Question

Let \(\mathbb{Q}\) have the relative topology induced from \(\mathbb{R}\). Then \(\mathbb{Q}\) is a topological group that is not locally compact, and there is no nonzero translation-invariant Borel measure on \(\mathbb{Q}\) that is finite on compact sets.

Short answer

\( \mathbb{Q} \) is not locally compact and lacks a finite translation-invariant Borel measure.

Step-by-step solution

Defining the Relative Topology

The rational numbers \( \mathbb{Q} \) are given the relative topology from \( \mathbb{R} \). This means that the open sets in \( \mathbb{Q} \) are intersections of open sets in \( \mathbb{R} \) with \( \mathbb{Q} \). As a subspace of \( \mathbb{R} \), which is a topological group, \( \mathbb{Q} \) inherits a group structure and two operations (addition and negation) which are continuous. Thus, \( \mathbb{Q} \) forms a topological group.

Understanding Local Compactness

To determine if \( \mathbb{Q} \) is locally compact, recall a space is locally compact if every point has a compact neighborhood. In \( \mathbb{R} \), intervals like \( [a, b] \) are compact. However, in \( \mathbb{Q} \), there exist no non-trivial intervals that are compact because any such interval would have to contain irrational numbers (which are not in \( \mathbb{Q} \)), hence fail to be closed in \( \mathbb{Q} \). Therefore, \( \mathbb{Q} \) is not locally compact.

Examining Borel Measures

A Borel measure on \( \mathbb{Q} \) should be translation-invariant, but for \( \mathbb{Q} \) neither being locally compact nor having compact elements restricts existence of such measures that are finite. As \( \mathbb{Q} \) is dense in \( \mathbb{R} \) and non-sigma compact, translation-invariance would imply that any nonzero measure would assign infinite measure to densely populated open sets, invalidating it being finite on compact subsets.

Key concepts

Relative Topology

In mathematics, when we talk about the **relative topology**, we're looking at a way to create a topology for a subset based on the topology of a larger space that contains it. Let's take a deeper look at this concept using the rational numbers, \( \mathbb{Q} \), as an example.

In this exercise, \( \mathbb{Q} \) is given the relative topology from the real numbers \( \mathbb{R} \). What this means is that the open sets in \( \mathbb{Q} \) are defined based on the open sets in \( \mathbb{R} \). More precisely, an open set in \( \mathbb{Q} \) would be the intersection of an open set in \( \mathbb{R} \) with \( \mathbb{Q} \) itself.

This concept helps us understand how \( \mathbb{Q} \), contained in \( \mathbb{R} \), can inherit certain properties like:

• **Continuity** of operations: Since \( \mathbb{R} \) is a topological group, \( \mathbb{Q} \) also inherits this group structure, including operations such as addition and negation.

These operations remain continuous in the relative topology, making \( \mathbb{Q} \) a topological group albeit with some unique features when compared to \( \mathbb{R} \).

This understanding is crucial when examining how smaller subsets of mathematical spaces behave under the influence of a larger set's topology.

Local Compactness

Local compactness is an important concept in topology. A space is **locally compact** if each point has a neighborhood that is compact. Compactness in a topological setup is similar to the idea of closeness and boundedness.

For \( \mathbb{R} \), many intervals like \( [a, b] \) are compact, meaning they are both closed and bounded. But, when we consider \( \mathbb{Q} \) with its relative topology, things change.

In \( \mathbb{Q} \), no non-trivial interval can truly be compact because it would include irrational numbers absent in \( \mathbb{Q} \), thus making it not closed. This absence of a compact neighborhood around each point implies \( \mathbb{Q} \) is not locally compact.

This nature has significant implications, particularly when considering structures and functions that rely on compactness, such as certain types of measures or transformations. Understanding why \( \mathbb{Q} \) is not locally compact helps to demystify why certain structures behave the way they do within the space.

Borel Measure

A **Borel measure** is a way to assign sizes or "measures" to sets in a topological space. In our context, we're exploring Borel measures specifically on the topological group of rational numbers \( \mathbb{Q} \).

For a measure on \( \mathbb{Q} \) to be both translation-invariant and finite on compact sets, several traits should align, yet they do not in \( \mathbb{Q} \).

Here’s why:

• **Translation-Invariance**: This property means that the measure does not change if the entire set is shifted. It's like the set being moved around but having the same "size."
• **Finite on Compact Sets**: Typically requires locally compact spaces, which \( \mathbb{Q} \) is not.
• **Density and Non-Sigma Compactness**: Since \( \mathbb{Q} \) is dense in \( \mathbb{R} \) and not sigma compact, meaning it cannot be expressed as a countable union of compact sets. This results in any translation-invariant measure becoming infinite when trying to measure non-trivial sets.

This disconnection makes it impossible for any nonzero measure to remain finite on compact sets, shedding light on the intricate relationship between measure theory and topological structures.