Question

If \(\mu\) is a Radon measure on the locally compact group \(G\) and \(f \in C_{c}(G)\), the functions \(x \rightarrow \int L_{x} f d \mu\) and \(x \rightarrow \int R_{x} f d \mu\) are continuous.

Short answer

The functions \( x \mapsto \int L_x f \, d\mu \) and \( x \mapsto \int R_x f \, d\mu \) are continuous due to properties of Radon measures and compact supports.

Step-by-step solution

Understand the Definitions

Here, \( G \) is a locally compact group, meaning it is both locally compact and a topological group. A Radon measure \( \mu \) is a measure defined on the Borel \( \sigma \)-algebra of \( G \). The function \( f \in C_c(G) \) is continuous and has compact support. Functions \( L_xf \) and \( R_xf \) are the left and right translations of \( f \) by \( x \) respectively, defined as \( L_xf(y) = f(x^{-1}y) \) and \( R_xf(y) = f(yx) \). We are to prove that the functions \( x \rightarrow \int L_x f \, d\mu \) and \( x \rightarrow \int R_x f \, d\mu \) are continuous.

Define Translations Under the Measure

The mappings are defined as follows: for every element \( x \in G \), the function \( L_xf(y) = f(x^{-1}y) \) and similarly for \( R_xf(y) = f(yx) \). We observe here that both \( L_xf \) and \( R_xf \) also belong to \( C_c(G) \) because translation by an element of \( G \) doesn't change the continuity or the compactness of the support of \( f \).

Use Properties of the Radon Measure

A Radon measure \( \mu \) integrates functions in \( C_c(G) \), and these functions are dense in \( G \). Integrals involving Radon measures combine continuity and compact support, ensuring the expression \( g \mapsto \int g \, d\mu \) is continuous when \( g \) varies continuously in \( C_c(G) \). Thus, for any \( x \in G \), \( \int L_xf \, d\mu = \int f(x^{-1}y) \, d\mu(y) \) and \( \int R_xf \, d\mu = \int f(yx) \, d\mu(y) \), both transformations preserve continuity.

Prove Continuity at a Point

Let \( x_0 \) be a point in \( G \). For continuity, consider any sequence \( x_n \to x_0 \) in \( G \). We must show that \( \int L_{x_n} f \, d\mu \to \int L_{x_0} f \, d\mu \) and similarly for \( R_xf \. d\mu \). This follows from the dominated convergence theorem and the properties of \( f \) having compact support. Since \( f \) and its translations are uniform in continuity and dominated by integrable functions, by convergence their integrals converge as well.

Conclude with Full Continuity

The conclusions of the previous steps ensure that both functions defined by \( x \mapsto \int L_x f \, d\mu \) and \( x \mapsto \int R_x f \, d\mu \) are continuous mappings from \( G \) to \( \mathbb{C} \). The use of translations does not disrupt the topology or measure space properties, affirming the initial claim of continuity.

Key concepts

Locally Compact Group

A locally compact group is a fascinating concept that brings together the features of local compactness and group theory. When we say a group is locally compact, we mean that each point within the group has a compact neighborhood. This is particularly important in analysis and topology.

Locally compact groups encompass many well-known structures, including:

• Lie groups, which are used to study continuous symmetry.
• Matrix groups, important in physics and engineering.
• Topological groups with additional properties relevant to measure theory.

These groups possess a topology where multiplication and inversion operations are continuous. Consequently, they play a pivotal role in harmonic analysis and representation theory, because the topological group structure aligns perfectly with analytical concepts. To sum it up, when working with locally compact groups, you're engaging with a rich field that melds algebraic and topological insights.

Continuous Functions

Continuous functions often form the backbone of analysis. Within a locally compact group, a continuous function is one where small changes in input result in small changes in output. This basic idea is easily visualized but holds deep implications when used in more complex mathematical settings.

In the context of a locally compact group:

• Continuous functions are smooth and have no abrupt changes.
• Their behavior remains manageable even over infinite domains, thanks to the compact support.
• They often map complex group elements into more tangible real or complex values, helping to bridge the gap between algebraic operations and analytical insights.

A significant property of such functions is their ability to integrate well under Radon measures, maintaining continuity when translated across the group. This attribute is crucial when determining the continuity of composed functions in measure-theoretic contexts.

Topological Group

Topological groups can be viewed as a harmonious blend of group theory and topology. These groups have a structure where the operation of addition or multiplication conforms to topological rules, offering continuity of both arithmetic operations.

Key features of a topological group include:

• The group operation (typically addition or multiplication) is continuous.
• Inverse functions within the group are also continuous, making the group operations compatible with the topological structure.
• They provide a flexible framework for analyzing properties such as homomorphisms, subgroup formations, and invariant measures.

Because of these properties, topological groups serve as an integrated setting for developing both algebraic and topological theorems. They are indispensable in deeper studies like that of Haar measure, which relies on the underlying topological group structure to enable measure definition across the entire group.

Borel σ-algebra

The Borel σ-algebra is a fundamental concept in the realm of measure theory and topology. Formed from the open sets within a topological space, it includes sets that can be constructed through countable unions, intersecting, and complementing of open sets.

Important aspects of Borel σ-algebras are:

• They provide the foundation for defining measures, such as Radon and Lebesgue measures, on a topological space.
• They encapsulate measurable functions, which are vital for integrating functions over given spaces.
• Borel sets are essential in probability theory, as they define the events for which probabilities are assigned.

When applied to locally compact groups, the Borel σ-algebra serves as the stage on which Radon measures act. This, in turn, plays a crucial role in establishing continuity for the integrals of translated functions within these groups.