“Haar measure”的意思、由来-开放百科全书

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

This measure was introduced by Alfréd Haar in 1933,^[1] though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral".^[2] Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

Preliminaries

Let

be a locally compact Hausdorff topological group. The

-algebra generated by all open subsets of

is called the Borel algebra. An element of the Borel algebra is called a Borel set. If

is an element of

and

is a subset of

, then we define the left and right translates of

as follows:

A measure

on the Borel subsets of

is called left-translation-invariant if for all Borel subsets

and all

one has

A measure

on the Borel subsets of

is called right-translation-invariant if for all Borel subsets

and all

one has

Haar's theorem

There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure

on the Borel subsets of

satisfying the following properties:

Such a measure on

is called a left Haar measure. It can be shown as a consequence of the above properties that

for every non-empty open subset

. In particular, if

is compact then

is finite and positive, so we can uniquely specify a left Haar measure on

by adding the normalization condition

Some authors define a Haar measure on Baire sets rather than Borel sets. This makes the regularity conditions unnecessary as Baire measures are automatically regular. Halmos^[3] rather confusingly uses the term "Borel set" for elements of the

-ring generated by compact sets, and defines Haar measure on these sets.

The left Haar measure satisfies the inner regularity condition for all

-finite Borel sets, but may not be inner regular for all 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 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.^[4] Weil's proof used the axiom of choice and Henri Cartan furnished a proof which avoided its use.^[5] Cartan's proof also establishes the existence and the uniqueness simultaneously. A simplified and complete account of Cartan's argument was given by Alfsen in 1963.^[6] The special case of invariant measure for second countable locally compact groups had been shown by Haar in 1933.^[1]

Construction of Haar measure

A construction using compact subsets

The following method of constructing Haar measure is essentially the method used by Haar and Weil.

For any subsets

with

nonempty define

to be the smallest number of left translates of

that cover

(so this is a non-negative integer or infinity). This is not additive on compact sets

, though it does have the property that

for disjoint compact sets

provided that

is a sufficiently small open neighborhood of the identity (depending on

and

). The idea of Haar measure is to take a sort of limit of

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

with non-empty interior (which exists as the group is locally compact) and for a compact set

define

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

is additive on disjoint compact subsets of

, which implies that it is a regular content. From a regular content one can construct a measure by first extending

to open sets by inner regularity, then to all sets by outer regularity, and then restricting it to Borel sets. (Even for open sets

, the corresponding measure

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

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

, and

are positive continuous functions of compact support rather than subsets of

. In this case we define

to be the infimum of numbers

such that

is less than the linear combination

of left translates of

for some

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

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

, the individual terms

are not usually linear in

A construction using mean values of functions

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

on a compact group, one can find a convex combination

(where

) of its left translates that differs from a constant function by at most some small number

. Then one shows that as

tends to zero the values of these constant functions tend to a limit, which is called the mean value (or integral) of the function

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

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

It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure

satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure

. 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

and

Indeed, for a Borel set

, let us denote by

the set of inverses of elements of

. If we define

then this is a right Haar measure. To show right invariance, apply the definition:

The modular function

The left translate of a right Haar measure is a right Haar measure. More precisely, if

is a right Haar measure, then

is also right invariant. Thus, by uniqueness of the Haar measure, there exists a function

from the group to the positive reals, called the Haar modulus, modular function or modular character, such that for every Borel set

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 into the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically

, 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|date=June 2018}} An example of a non-unimodular group is the group of affine transformations

on the real line. This example shows that a solvable Lie group need not be unimodular.

Measures on homogeneous spaces

If the locally compact group

acts transitively on a homogeneous space

, one can ask if this space has an invariant measure, or more generally a semi-invariant measure with the property that

for some character

. A necessary and sufficient condition for the existence of such a measure is that the restriction

is equal to

, where

and

are the modular functions of

and

respectively.

In particular an invariant measure on

exists if and only if the modular function

restricted to

is the modular function

Example

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

. This integral is called the Haar integral. If

is a left Haar measure, then

for any Haar integrable function

. This is immediate for indicator functions, being essentially the definition of left invariance.

Examples

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 for compact groups by John von Neumann.^[7]

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.^[8]^[9]^[10] 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 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.^[11] 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.^[12]

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 distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a maximal invariant, so that by itself a statistical principle of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed.

For non-compact groups, statisticians have extended Haar-measure results using amenable groups.^[13]

Weil's converse theorem

In 1936 Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure for which one can define a convolution product, then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as Haar measure on this completion.

Preliminaries

Haar's theorem

Construction of Haar measure

A construction using compact subsets

A construction using compactly supported functions

A construction using mean values of functions

A construction on Lie groups

The right Haar measure

The modular function

Measures on homogeneous spaces

Example

Haar integral

Examples

Uses

Abstract harmonic analysis

Mathematical statistics

Weil's converse theorem

See also

Notes

Further reading

External links