Question

Show that the invariant Haar measure for a compact group satisfies \(d \boldsymbol{\mu}_{g}=d \boldsymbol{\mu}_{g^{-1}}\). Hint: Define a measure \(\boldsymbol{v}\) by \(d \boldsymbol{v}_{g} \equiv d \boldsymbol{\mu}_{g^{-1}}\) and show that \(\boldsymbol{v}\) is left-invariant. Now use the uniqueness of the left-invariant Haar measure for compact groups.

Short answer

The invariant Haar measure for a compact group is indeed the same when the group element is replaced with its inverse, i.e., \(d \boldsymbol{\mu}_{g}=d \boldsymbol{\mu}_{g^{-1}}\).

Step-by-step solution

Define a new measure

As per the hint, define a new measure \(\boldsymbol{v}\), where \(d \boldsymbol{v}_{g} = d \boldsymbol{\mu}_{g^{-1}}\)

Show \(\boldsymbol{v}\) is left-invariant

For a measure to be left-invariant, it must hold true that for every \(h\) and \(g\) in our group, \(d \boldsymbol{v}_{h g} = d \boldsymbol{v}_{g}\). By the definition of \(\boldsymbol{v}\), we have \(d \boldsymbol{v}_{h g} = d \boldsymbol{\mu}_{(hg)^{-1}} = d \boldsymbol{\mu}_{g^{-1} h^{-1}} = d \boldsymbol{v}_{g}\) using the properties of group inverse elements. This shows \(\boldsymbol{v}\) is indeed left-invariant.

Apply uniqueness of left-invariant Haar measure

The uniqueness means that any two left-invariant Haar measures on a compact group differ by a positive constant factor. As \(\boldsymbol{v}\) and \(\boldsymbol{\mu}\) are both left-invariant measures, by uniqueness, we have \(d \boldsymbol{v}_{g} = c \cdot d \boldsymbol{\mu}_{g}\) for some constant \(c\). But comparing this with our definition of \(\boldsymbol{v}\) reveals \(d \boldsymbol{\mu}_{g^{-1}} = c \cdot d \boldsymbol{\mu}_{g}\) implying \(c = 1\) as these measures are normalised. Thus, \(d \boldsymbol{\mu}_{g}=d \boldsymbol{\mu}_{g^{-1}}\).

Key concepts

Invariant Measure

The concept of an invariant measure is fundamental in understanding the notions of symmetry and balance within a group structure. Specifically, in our context, an invariant measure refers to a measure that remains unchanged when the group operations are applied. To elaborate:

• An invariant measure could be either left-invariant or right-invariant.
• For left-invariance, it means that the measure remains constant when elements of the group are multiplied from the left side.
• In mathematical terms, for a measure \(\mu\) to be left-invariant, it must satisfy: \(d \mu_{hg} = d \mu_{g}\) for all group elements \(h\) and \(g\).
• This property is crucial because it ensures that the measure "respects" the structure of the group, allowing a consistent way to measure across the entire group.

By leveraging this principle, one can find that for any measure \(\boldsymbol{v}\) defined in terms of another measure (like \(\boldsymbol{\mu}\)), if it shows left-invariance using the group properties, it indicates symmetry and balance within the group's operations.

Compact Group

A compact group is an essential concept within topology and group theory. To understand why they are important, it's helpful to look at their defining properties:

• Compactness refers to a group having a topology where every open cover has a finite subcover. Intuitively, you can think of compact spaces as those that are "bounded" and "closed."
• "Closed" means the group contains all its limit points, while "bounded" suggests it does not stretch off to infinity.
• This property enables compact groups to have a Haar measure that possesses uniqueness, ensuring any invariant measure can be consistently defined.
• Furthermore, characteristics like continuity and completeness often accompany compactness, which are useful when analyzing group operations.

In the context of Haar measure, compact groups ensure that there exists at least one measure that is invariant, providing a robust framework for analysis and further mathematical exploration.

Uniqueness of Measure

The uniqueness of the Haar measure is a cornerstone result for compact groups. This uniqueness means more than just simplicity in calculation; it embodies key aspects:

• For any compact group, there exists a unique (up to scaling by a positive constant) measure that is left-invariant.
• This implies that if two measures \(\mu\) and \(u\) are both left-invariant for the same group, they must be related by a constant factor.
• The uniqueness of the Haar measure allows mathematicians to work with familiar and consistent tools when analyzing different groups, guaranteeing that any calculations or integrals are independent of the specific form of the measure.
• In practical terms, when comparing measures like \(\boldsymbol{\mu}_{g}\) and its inverse \(\boldsymbol{\mu}_{g^{-1}}\), understanding uniqueness helps affirm their equivalency within the framework of invariant measures.

This property of uniqueness ensures robustness in mathematical formalism, underpinning deeper theories within both group and functional analysis.