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Chapter 2: Problem 5

Es seien \(X\) ein metrischer (oder topologischer) Raum und \(\mu\) ein endlicher Inhalt auf dem Halbring \(\mathfrak{H}\) 眉ber \(X\). Dann hei脽t \(\mu\) von innen regul盲r, wenn zu jedem \(\varepsilon>0\) und jedem \(A \in \mathfrak{H}\) ein \(K \in \mathfrak{H}\) existiert, so da脽 gilt: \(\bar{K}\) ist kompakt, \(\bar{K} \subset A, \mu(A) \leq \mu(K)+\varepsilon .\) Zeigen Sie: Ist \(\mu\) von innen regul盲r, so ist \(\mu\) ein Pr盲ma脽. (Hinweis: Ist \(\mu\) von innen regul盲r, so auch die Fortsetzung \(\nu\) von \(\mu\) auf den von \(\mathfrak{H}\) erzeugten Ring. \(\nu\) gen眉gt der Bedingung d) aus Satz 1.10.)

Short Answer

Expert verified
The given internally regular measure \(\mu\) is indeed a pre-measure. The proof follows from the definitions of regularity and pre-measure, generation of the extension \(\nu\) of \(\mu\), and the application of condition d) from theorem 1.10 to \(\nu\).

Step by step solution

01

Recall the definitions

A measure \(\mu\) is said to be internally regular if, for every \(\varepsilon>0\) and every \(A \in \mathfrak{H}\), there exists \(K \in \mathfrak{H}\) such that \(\bar{K}\) is compact, \(\bar{K} \subseteq A\), and \(\mu(A) \leq \mu(K)+\varepsilon\). A pre-measure \(\mu\) on a semi-ring \(\mathfrak{H}\) is said to be a function from \(\mathfrak{H}\) to the non-negative extended real numbers, satisfying \(\mu(\emptyset) = 0\) and for every countable collection \((A_i)_{i=1}^{\infty}\) of pairwise disjoint sets in \(\mathfrak{H}\) with \(\bigcup_{i=1}^{\infty}A_i \in \mathfrak{H}\), \(\mu(\bigcup_{i=1}^{\infty}A_i) = \sum_{i=1}^{\infty}\mu(A_i)\).
02

Assume \(\mu\) is internally regular

Suppose \(\mu\) is an internally regular measure on the semi-ring \(\mathfrak{H}\).
03

Consider the extension of \(\mu\)

Assume \(\nu\) is the extension of \(\mu\) to the ring generated by \(\mathfrak{H}\). If \(\mu\) is internally regular, so is the extension \(\nu\), as suggested by the hint.
04

Apply condition d) from theorem 1.10 to \(\nu\)

Condition d) from theorem 1.10 states that if \(\mu\) is a pre-measure and \(\mu(R) < \infty\) for all \(R \in \mathfrak{H}\), then \(\mu\) can be extended to a measure on the smallest \(\sigma\)-algebra containing \(\mathfrak{H}\) and still be regular. Since \(\nu\) satisfies this condition, it can be concluded that \(\mu\) as well is a pre-measure.

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Key Concepts

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

Internally Regular Measure
In measure theory, an **internally regular measure** is an important concept when working with measures in mathematics. Imagine you have a measure \( \mu \) defined on a semi-ring \( \mathfrak{H} \) over a topological space. The measure \( \mu \) is called internally regular if, for any small tolerance \( \varepsilon > 0 \) and for every set \( A \) within \( \mathfrak{H} \), there exists another set \( K \) such that the closure of \( K \) is compact and it fits nicely inside \( A \). Thus:
  • The closure of \( K \) is compact, signifying it has nice boundary properties.
  • It should sit completely inside \( A \).
  • The measure of \( A \) is within \( \varepsilon \) of the measure of \( K \), i.e., \( \mu(A) \leq \mu(K) + \varepsilon \).
This characteristic indicates that \( \mu \) behaves well within its domain and can be extended beyond its current scope in a stable manner.
Topological Space
A **topological space** is a fundamental concept in mathematics that generalizes the idea of space, geometry, and continuity. It's where we examine how different points or subsets can relate and interact. A set \( X \) equipped with a collection of open sets forms a topological space if:
  • The empty set and the whole set \( X \) itself are considered open sets.
  • The union of any number of open sets is also open.
  • The intersection of a finite number of open sets is open as well.
These properties help form a structure that allows mathematicians to work with concepts like convergence, continuity, and compactness in very flexible and abstract ways. Topological spaces form the bedrock for numerous branches of mathematics, including analysis and topology.
Compact Set
A **compact set** is a powerful concept in topology and real analysis that extends the notion of closed and bounded sets. In essence, a set is compact if every open cover has a finite subcover. Here's what that means:
  • Imagine covering a set with numerous overlapping open sets, much like laying a patchwork quilt completely over a shape.
  • A set is compact if, from this infinite collection of open sets, we can pull out a finite number and still cover the whole set.
This property is hugely beneficial because it allows for the transfer of local properties to global properties. Compactness is used widely in mathematical analysis, where it often helps simplify complex problems and ensure certain desirable conditions.
Measure Theory
**Measure theory** is a rich field of mathematics focused on investigating sizes, lengths, areas, and volumes. At its core, it allows generalizing the concept of integration beyond simple functions. Here's why measure theory is significant:
  • It defines how to measure subsets of a given set \( X \), especially when things aren't as straightforward as checking lengths or areas.
  • Unlike regular measurements, it accounts for complexities in shapes, borders, and infinite dimensions.
  • Measure theory forms the foundation for integration, probability, and real analysis by providing tools to handle abstract, complex spaces systematically.
It captures concepts of size in mathematical spaces regardless of their dimensions, aiding in deeper explorations of mathematical relationships involving values, structures, and applications across various technical fields.

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Most popular questions from this chapter

Konstruieren Sie eine \(F_{\sigma}\)-Menge \(A \subset[0,1]\), so da脽 f眉r jede nicht-leere offene Menge \(U \subset[0,1]\) gilt \(0<\lambda(A \cap U)<\lambda(U)\). (Anleitung: Es seien \(\left(I_{n}\right)_{n \geq 1}\) eine Abz盲hlung der abgeschlossenen Teilintervalle von \([0,1]\) mit rationalen Endpunkten und \(A_{1} \subset I_{1}\) eine nirgends dichte perfekte Menge positiven Ma脽es (s. Aufgabe 8.1). Es gibt eine nirgends dichte perfekte Menge \(B_{1} \subset I_{1} \backslash A_{1}\) mit \(\lambda\left(B_{1}\right)>0 .\) Sind \(A_{1}, \ldots, A_{n-1}, B_{1}, \ldots, B_{n-1}(n \geq 1)\) schon als disjunkte nirgends dichte perfekte Mengen positiven Ma脽es gew盲hlt, so da脽 \(A_{k} \subset I_{k}\) und \(B_{k} \subset I_{k} \backslash A_{k}\) f眉r \(k=1, \ldots, n-1\), so enth盲lt \(I_{n} \backslash\left(A_{1} \cup \ldots \cup A_{n-1} \cup B_{1} \cup \ldots \cup B_{n-1}\right)\) ein Intervall, und die Konstruktion l盲脽t sich fortsetzen. \(A:=\bigcup_{k=1}^{\infty} A_{k}\) leistet das Verlangte.)

Es seien \(F: \mathbb{R} \rightarrow \mathbb{R}\) die Cantorsche Funktion und \(x, y \in C, x

Eine Menge \(A \subset \mathbb{R}^{p}\) hei脽t Jordan-me脽bar, wenn \(A\) beschr盲nkt und \(\sup \left\\{\lambda^{p}(M): M \in\right.\) \(\left.\mathfrak{F}^{p}, M \subset A\right\\}=\inf \left\\{\lambda^{p}(N): N \in \mathfrak{F}^{p}, N \supset A\right\\}\) ist. F眉r Jordan-me脽bares \(A\) hei脽t \(\iota^{p}(A):=\) \(\sup \left\\{\lambda^{p}(M): M \in \mathfrak{F}^{p}, M \subset A\right\\}\) das Jordan-Ma脽 von \(A\). (Diese Begriffe sind benannt nach dem franz枚sischen Mathematiker C. JORDAN, dessen einflu脽reicher Cours d'analyse lange Zeit ein Ma脽stab f眉r Strenge auf dem Gebiet der Analysis war. Unabh盲ngig vom italienischen Mathematiker G. PEANO entwickelte JORDAN um 1890 eine Inhaltslehre f眉r Teilmengen des \(\mathbb{R}^{p}\) und einen Integralbegriff, der dem Riemannschen Integralbegriff analog ist. Eine genauere Diskussion des Jordan-Ma脽es und des entsprechenden Integrals findet man im Grundwissen-Band Analysis II von W. WALTER und bei MayRhoFeR \([1] .)\) Ist \(A\) Jordan-me脽bar mit \(\iota^{p}(A)=0\), so hei脽t \(A\) eine Jordan-Nullmenge. a) Ist \(A\) Jordan-me脽bar, so gilt \(A \in \mathfrak{L}^{p}\) und \(\lambda^{p}(A)=\iota^{p}(A)\). b) Eine Menge \(A \subset \mathbb{R}^{p}\) ist genau dann Jordan-me脽bar, wenn \(A\) beschr盲nkt und der Rand von \(A\) eine Jordan-Nullmenge ist. c) Das System \(\mathfrak{J}^{p}\) der Jordan-me脽baren Teilmengen des \(\mathbb{R}^{p}\) ist ein Ring und \(\iota^{p}: \mathfrak{J}^{p} \rightarrow \mathbb{R}\) ein Inhalt. d) F眉r jedes \(A \in \mathfrak{J}^{p}\) gilt \(\stackrel{\circ}{A} \in \mathfrak{J}^{p}, \bar{A} \in \jmath^{p}\) und \(\iota^{p}(\AA{A})=\iota^{p}(A)=\iota^{p}(\bar{A})\). e) Eine kompakte Menge \(K \subset \mathbb{R}^{p}\) ist genau dann eine Lebesguesche Nullmenge, wenn \(K\) eine Jordan-Nullmenge ist. f) Eine beschr盲nkte Menge \(A \subset \mathbb{R}^{p}\) ist genau dann Jordan- me脽bar, wenn \(\lambda^{p}(\hat{A})=\lambda^{p}(\bar{A})\) ist, und dann ist \(\iota^{p}(A)=\lambda^{p}(\stackrel{\circ}{A})=\lambda^{p}(\bar{A})\) g) Die Menge \(\mathbb{Q}^{p} \cap[0,1]^{p}\) ist eine beschr盲nkte Lebesguesche Nullmenge, aber keine JordanNullmenge. h) Es seien \(f:[a, b] \rightarrow \mathbb{R}, f \geq 0\) und \(\mathcal{O}(f):=\left\\{(x, y)^{t} \in \mathbb{R}^{2}: x \in[a, b], 0 \leq y \leq f(x)\right\\}\) die Ordinatenmenge von \(f\). Dann ist \(f\) Riemann- integrierbar genau dann, wenn \(\mathcal{O}(f)\) Jordanme脽bar ist, und in diesem Falle gilt \(\int_{a}^{b} f(x) d x=\iota^{2}(\mathcal{O}(f))\). i) Ist \(K \subset \mathbb{R}^{k}\) kompakt und \(f: K \rightarrow \mathbb{R}^{n}\) stetig, so ist der Graph \(G:=\left\\{(x, f(x))^{t}: x \in K\right\\}\) eine Jordansche Nullmenge des \(\mathbb{R}^{k+n}\). j) Es seien \(M \subset \mathbb{R}^{k+n}\) offen und \(g: M \rightarrow \mathbb{R}^{n}\) stetig differenzierbar. Ferner sei \(F:=\\{x \in\) \(M: g(x)=0\\} \neq \emptyset\), und der Rang der Funktionalmatrix von \(g\) sei in allen Punkten von \(F\) gleich \(n\). Dann hei脽t \(F\) eine stetig differenzierbare \(k\)-dimensionale Fl盲che im \(\mathbb{R}^{k+n}\). Zeigen Sie: Jede kompakte Teilmenge von \(F\) ist eine Jordan-Nullmenge. (Hinweis: Satz 眉ber implizite Funktionen.) k) Jede (offene oder abgeschlossene) Kugel im \(\mathbb{R}^{p}\) ist Jordan- me脽bar.

Es seien \(\mu: \mathfrak{f} \rightarrow \overline{\mathbb{R}}\) ein Pr盲ma脽 auf dem Halbring \(\mathfrak{H}\) 眉ber \(X\) und \(\eta\) das 盲u脽ere \(\mathrm{MaB} \mathrm{zu}\) \(\mu\). Zeigen Sie: a) Eine Teilmenge \(A \subset X\) ist genau dann \(\eta\)-me脽bar, wenn f眉r alle \(M \in \mathfrak{H}(!)\) mit \(\mu(M)<\infty\) gilt: \(\mu(M)=\eta(M \cap A)+\eta\left(M \cap A^{c}\right)\). (Bemerkung: Im Falle des Lebesgueschen Pr盲ma脽es auf \(\mathbb{R}\) ist dieses die urspr眉ngliche Me脽barkeitsdefinition von LEBESGUE [1], S. 209-210.) b) Eine Menge \(M \subset X\) ist genau dann \(\eta\)-me脽bar, wenn \(M \cap A \eta\)-me脽bar ist f眉r alle \(A \in \mathfrak{H}\) \(\operatorname{mit} \mu(A)<\infty\)

Es seien \(\mu\) ein Pr盲ma脽 auf dem \(\sigma\)-Ring \(\mathfrak{S}\) und \(\left(A_{n}\right)_{n \geq 1}\) eine Folge von Mengen aus \(\mathrm{S}\). Zeigen Sie: a) \(\mu\left(\varliminf_{n \rightarrow \infty} A_{n}\right) \leq \varliminf_{n \rightarrow \infty} \mu\left(A_{n}\right)\). b) Gibt es ein \(p \in \mathrm{N}\), so da脽 \(\mu\left(\bigcup_{k=p}^{\infty} A_{k}\right)<\infty\), so gilt \(\mu\left(\varlimsup_{n \rightarrow \infty} A_{n}\right) \geq \varlimsup_{n \rightarrow \infty} \mu\left(A_{n}\right)\) c) Gibt es ein \(p \in \mathbb{N}\), so da脽 \(\mu\left(\bigcup_{k=p}^{\infty} A_{k}\right)<\infty\), und konvergiert \(\left(A_{n}\right)_{n \geq 1}\), so gilt \(\lim _{n \rightarrow \infty} \mu\left(A_{n}\right)=\) \(\mu\left(\lim _{n \rightarrow \infty} A_{n}\right)\)
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