of left invariance.

Applications

De Haar measures are used in harmonic analysis on arbitrary locally compact groups, consider Pontryagin's duality. A technique frequently used to prove the existence of a Haar measure on a locally compact group G is by proving the existence of a left invariant Radon measure on G.

Note that, unless G is a discrete group, it is impossible to define a countably-additive right invariant measure on all subsets of G, assuming the axiom of choice. See non-measurable sets.

Examples

• The measurement of Haar in the topological group (R, +) that takes the value 1 in the interval [0, 1] is equal to the restriction of the measure of Lebesgue to the subsets of Borel of R. This can be generalized to (Rⁿ, +).

• Yeah. G is the group of positive real numbers with multiplication as operation, then the measurement of Haar μ(S) is given by

μ μ (S)=∫ ∫ S1tdt{displaystyle mu (S)=int _{S}{frac {1}{t}{,dt}

for any subset S of Borel in the real positives.

This generalizes to the following:

• Stop. G = GL(n, R) the left and right measurements of Haar are proportional and

μ μ (S)=∫ ∫ S1日本語det(X)日本語ndX{displaystyle mu (S)=int _{S}{1 over Δdet(X)Δ^{n},dX}

where dX denotes the measure of Lebesgue to R^{n2{displaystyle n^{2}}}The set of all the matrices n × n. This follows from the variable change formula.

• More generally, in any dimension Lie group d a left measurement of Haar can be associated with any d-form ω invariant left different from zero, like the measurement of Lebesgue LICENSITY; and similarly for the right measures of Haar. This also means that the modular function can be computed, such as the absolute value of the determinant of the attached representation.

The modular function

Note that the left translation of a right Haar measure is a right Haar measure. More exactly, if ν is a right Haar measure, then

A μ μ (t− − 1A){displaystyle Amapsto mu (t^{-1}A)quad }

is also right invariant. Thus, there exists a unique function such that for every Borel set A

μ μ (t− − 1A)=Δ Δ (t)μ μ (A).{displaystyle mu (t^{-1}A)=Delta (t)mu (A).quad }

A group is unimodular if the modular function is identically 1. Examples of unimodular groups are compact groups and abelian groups. An example of a nonunimodular group is the group of transformations of the form

x ax+b{displaystyle xmapsto ax+bquad }

on the real line.