Hairy measure

The Haar measure was introduced into mathematics by Alfréd Haar in order to make the results of the measure theory applicable in group theory .

It is a generalization of the Lebesgue measure . The Lebesgue measure is a measure in Euclidean space that is invariant under translations. The Euclidean space is a locally compact topological group with respect to addition. The Haar measure can be defined for every locally compact (in the following always assumed as Hausdorffsch ) topological group, in particular for every Lie group . Locally compact groups with their Haar's dimensions are examined in the harmonic analysis .

definition

A (left) Haar's measure of a locally compact group is a left-invariant regular Borel measure , which is positive on non-empty open subsets. ${\displaystyle G}$

A measure is called left invariant if for every Borel set and every group element ${\displaystyle \mu }$ ${\displaystyle A}$ ${\displaystyle g}$

${\displaystyle \mu (gA)=\mu (A)}$,

or in integral notation

${\displaystyle \int _{G}f(gx)\,\mathrm {d} \mu =\int _{G}f(x)\,\mathrm {d} \mu }$

applies to integrable functions and group elements . ${\displaystyle f}$ ${\displaystyle g}$

Replacing “left invariant” with the analogous term “right invariant” results in the term right hair measure. The left and right hair measure exist in every locally compact topological group and are each uniquely determined except for one factor. If they both match, the group is called unimodular. Abelian (locally compact) groups as well as compact groups are unimodular.

Proof of existence

According to a variant of Riesz-Markow's representation theorem, it is sufficient to show the existence of a continuous, positive, left-invariant, linear functional on the non-negative, real-valued, continuous functions with compact support on a locally compact group . In the real case, an example of this would be the Riemann integral, which can be continued into the Lebesgue integral and thus induces the Lebesgue measure . The proof of existence is possible non-constructively via Tychonoff's theorem . ${\displaystyle G}$

For this purpose, one defines first for every two non-negative, continuous functions with compact support with the coverage number as ${\displaystyle (f,\varphi )}$ ${\displaystyle \varphi \neq 0}$ ${\displaystyle (f:\varphi )}$

${\displaystyle (f:\varphi ):=\inf \left\{\sum _{i}c_{i}\mid c\in {\mathbb {R} ^{+}}^{n},\ n\in \mathbb {N} ,\ \exists g\in G^{n}\ \forall h\in G\ f(h)\leq \left(\sum _{i}c_{i}L_{g_{i}}\varphi \right)(h)\right\}}$,

where the left shift denotes, that is . For more and more "finer" , the overlap then becomes more and more "more accurate", although the overlap number usually increases. This can be brought under control by standardization, one defines ${\displaystyle L_{g}f}$ ${\displaystyle L_{g}f(h)=f(g^{-1}h)}$ ${\displaystyle \varphi }$

${\displaystyle I_{\varphi }(f):={\frac {(f:\varphi )}{(f_{0}:\varphi )}}}$

for any non-zero. However, this functional is still generally not linear - it is homogeneous , but generally only subadditive and not additive. The following inequality is then decisive for the further proof: ${\displaystyle f_{0}}$

${\displaystyle 0<(f_{0}:f)^{-1}\leq I_{\varphi }(f)\leq (f:f_{0})}$.

Now consider the environment filter of the neutral element in and form the image filter below the figure, which assigns to each the set of all for which the carrier of in is contained. In this way, thanks to the estimation, one obtains a filter in space and this space is compact according to Tychonoff's theorem. The filter thus has a point of contact; it is calculated that such a point of contact has all the desired properties, in particular is linear, that is to say is a left Haar integral. ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle I_{\varphi }}$ ${\displaystyle \varphi }$ ${\displaystyle V}$ ${\displaystyle \textstyle \prod _{f}[(f_{0}:f)^{-1},(f:f_{0})]}$

properties

The Haar measure of a locally compact topological group is finite if and only if the group is compact . This fact makes it possible to carry out an averaging over infinite compact groups by integration with regard to this measure. One consequence is, for example, that every finite-dimensional complex representation of a compact group is unitary with respect to a suitable scalar product . A one-element set has a non-zero hair measure if and only if the group is discrete.

Examples

• The Lebesgue measure on and is the Haarsche measure on the additive groups respectively . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle (\mathbb {R} ^{n},+)}$ ${\displaystyle (\mathbb {C} ^{n},+)}$
• Let the circle group , i.e. the compact group of complex numbers of magnitude 1 with the usual multiplication of complex numbers as a link. If the Lebesgue measure denotes the interval and the function , the Haar measure is given by the image measure , that is, for each Borel set . ${\displaystyle T}$ ${\displaystyle \lambda }$ ${\displaystyle [0,1]}$ ${\displaystyle f}$ ${\displaystyle [0,1]\rightarrow T,x\mapsto e^{2\pi ix}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda \circ f^{-1}}$ ${\displaystyle \mu (A)\,=\,\lambda (f^{-1}(A))}$ ${\displaystyle A\subset T}$
• If the general linear group , the Haar measure is given by, where the Lebesgue measure is on. ${\displaystyle GL(n,\mathbb {R} )}$ ${\displaystyle \mu (A)=\int _{A}{\frac {1}{|\det(u)|^{n}}}\,\mathrm {d} \lambda (u)}$ ${\displaystyle \lambda }$ ${\displaystyle \mathbb {R} ^{n^{2}}}$
• For a discrete group, the counting measure is Haar's measure.
• The Haar's measure on the multiplicative group is given by the formula , where the Lebesgue measure is. ${\displaystyle \mathbb {R} ^{\times }=(\mathbb {R} \setminus \{0\},\cdot )}$ ${\displaystyle \mu (A)=\int _{A}{\frac {1}{|x|}}\,\mathrm {d} \lambda (x)}$ ${\displaystyle \lambda }$

The modular function

→ Main article : Modular function (harmonic analysis)

If a (left-invariant) Haar's measure, then also the assignment , where is a fixed group element. Because of the uniqueness of the Haar measure, there is a positive real number such that ${\displaystyle \mu }$ ${\displaystyle A\mapsto \mu (Ag)}$ ${\displaystyle g}$ ${\displaystyle \Delta (g)}$

${\displaystyle \mu (Ag)=\Delta (g)\mu (A)}$.

${\displaystyle \Delta }$is a continuous group homomorphism from the group to the multiplicative group of positive real numbers called a modular function . measures how much a (left) hair measure is also right invariant; and a group is unimodular if and only if its modular function is constant. ${\displaystyle \Delta }$

literature

• Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis. D. van Nostrand Co., Toronto et al. 1953.
• Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 .

Individual evidence

1. ↑ Nicolas Bourbaki : VI. Integration (=  Elements of Mathematics ). Springer , Berlin 2004, ISBN 3-540-20585-3 , VII, p. 6 ff .
2. ^ Gerald B. Folland : Real Analysis . Modern Techniques and Their Applications. 2nd Edition. John Wiley & Sons, New York 1999, ISBN 0-471-31716-0 , pp. 342 ff .