14.3 Haar measure and invariant integration

Haar measure and invariant integration are crucial concepts in harmonic analysis on groups. They provide a way to measure sets and integrate functions that respects the group structure. This allows us to extend familiar notions from calculus to more abstract settings.

The Haar measure is unique up to a constant and has important properties like regularity and local finiteness. Invariant integration, based on the Haar measure, preserves the group's symmetry. These tools are fundamental for studying harmonic functions on groups.

Haar Measure and Invariance

Definition and Properties of Haar Measure

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• Haar measure is a non-zero, left-invariant, countably additive measure defined on a locally compact group
• Unique up to a positive multiplicative constant, meaning any two Haar measures on a group differ only by a positive scalar factor
• Haar measure is regular, which means that the measure of any set can be approximated from above by open sets and from below by compact sets
• Haar measure is not necessarily finite, but it is always locally finite (every point has a neighborhood with finite measure)

Left and Right Invariance

• Left-invariant measure satisfies $\mu(gE) = \mu(E)$ for all measurable sets $E$ and group elements $g$, where $gE = \{gx : x \in E\}$
• Right-invariant measure satisfies $\mu(Eg) = \mu(E)$ for all measurable sets $E$ and group elements $g$, where $Eg = \{xg : x \in E\}$
• On a locally compact abelian group, the left and right Haar measures coincide, making the Haar measure both left and right-invariant
• For non-abelian groups, left and right Haar measures may differ, but they are always equivalent (mutually absolutely continuous)

Invariant Integration

• Invariant integration is the process of integrating functions with respect to a Haar measure
• For a left-invariant Haar measure $\mu$ and a function $f$, the left-invariant integral is defined as $\int f(x) d\mu(x)$
• Left-invariant integrals satisfy the invariance property $\int f(gx) d\mu(x) = \int f(x) d\mu(x)$ for all group elements $g$
• Right-invariant integrals can be defined similarly and satisfy the corresponding invariance property $\int f(xg) d\mu(x) = \int f(x) d\mu(x)$

Modular Function and Unimodularity

Modular Function

• The modular function $\Delta$ of a locally compact group $G$ is a homomorphism from $G$ to the multiplicative group of positive real numbers, defined by the relation $\mu(Eg) = \Delta(g)\mu(E)$ for all measurable sets $E$ and group elements $g$
• Measures the discrepancy between left and right Haar measures, as the left Haar measure $\mu_L$ and the right Haar measure $\mu_R$ are related by $\mu_R(E) = \Delta(g^{-1})\mu_L(E)$
• The modular function is continuous and satisfies the cocycle condition $\Delta(gh) = \Delta(g)\Delta(h)$ for all group elements $g$ and $h$
• Examples of groups with non-trivial modular functions include the affine group of the real line and the group of invertible upper triangular matrices

Unimodular Groups

• A locally compact group is called unimodular if its modular function is identically equal to 1, meaning that the left and right Haar measures coincide
• Compact groups, abelian groups, and discrete groups are always unimodular
• Examples of unimodular groups include the real numbers under addition, the circle group, and the group of invertible matrices with determinant 1 (special linear group)
• Many important results in harmonic analysis and representation theory simplify for unimodular groups due to the symmetry between left and right invariance

Representation Theorem

Riesz Representation Theorem

• The Riesz representation theorem establishes a correspondence between positive linear functionals on the space of continuous functions with compact support and Radon measures
• Specifically, for a locally compact Hausdorff space $X$, every positive linear functional $\Lambda$ on the space $C_c(X)$ of continuous functions with compact support can be uniquely represented as integration against a Radon measure $\mu$, i.e., $\Lambda(f) = \int f d\mu$ for all $f \in C_c(X)$
• The representing measure $\mu$ is uniquely determined by the functional $\Lambda$ and is called the Riesz representer of $\Lambda$
• The Riesz representation theorem is a key tool in the construction and study of Haar measures, as it allows one to define a Haar measure by specifying a suitable invariant positive linear functional
• Applications of the Riesz representation theorem extend beyond harmonic analysis to areas such as probability theory, where it is used to characterize expectation functionals, and functional analysis, where it relates linear functionals to measures