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Chapter 6: Problem 5

We wish to consider the complex linear span of Radon integrals on \(X\). This will be referred to as the space of Radon charges on \(X\), and consists of certain linear functionals on \(C_{c}(X)\). Equipped with the weak topology \(\tau\) induced by the seminorms \(f \rightarrow\left|\int f\right|\), where \(\int\) ranges over all Radon integrals on \(X\), it follows from \(2.4 .1\) and \(2.4 .4\) that the Radon charges constitute the dual space of the topological vector space \(\left(C_{c}(X), \tau\right) .\) Clearly, a more constructive description of these elements is desirable. We therefore define a Radon charge to be a functional \(\Phi\) on \(C_{c}(X)\), which for every \(f\) in \(C_{c}(X)_{+}\)satisfies $$ \sup \left\\{\left|\Phi(g) \| g \in C_{c}(X),\right| g \mid \leq f\right\\}<\infty . $$

Short Answer

Expert verified
Radon charges are linear functionals on \( C_c(X) \) satisfying boundedness criteria with respect to a weak topology induced by Radon integrals.

Step by step solution

01

Understanding Radon Charges

A Radon charge on a space \( X \) is defined as a linear functional \( \Phi \) on the space of continuous, compactly supported functions \( C_c(X) \). To be considered a Radon charge, \( \Phi \) must satisfy the condition: for each non-negative function \( f \in C_c(X)_+ \), the supremum of \( |\Phi(g)| \) over functions \( g \in C_c(X) \) with \( |g| \leq f \) is finite.
02

Relating to the Weak Topology

The weak topology \( \tau \) on \( C_c(X) \) is defined by the seminorms \( f \rightarrow \left| \int f \right| \), where the integrals range over all Radon integrals on \( X \). This implies that the Radon charges form the continuous dual space of the topological vector space \( (C_c(X), \tau) \).
03

Using Conditions from Given Theorems

From theorems 2.4.1 and 2.4.4, it follows that the Radon charges are precisely those linear functionals on \( C_c(X) \) that respect the constraints imposed by the weak topology. The condition given in the problem ensures that \( \Phi \) remains bounded over functions whose absolute value is dominated by a given function \( f \).
04

Constructive Description of Radon Charges

To describe Radon charges constructively, identify linear functionals that are bounded for all test functions \(g\), such that any \(g\) with \( |g| \leq f \) for some \( f \in C_c(X)_+ \) satisfies the given supremum condition. This ultimately means \( \Phi \) measures how \(|\Phi(g)|\) scales with \(f\), aligning with how integration measures scale over domain complexity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weak Topology
In mathematics, weak topology is often used to understand the behavior of spaces by observing continuous mappings. Rather than focusing on the space itself, weak topology emphasizes the collection of seminorms or linear functionals that define it. For the context of Radon charges, the weak topology \( \tau \) on \( C_c(X) \) is determined by seminorms of the form \( f \rightarrow \left| \int f \right| \), where \( \int \) represents various Radon integrals.
This approach is useful as it allows the space to be studied via its continuous dual space. Specifically, the weak topology provides a framework where we study how close functions are by observing their behavior through integrals. It essentially captures convergences that might not be apparent by looking at the functions alone.
Key points to remember about weak topology include:
  • **It is defined by seminorms and focuses on continuous mappings.**
  • **It allows a different form of convergence than traditional topology, which is often more practical in functional analysis.**
  • **For Radon charges, it provides the structure to understand them as the dual of the space \( C_c(X) \).**
Topological Vector Space
A topological vector space is a combination of algebraic and topological structures. It seamlessly blends the notions of vector spaces and topological spaces, making it a crucial concept in functional analysis. This allows mathematicians to talk about continuous linear transformations in a geometrically intuitive way.
In the case of Radon charges, the space \( (C_c(X), \tau) \) forms a topological vector space. Here, \( C_c(X) \) is the vector space of continuous functions with compact support on \( X \), while \( \tau \) is the weak topology imposed by Radon integrals.
How this concept helps:
  • **It allows the study of limits and continuity in infinite-dimensional spaces.**
  • **Combines algebraic and geometric intuitions, making abstract concepts easier to understand.**
  • **Facilitates the description of dual spaces, which is essential in understanding Radon charges as demonstrated by the weak topology \( \tau \).**
Linear Functional
Linear functionals are fundamental tools in abstract analysis, connecting algebraic properties with geometry. A linear functional is a function that maps a vector space to its underlying field鈥攗sually the real or complex numbers鈥攁nd satisfies the properties of additivity and homogeneity.
For Radon charges, each \( \Phi \) is a linear functional on \(C_c(X)\). This means \( \Phi(f + g) = \Phi(f) + \Phi(g) \) and \( \Phi(\alpha f) = \alpha \Phi(f) \) for functions \(f, g\) and scalar \(\alpha\). The uniqueness of Radon charges lies in their additional boundedness condition: the supremum of \( |\Phi(g)| \) over certain functions must be finite.
Benefits of understanding linear functionals include:
  • **They provide a simple way to define dual spaces.**
  • **Link algebraic operations with continuity, particularly in spaces of functions.**
  • **Facilitate complex analysis and integration within topological vector spaces.**
Radon Integral
The Radon integral is a generalization of the classical Lebesgue integral, designed to handle measures in more general topological spaces. It's useful in the context of locally compact spaces, like those dealt with in Radon charges.
In Radon charges, integrals of the form \( \int f \) with respect to Radon measures define a significant class of seminorms. These integrals assess the cumulative 'size' or 'mass' of a function over its domain. They are incredibly powerful as they remain well-defined across any compact subsets of the space, enabling a robust theory of measure and integration.
Here are some pivotal features of Radon integrals:
  • **They extend the reach of integration theory to more complex spaces.**
  • **Its structure allows working with "charges" or linear functionals that mimic measures.**
  • **Directly influences the nature of weak topology on \(C_c(X)\), thereby influencing Radon charges.**
By understanding Radon integrals, we grasp the underpinning mechanics of Radon charges and their place in functional analysis.

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Most popular questions from this chapter

Lemma. For a real function \(f\) on \(X \times Y\) and \(y\) in \(Y\), consider the function \(f(\cdot, y)\) on \(X .\) If \(f \in C_{c}(X \times Y)\), then \(f(\cdot, y) \in C_{c}(X) .\) If \(f \in C_{c}(X \times Y)^{m}\), then \(f(\cdot, y) \in C_{c}(X)^{m} .\) If \(f \in \mathscr{B}(X \times Y)\), then \(f(\cdot, y) \in \mathscr{B}(X)\). Similar results hold for each function \(f\left(x,^{\circ}\right), x \in X\). Proof. Let \(t_{y}: X \rightarrow X \times Y\) be the continuous function given by \(l_{y}(x)=(x, y)\). Then \(f(\cdot, y)=f \circ t_{y}\). From this we see immediately that if \(f \in C_{c}(X \times Y)\), then \(f(\cdot, y)\) is continuous. Furthermore, it is clear that the support of \(f(\cdot, y)\) is contained in the projection from \(X \times Y\) onto \(X\) of the support of \(f\), so that \(f(\cdot, y) \in C_{c}(X)\). It follows from the definitions that \(f(\cdot, y) \in C_{c}(X)^{m}\) if \(f \in\) \(C_{c}(X \times Y)^{m}\). Finally, since composition of Borel functions again produce Borel functions, we see from the identity \(f(\cdot, y)=f \circ t_{y}\) that \(f(\cdot, y) \in \mathscr{B}(X)\) if \(f \in \mathscr{B}(X \times Y)\).

For a topological space \(X\) the system \(\mathscr{B}\) of Borel sets is defined as the smallest \(\sigma\)-algebra of subsets of \(X\) that contains all open sets (equivalently, contains all closed sets). The theory of Borel sets is due to Baire and Lebesgue. Taking \(X=\mathbb{R}^{n}\) we know that every open set is the countable union of open boxes, and each such is the intersection of \(2 n\) open half-spaces. Thus, the system of Borel subsets of \(\mathbb{R}^{n}\) is the \(\sigma\)-ring generated by all half-spaces of the form \(\left\\{x \in \mathbb{R}^{n} \mid x_{k}>t\right\\}\), where \(1 \leq k \leq n\) and \(t \in \mathbb{R}\).

Theorem. Given a Radon integral \(\int\) on a locally compact Hausdorff space \(X\), the class \(\mathscr{L}^{1}(X)\) of integrable functions is a vector space containing \(C_{e}(X)\), which is closed under the lattice operations \(\bigvee\) and \(\bigwedge\). Moreover, \(\int: \mathscr{L}^{1}(X) \rightarrow \mathbb{R}\) is a positive functional that extends the original integral on \(C_{c}(X)\). Proof. Simple manipulations with the definition of integrability, in connection with \(6.1 .6\) (and the corresponding result for lower integrals), show that if \(f\) and \(g\) both belong to \(\mathscr{L}^{1}(X)\) and \(t \geq 0\), then \(t f+g \in \mathscr{L}^{1}(X)\) with $$ \int(t f+g)=t \int f+\int g \text {. } $$ Since \(-C_{c}(X)_{m}=C_{c}(X)^{m}\), it is also easy to verify that if \(f \in \mathscr{L}^{1}(X)\), then \(-f \in \mathscr{L}^{1}(X)\) with \(\int-f=-\int f .\) Taken together this means that \(\mathscr{L}^{1}(X)\) is areal vector space, and that \(\int\) is a functional on it. Since \(\int^{*} f=\int_{*} f=\int f\) if \(f \in C_{c}(X)\), cf. 6.1.2, it follows that \(C_{c}(X) \subset \mathscr{L}^{1}(X)\) and that \(\int\) extends the original integral on \(C_{c}(X) .\) Moreover, if \(f \geq 0\), then \(\int * f \geq 0\), so that \(\int\) is a positive functional on \(\mathscr{L}^{1}(X)\). Take \(f_{1}\) and \(f_{2}\) in \(\mathscr{L}^{1}(X)\) and for \(\varepsilon>0\) choose \(g_{1}, g_{2}\) in \(C_{e}(X)^{m}, h_{1}, h_{2}\) in \(C_{c}(X)_{m}\), such that \(h_{k} \leq f_{k} \leq g_{k}\) and \(\int * g_{k}-\int_{*} h_{k}<\varepsilon\) for \(k=1,2 ;\) cf. (*) in 6.1.9. Note that $$ g_{1} \vee g_{2}-h_{1} \vee h_{2} \leq\left(g_{1}-h_{1}\right)+\left(g_{2}-h_{2}\right) $$ so that \(\int^{*} g_{1} \vee g_{2}-\int_{*} h_{1} \vee h_{2}<2 \varepsilon\). It follows that \(f_{1} \vee f_{2} \in \mathscr{L}^{1}(X)\). Similarly, or by exploiting the identity \(f_{1} \wedge f_{2}+f_{1} \vee f_{2}=f_{1}+f_{2}\), we see that \(f_{1} \wedge f_{2} \in \mathscr{L}^{1}(X)\).

Remark. Riesz' representation theorem \((6.3 .4)\) shows that the discussion about what is more important, the measure or the integral, is not of mathematical nature. It may, however, be waged on aestetical, proof- technological, yea even pedagogical, assumptions. There is little doubt (in this author) that the Lebesgue measure on \(\mathbb{R}\) is more basic than the Lebesgue integral. This is marked by using the standard notation \(\int f(x) d x\) for the Lebesgue integral of a function \(f\). This symbolism (going back to Leibniz) is versatile and intuitive [derived from \(\sum f(x) \Delta x\) ]. In particular, it is well suited to describe a change of variables and to distinguish the variables in multiple integrals. It is much more doubtful (for this author), whether general (Radon) measures are more basic than general (Radon) integrals. Moreover, the integral notation is clearly superior to measure notation when we have to derive one measure/integral from another (cf. \(6.5 .4\) and \(6.5 .6\) ). For these reasons the reader will not find the traditional notation \(\int f(x) d \mu(x)\) [or the slightly more "logical" \(\int f(x) \mu(d x)\); not to mention the rather masochistic \(\int d \mu(x) f(x)\) ] for the integral of a function with respect to a measure. The exception being the case of translation invariant, "Lebesgue-like" measures; see 6.6.15.

Lemma. If A is an algebra of real functions on \(X\) that is stable under the lattice operations \(\bigvee\) and \(\bigwedge\), then the monotone sequential completion \(\mathscr{B}(\Omega)\) of A is a sequentially complete algebra of functions, stable under \(\bigvee\) and \(\bigwedge\). Proof. Take \(f\) in \(\mathscr{A}\) and let \(\mathscr{B}_{1}\) denote the class of functions \(g\) in \(\mathscr{B}(\mathscr{A})\) such that \(f+g \in \mathscr{B}(\mathscr{A})\). Since \(\mathscr{A} \subset \mathscr{B}_{1}\) and \(\mathscr{B}_{1}\) is monotone sequentially complete, it follows from the minimality of \(\mathscr{B}(\mathscr{A})\) that \(\mathscr{B}_{1}=\mathscr{B}(\mathscr{A})\). Thus, \(\mathscr{A}+\mathscr{B}(\mathscr{A}) \subset \mathscr{B}(\mathscr{A}) .\) Now take \(g\) in \(\mathscr{B}(\mathscr{A})\), and let \(\mathscr{B}_{2}\) denote the class of functions \(f\) in \(\mathscr{B}(\mathscr{A})\) such that \(f+g \in \mathscr{B}(\mathscr{A})\). We just proved that \(\mathscr{A} \subset \mathscr{B}_{2}\), and since evidently \(\mathscr{B}_{2}\) is monotone sequentially complete, it follows that \(\mathscr{B}_{2}=\mathscr{B}(\mathscr{A})\). Thus, \(\mathscr{B}(\mathscr{A})+\mathscr{B}(\mathscr{A}) \subset \mathscr{B}(\mathscr{A}) .\) The proof that \(R \mathscr{B}(\mathscr{A}) \subset \mathscr{B}(\mathscr{A})\) is similar (but simpler), and we see that \(\mathscr{B}(\mathscr{A})\) is a vector space. With + replaced by \(\bigvee\) or by \(\bigwedge\), the argument above shows that \(\mathscr{B}(\mathscr{A})\) is stable under maximum and minimum. To show that \(\mathscr{B}(\mathscr{A})\) is sequentially complete, let \(\left(f_{n}\right)\) be a sequence in \(\mathscr{B}(\mathscr{A})\) To show that \(\mathscr{B}(\mathscr{A})\) is sequentially complete, let \(\left(f_{n}\right)\) be a sequence in \(\mathscr{A}(\mathscr{A})\) at converges pointwise to some function \(f\) on \(X .\) Since \(f_{n} \vee 0 \rightarrow f \vee 0\) and that converges pointwise to some function \(f\) on \(X .\) Since \(f_{n} \vee 0 \rightarrow f \vee 0\) and \(f_{n} \wedge 0 \rightarrow f \wedge 0\), it suffices to consider the case where \(f_{n} \geq 0\) for all \(n .\) For fixed \(n\) and some \(m>n\) define $$ g_{m m}=f_{n} \wedge f_{n+1} \wedge \cdots \wedge f_{m} $$ Then \(g_{n m} \in \mathscr{B}(\mathscr{A})\), and since the sequence \(\left(g_{n m}\right)\) is monotone decreasing (in \(m\) ) and positive, it has a pointwise limit \(g_{n}\), and \(g_{n} \in \mathscr{B}(\mathscr{A}) .\) Moreover, the sequence \(\left(g_{n}\right)\) is monotone increasing (since \(g_{n_{1} m} \leq g_{n_{2} m}\) if \(\left.n_{1} \leq n_{2}
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