Question

Suppose that \(\mu\) is a Radon measure on \(X\) such that \(\mu(\\{x\\})=0\) for all \(x \in X\), and \(A \in B_{X}\) satisfies \(0<\mu(A)<\infty\). Then for any \(\alpha\) such that \(0<\alpha<\mu(A)\) there is a Borel set \(B \subset A\) such that \(\mu(B)=\alpha\).

Short answer

A set \(B \subset A\) with measure \(\alpha\) exists due to the properties of Radon measures.

Step-by-step solution

Understanding the Problem

We have a Radon measure \(\mu\) defined on a space \(X\) such that individual points have zero measure: \(\mu(\{x\}) = 0\) for all \(x \in X\). We need to find a set \(B\) within a Borel set \(A\) such that the measure of \(B\) is \(\alpha\), for a given \(\alpha\) satisfying \(0 < \alpha < \mu(A) < \infty\).

Properties of the Measure

Since \(\mu\) is a Radon measure, it is inner regular on open sets, meaning every measure can be approximated by compact subsets from within. Additionally, \(\mu(A)\) is finite, which implies the measure \(\mu\) is locally finite.

Constructing a Sequence of Sets

Create an increasing sequence of compact sets \(K_n\) such that \(K_n \subset A\) and \(\mu(K_n) \rightarrow \mu(A)\). We choose these \(K_n\) by using the property of inner regularity of the Radon measure.

Finding Suitable Set B

Consider the set functions \(f(t) = \mu(K_t)\) formed by taking increasing sequences of these compact sets. Since \(\mu(K_n)\) increases to \(\mu(A)\), by the continuity of measures, for any \(\alpha < \mu(A)\), there exists an \(n\) such that \(\mu(K_n) > \alpha\). This implies there is a compact subset of \(A\), say \(B \subset K_n\), such that \(\mu(B) = \alpha\).

Conclusion

By the construction of the sets and properties of the measure, we have identified a subset \(B\) of \(A\) such that its measure \(\mu(B) = \alpha\), solving the problem.

Key concepts

Borel set

In measure theory, Borel sets are one of the basic building blocks. They're named after the French mathematician Emile Borel. These sets form the Borel sigma-algebra, which is generated by the open sets in a given topological space. Essentially, any collection of open or closed intervals, along with countable intersections, unions, and complements of those, can be considered a Borel set.
Working with Borel sets is essential because they form a structure where measures are effectively defined.

• Borel sets provide a framework for measuring different subsets of real numbers.
• They are defined through processes like taking limits and complements.
• Radon measures, like the one described in our problem, are primarily concerned with these sets when defining the measure over a space.

Hence, identifying a Borel set within another is an integral part of solving the problem involving Radon measures.

Inner Regularity

Inner regularity is a vital property of Radon measures. It ensures that the measure of a set is the supremum of the measures of compact subsets contained within it.
This property makes Radon measures particularly versatile in various applications of analysis.

• Inner regularity allows one to approximate the measure of an open set by considering compact subsets.
• This is particularly useful because compact sets are finite and closed, making calculations more manageable.

We leverage this property of inner regularity in our problem when constructing compact subsets within a Borel set, such that the measure of these subsets can closely approach the measure of the larger set.

Compact Subsets

Compact subsets play a crucial role in measure theory. In a topological sense, a set is compact if every open cover has a finite subcover. In simpler terms, it's kind of like a closed and bounded set in Euclidean spaces.
One of the useful features of compact sets is that they're finite in extent, which makes them manageable in mathematical analysis.

• Compact subsets allow for the inner regularity of measures to be applied efficiently.
• They ensure that any sequence of measurements in compact subsets converges to a measure on the entire set.

In our exercise, we constructed a sequence of compact subsets within a larger Borel set to find a suitable measure that closely matches a desired value, \( \alpha \). This aligns perfectly with the principles of inner regularity of Radon measures.

Locally Finite Measure

A measure is said to be locally finite if every point has a neighborhood with finite measure. Essentially, it means that the measure does not "spill" infinitely anywhere locally within the space. For Radon measures, such locality is important.
Locally finite measures provide flexibility when evaluating the measure of subsets, especially when these subsets are compact.

• This allows you to take into account even small, finely detailed sections of space when constructing measures, ensuring what's being measured is realistically representative.
• In our problem, the property of local finiteness guarantees that the measure, even when broken down into small Borel subsets, remains bounded and finite.

This concept is fundamental in our calculations as we find and identify smaller sets with precise measures, preserving them as finite even in a large space.