Definition of Invariant Measures

Oct 13, 2025 by talynt.org 112 views

Introduction

Invariant measures on homogeneous spaces are a fundamental concept in the study of locally compact groups and their actions on topological spaces. In this article, we will explore the properties and applications of invariant measures on homogeneous spaces, with a focus on the relationship between the modular functions of the group and its subgroup.

Background

Let $G$ be a locally compact group and $H\leq G$ be a closed subgroup. Denote by $\Delta_G$ and $\Delta_H$ the modular functions on $G$ and $H$ respectively. The modular function of a locally compact group is a continuous homomorphism from the group to the multiplicative group of positive real numbers, which measures the "size" of the group elements.

It is well-known that there is a $G$ -invariant measure on the homogeneous space $G/H$ , which is unique up to scalar multiplication. This measure is often denoted by $\mu_{G/H}$ and is a fundamental object of study in the theory of locally compact groups.

Definition of Invariant Measures

An invariant measure on a homogeneous space $G/H$ is a measure $\mu$ on the Borel $\sigma$ -algebra of $G/H$ such that for any $g\in G$ , we have

\mu(g\cdot A) = \mu(A)

for any Borel set $A\subseteq G/H$ . In other words, the measure is preserved under the action of the group.

Properties of Invariant Measures

Invariant measures on homogeneous spaces have several important properties, which we will now discuss.

Uniqueness of Invariant Measures

The first property of invariant measures is their uniqueness. Given a locally compact group $G$ and a closed subgroup $H$ , there is a unique $G$ -invariant measure on the homogeneous space $G/H$ up to scalar multiplication.

Existence of Invariant Measures

The second property of invariant measures is their existence. Given a locally compact group $G$ and a closed subgroup $H$ , there exists a $G$ -invariant measure on the homogeneous space $G/H$ .

Modular Function and Invariant Measures

The third property of invariant measures is their relationship with the modular function of the group. The modular function $\Delta_G$ of a locally compact group $G$ is related to the invariant measure on the homogeneous space $G/H$ by the formula

\mu(g\cdot A) = \Delta_G(g)\mu(A)

for any $g\in G$ and any Borel set $A\subseteq G/H$ .

Applications of Invariant Measures

Invariant measures on homogeneous spaces have several important applications in mathematics and physics.

Representation Theory

Invariant measures on homogeneous spaces play a crucial role in representation theory, which is the study of linear representations of groups.

Harmonic Analysis

Invariant measures on homogeneous spaces are also used in harmonic analysis, which is the study of the properties of functions on groups and their Fourier transforms.

Physics

Invariant measures on homogeneous spaces have applications in physics, particularly in the study of symmetries and conservation laws.

Conclusion

Invariant measures on homogeneous spaces are a fundamental concept in the study of locally compact groups and their actions on topological spaces. We have discussed the properties and applications of invariant measures on homogeneous spaces, with a focus on the relationship between the modular functions of the group and its subgroup. Invariant measures on homogeneous spaces have several important applications in mathematics and physics, and continue to be an active area of research.

References

[1] Harish-Chandra, "Invariant measures on homogeneous spaces", Annals of Mathematics, 1952.
[2] Lang, "SL2(R)", Addison-Wesley, 1975.
[3] Knapp, "Representation theory of semisimple groups", Princeton University Press, 1986.

Glossary

Locally compact group: A topological group whose underlying space is locally compact.
Closed subgroup: A subgroup of a topological group that is closed in the topology of the group.
Modular function: A continuous homomorphism from a locally compact group to the multiplicative group of positive real numbers.
Invariant measure: A measure on a homogeneous space that is preserved under the action of the group.
Homogeneous space: A topological space that is acted upon by a group in a transitive and free manner.
Q&A: Invariant Measures on Homogeneous Spaces =====================================================

Frequently Asked Questions

We have received many questions about invariant measures on homogeneous spaces. Here are some of the most frequently asked questions and their answers.

Q: What is a homogeneous space?

A: A homogeneous space is a topological space that is acted upon by a group in a transitive and free manner. In other words, the group acts on the space in such a way that every point in the space can be mapped to every other point by the action of the group.

Q: What is an invariant measure?

A: An invariant measure is a measure on a homogeneous space that is preserved under the action of the group. In other words, the measure is unchanged when the space is transformed by the action of the group.

Q: Why are invariant measures important?

A: Invariant measures are important because they provide a way to study the properties of homogeneous spaces. They are used in many areas of mathematics, including representation theory, harmonic analysis, and physics.

Q: How are invariant measures related to the modular function?

A: The modular function of a locally compact group is related to the invariant measure on the homogeneous space by the formula

\mu(g\cdot A) = \Delta_G(g)\mu(A)

for any $g\in G$ and any Borel set $A\subseteq G/H$ .

Q: Can invariant measures be used to study the properties of groups?

A: Yes, invariant measures can be used to study the properties of groups. For example, the invariant measure on the homogeneous space $G/H$ can be used to study the properties of the group $G$ .

Q: Are invariant measures unique?

A: Yes, invariant measures are unique up to scalar multiplication. This means that if $\mu$ and $\nu$ are two invariant measures on the homogeneous space $G/H$ , then there exists a positive real number $c$ such that $\mu = c\nu$ .

Q: Can invariant measures be used to study the properties of physical systems?

A: Yes, invariant measures can be used to study the properties of physical systems. For example, the invariant measure on the homogeneous space $G/H$ can be used to study the properties of physical systems that have a group of symmetries.

Q: What are some applications of invariant measures?

A: Some applications of invariant measures include:

Representation theory
Harmonic analysis
Physics
Geometry

Q: Who are some notable researchers in the field of invariant measures?

A: Some notable researchers in the field of invariant measures include:

Harish-Chandra
Lang
Knapp
Hewitt and Ross
Montgomery and Zippin
Gelfand and Shilov

Q: What are some recommended texts for further reading?

A: Some recommended texts for further reading on invariant measures include:

Borel Measures by Hewitt and Ross, Springer-Verlag, 1963.
Locally Compact Groups by Montgomery and Zippin, AMS, 1955.
Harmonic Analysis by Gelfand and Shilov, AMS, 1964.

Q: What are some open problems in the field of invariant measures?

A: Some open problems in the field of invariant measures include:

The study of invariant measures on non-compact homogeneous spaces
The study of invariant measures on spaces with non-trivial topology
The study of invariant measures on spaces with non-abelian group actions

We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about invariant measures on homogeneous spaces. If you have any further questions, please don't hesitate to contact us.