Question

Let \(\mu\) be a Radon measure on \(X\). a. Let \(N\) be the union of all open \(U \subset X\) such that \(\mu(U)=0\). Then \(N\) is open and \(\mu(N)=0\). The complement of \(N\) is called the support of \(\mu\). b. \(x \in \operatorname{supp}(\mu)\) iff \(\int f d \mu>0\) for every \(f \in C_{c}(X,[0,1])\) such that \(f(x)>0\).

Short answer

a) \( N \) is open with \( \mu(N)=0 \). b) \( x \in \operatorname{supp}(\mu) \) iff \( \int f \, d\mu>0 \) for certain \( f \).

Step-by-step solution

Understanding Radon Measure

A Radon measure \( \mu \) on \( X \) assigns a non-negative value to each open set in a \( \sigma \)-algebra \( \mathcal{B}(X) \) and satisfies measure-related properties such as being a Borel measure and regular.

Analyzing Part (a): Defining Set N

Let \( N \) be defined as the union of all open sets \( U \subset X \) for which \( \mu(U) = 0 \). Since \( N \) is a union of open sets, \( N \) itself is open. Because each \( U \) in this union has measure zero, their union \( N \) also has measure zero, as \( \mu \) is countably additive.

Defining Support of the Measure

The support of a measure \( \mu \), denoted \( \operatorname{supp}(\mu) \), is the set of points not contained in \( N \), i.e., the complement of \( N \) in \( X \). Thus, \( \operatorname{supp}(\mu) = X \setminus N \).

Analyzing Part (b): Using Test Functions

To show that \( x \in \operatorname{supp}(\mu) \), consider test functions \( f \in C_c(X, [0,1]) \), continuous with compact support and taking values in \([0,1]\). We need that for all such \( f \), if \( f(x) > 0 \), then \( \int f \, d\mu > 0 \).

Checking the Condition for Support

Suppose \( x \in \operatorname{supp}(\mu) \), implying \( x otin N \). Thus, there is no open neighborhood of \( x \) with measure zero. For any function \( f \) with \( f(x) > 0 \), \( f \) must be positive on a neighborhood of \( x \), forcing \( \int f \, d\mu > 0 \). Conversely, if some \( f \) with \( f(x) > 0 \) had \( \int f \, d\mu = 0 \), then \( x \) could be in an open set of measure zero, contradicting \( x \in \operatorname{supp}(\mu) \).

Key concepts

Support of a Measure

In measure theory, the support of a measure is a crucial concept that helps us understand where the measure is "active" or "significant" on a given space. To define it formally, consider a Radon measure \( \mu \) on a topological space \( X \). The support, denoted \( \operatorname{supp}(\mu) \), is defined as the complement of the largest open set \( N \) for which \( \mu(N) = 0 \). In simpler terms, \( \operatorname{supp}(\mu) = X \setminus N \), where \( N \) is the union of all open sets with measure zero.

Understanding the support is important because it indicates the least subset of \( X \) outside of which the measure \( \mu \) is zero. It means that outside the support, the measure does not "recognize" or "feel" the set: any test function or continuous function with compact support that is non-zero at any point on the support will have a non-zero integral with respect to the measure. Conversely, on the support, the measure is positive for at least some local "activities."

• Closed set in the smallest \( \sigma \)-algebra.
• Defines "boundaries" of where \( \mu \) can be active.

Borel Measure

A Borel measure is a type of measure defined on the Borel \( \sigma \)-algebra of a topological space \( X \). The Borel \( \sigma \)-algebra, \( \mathcal{B}(X) \), is generated by the open sets of \( X \) and includes all the open and closed sets, as well as any countable intersections and unions thereof. This makes it essential for discussing measures in a way that respects the topology of the space.

Borel measures are foundational in probability theory and real analysis because they align measures with functions and sets that we can analyze through limits and continuity. A Radon measure, which is a Borel measure as described in the exercise, is not just defined on Borel sets but also regular. This means:

• Inner regularity: \( \mu(A) = \sup\{ \mu(K) : K \subset A, K \text{ compact}\} \).
• Outer regularity: \( \mu(A) = \inf\{ \mu(U) : A \subset U, U \text{ open}\} \).

These properties allow Radon measures to be extended from Borel measures with detailed and precise limitations, thus interacting properly with the space's topology.

Measure Zero

In the context of measure theory, a set having measure zero is intuitively "negligible" with respect to a given measure. If a set \( U \) in a space \( X \) is such that \( \mu(U) = 0 \), then it means that \( U \) essentially makes no contribution to the overall measure in the space. For example, although the rational numbers in the interval \([0, 1]\) are dense, they have Lebesgue measure zero.

The idea of measure zero is essential when dealing with Radon and Borel measures because it helps segregate important regions in a space from negligible ones. Sets of measure zero become especially meaningful in contrast with the support of the measure:

• Parts of the space where the measure "ignores" or is inactive.
• Relevant in integration: functions supported on measure zero sets integrate to zero.

A key property is their countable additivity, which means any countable union of sets with measure zero still has measure zero— crucial for demonstrating various properties of functions and sets in analysis.