Haar Measure

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tags: - colorclass/functional analysis ---see also: - Measure

The Haar measure is a fundamental concept in the field of analysis on topological groups, named after Alfréd Haar, a Hungarian mathematician who introduced the concept in the early 20th century. It provides a way to define a “volume” for subsets of a topological group, extending the idea of length, area, and volume in Euclidean spaces to a much broader context. The Haar measure is particularly notable for being invariant under the group operations, making it an indispensable tool in harmonic analysis, representation theory, and other areas of mathematics and physics.

Definition and Properties

A Haar measure on a locally compact topological group $G$ is a measure $\mu$ that satisfies the following properties:

1. Left Invariance: For any measurable set $A\subseteq G$ and any element $g\in G$, the measure of $A$ and the measure of its left translate $gA=\{ga:a\in A\}$ are the same. That is, $\mu (gA)=\mu (A)$ for all $g\in G$ and for all measurable sets $A$.

2. Regularity: The Haar measure is regular, meaning that the measure of any measurable set can be approximated both from above and from below by open and compact sets, respectively.

3. Non-triviality: The Haar measure is not identically zero and does not assign infinite measure to non-empty open sets, ensuring that it meaningfully distinguishes between different sizes of sets.

The Haar measure is unique up to a multiplicative constant, meaning that if $\mu$ and $\nu$ are two Haar measures on $G$, then there exists a positive constant $c$ such that $\nu =c\mu$.

Existence

The existence of the Haar measure is guaranteed for every locally compact topological group, a result that significantly generalizes the concept of volume and enables the integration of functions over groups. This broad applicability makes the Haar measure a central object in analysis and its related fields.

Applications

- Harmonic Analysis: The Haar measure allows for the formulation and study of Fourier transforms and other analytical tools on groups, facilitating the analysis of functions defined on the group.

- Representation Theory: In studying the representations of groups, especially Lie groups, the Haar measure provides a way to define and work with integrals over the group, which are crucial for understanding the structure of representations.

- Ergodic Theory: The Haar measure plays a role in ergodic theory, where it is used to study dynamical systems with a group action. It is essential for the formulation of ergodic theorems, which relate time averages and space averages of functions.

- Probability Theory: In probability, the Haar measure can be used to define uniform distributions on groups, providing a foundational tool for the study of random processes with symmetries.

- Physics: In theoretical physics, especially in the study of quantum mechanics and field theory, the Haar measure is used in path integrals and to define measures on the space of states or paths, reflecting physical symmetries.

The Haar measure’s invariance property makes it particularly suited to studying systems and functions that exhibit symmetrical behavior, providing a powerful and versatile tool across a wide range of mathematical and physical disciplines.