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Chapter 2: Problem 6

Es sei \(\mu\) ein Inhalt auf dem Ring \(\Re\) 眉ber \(X\). Zeigen Sie: a) Durch \(A \sim B: \Longleftrightarrow \mu(A \triangle B)=0 \quad(A, B \in \Re)\) wird eine 脛quivalenzrelation auf \(\Re\) definiert. b) Die 脛quivalenzklasse \(\mathfrak{N}:=\\{A \in \mathfrak{R}: A \sim \emptyset\\}\) der leeren Menge enth盲lt genau die \(\mu\) Nullmengen, und \(\mathfrak{N}\) ist ein Ideal in \(\mathfrak{R}\) (d.h. \(\mathfrak{N}\) ist ein Unterring von \(\mathfrak{R}\), und f眉r alle \(A \in\) \(\mathfrak{R}, B \in \mathfrak{N}\) gilt \(A \cap B \in \mathfrak{N})\). c) F眉r alle \(A, B \in \Re\) mit \(A \sim B\) gilt \(\mu(A)=\mu(B)=\mu(A \cap B)=\mu(A \cup B)\). \({ }^{3}\) Diese Arbeit von Herrn Young ist die erste unter denjenigen, die zum endgultigen Verst盲ndnis dessen, was das Stieltjessche Integral ist, gef眉hrt haben. Man ist [aber erst] Dank der Definition von Herrn Radon ... und der Arbeiten von Herrn de la Vall茅e Poussin 眉ber die Ausdehnung des Ma脽begriffes ... wirklich zum Kern dieses Begriffes vorgedrungen. \(\mu\) gleichm盲脽ig stetig bez. \(\delta\). Die Mengenoperationen \(\cap, \cup, \backslash, \triangle\) sind bez. \(\delta\) gleichm盲脽ig stetige Abbildungen von \(\Re \times \Re\) in \(\Re\). e) \(\mu\) sei endlich, \(\hat{\Re}\) bezeichne die Menge der 脛quivalenzklassen \(\hat{A}:=\\{B \in \Re: B \sim A\\} \quad(A \in\) \(\mathfrak{R})\), und f眉r \(A, B \in \mathfrak{R}\) sei \(d(\hat{A}, \hat{B}):=\delta(A, B)\). Zeigen Sie: \((\hat{\Re}, d)\) ist ein metrischer Raum. f) Es sei \(\mu\) ein endliches Pr盲ma脽 auf dem \(\sigma\)-Ring \(\mathfrak{S} .\) Dann ist der metrische Raum \((\hat{\mathrm{S}}, d)\) vollst盲ndig. (Hinweise: Es seien \(A_{n} \in \mathfrak{S}\) und \(\left(\hat{A}_{n}\right)_{n \geq 1}\) eine Cauchy-Folge in \((\hat{\mathfrak{S}}, d)\). W盲hlen Sie eine Teilfolge \(B_{k}=A_{n_{k}} \quad(k \geq 1)\), so da脽 \(d\left(\hat{B}_{k}, \hat{B}_{k+1}\right) \leq 2^{-k} \quad(k \geq 1)\). Folgern Sie aus \(B_{p} \triangle\) \(\bigcup_{k=p}^{q} B_{k} \subset \bigcup_{k=p}^{q-1} B_{k} \Delta B_{k+1}(q>p \geq 1)\), da脽 f眉r alle \(q \geq p\) gilt \(\mu\left(B_{p} \Delta \bigcup_{k=p}^{q} B_{k}\right)<2^{-(p-1)}\) und folgern Sie aus der Stetigkeit des Pr盲ma脽es von unten, da脽 f眉r \(C_{p}:=\bigcup_{k=p}^{\infty} B_{k} \in \mathcal{G}\) gilt \(d\left(\hat{B}_{p}, \hat{C}_{p}\right) \leq 2^{-(p-1)} .\) F眉r \(B:=\varlimsup_{k \rightarrow \infty} B_{k}\) gilt \(C_{p} \downarrow B\). Schlie脽en Sie nun aus der Stetigkeit des Pr盲ma脽es von oben und aus \(d\left(\hat{B}_{p}, \hat{B}\right) \leq d\left(\hat{B}_{p}, \hat{C}_{p}\right)+d\left(\hat{C}_{p}, \hat{B}\right)\), da脽 \(\left(\hat{B}_{k}\right)_{k \geq 1}\) gegen \(\hat{B}\) konvergiert. Warum folgt hieraus die Konvergenz der Folge \(\left(\hat{A}_{n}\right)_{n \geq 1}\) gegen \(\hat{B}\) ?) d) Es seien zus盲tzlich \(\mu\) endlich und \(\delta: \Re \times \Re \rightarrow \mathbb{R}, \delta(A, B):=\mu(A \triangle B)\) f眉r \(A, B \in \Re\). Zeigen Sie: \(\delta\) ist eine Halbmetrik auf \(\Re\) (d.h. es gilt f眉r alle \(A, B, C \in \Re: \delta(A, A)=0, \delta(A, B)=\) \(\delta(B, A), \delta(A, C) \leq \delta(A, B)+\delta(B, C)) .\) Es ist \(|\mu(A)-\mu(B)| \leq \delta(A, B) \quad(A, B \in \Re)\) und daher

Short Answer

Expert verified
The equivalence relation and class are clarified which relate to the sets with measure 0 in a ring of sets. Then, it is demonstrated how measures behave under the relation of equivalence. A metric structure is introduced on the set of equivalence classes and the completeness of the created metric space is probed. Finally, a semi-metric is defined on \(\Re\), providing an alternative way to compare the distance between sets.

Step by step solution

01

Understanding the equivalence relation and properties of measure

The equivalence relation \(A \sim B\) is defined by the condition \(\mu(A \triangle B)=0\). This means sets A and B are considered equivalent if the measure of their symmetric difference (the set of elements which are in either of the sets and not in their intersection) is 0.
02

Characterizing the equivalence class containing the empty set

The equivalence class \(\mathfrak{N}\) of the empty set contains exactly the \(\mu\) null sets (sets with measure 0). In the ring \(\Re\), \(\mathfrak{N}\) is an ideal, meaning it is a subring and for each pair \(A \in \Re, B \in \mathfrak{N}\), \(A \cap B \in \mathfrak{N}\), emphasizing the closure of intersection within an ideal.
03

Understanding equivalence in terms of measures

For any \(A, B \in \Re\) with \(A \sim B\), then \(\mu(A)=\mu(B)=\mu(A \cap B)=\mu(A \cup B)\). This condition reiterates the idea of measure preserving the equivalence relation, meaning the measure of the union or intersection of equivalent sets equals the measure of the individual sets.
04

Introducing a metric on the equivalence classes

If the measure \(\mu\) is finite, we can introduce a metric \(d\) on the set of equivalence classes. For each \(A, B \in \Re\), define \(d(\hat{A}, \hat{B}):=\delta(A, B)\). The pair \((\hat{\Re}, d)\) forms a metric space.
05

The completion of the metric space

If \(\mu\) is a finite pre-measure on the sigma-ring \(\mathfrak{S}\), then the metric space \((\hat{\mathrm{S}}, d)\) can be shown to be complete. This involves showing any Cauchy sequence in this space converges to a limit in the same space.
06

Identifying the semi-metric \(\delta\)

If \(\mu\) is finite, a semi-metric \(\delta\) is introduced on \(\Re\). For each \(A, B \in \Re\), \(\delta(A, B):=\mu(A \triangle B)\).

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Key Concepts

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

Equivalence Relation
An equivalence relation is a way of grouping objects into categories where each object in a category is considered as equivalent to the others. To be considered an equivalence relation, a rule must be reflexive, symmetric, and transitive. In measure theory, such a relation can be defined on a collection of sets using a measure. The relation \(A \sim B\) is defined by \(\mu(A \triangle B) = 0\), meaning that two sets, \(A\) and \(B\), are considered equivalent if the measure of their symmetric difference is zero.
This means that, with respect to measure, they are indistinguishable from one another. Reflexivity ensures that any set is equivalent to itself since \(\mu(A \triangle A) = 0\). Symmetry ensures if \(A\) is equivalent to \(B\), then \(B\) is equivalent to \(A\). Finally, transitivity ensures that if \(A\) is equivalent to \(B\) and \(B\) is equivalent to \(C\), then \(A\) is equivalent to \(C\). These properties make equivalence relations foundational in building complex mathematical structures.
Sigma-Ring
A sigma-ring is a collection of sets that is closed under the operations of finite union and relative complement. It is a weaker type of structure compared to a sigma-algebra, which is also closed under countable unions of sets. Sigma-rings are pivotal in the field of measure theory, especially when dealing with measures that need only be finitely additive rather than countably additive.

In essence, a sigma-ring is a useful tool for dealing with measures like the pre-measure \( \mu \) described in the original exercise. Extensions of these concepts lead to important results such as the completion of metric spaces, enabling deeper insights into the convergence of Cauchy sequences within these structures. While not as robust as sigma-algebras, sigma-rings provide sufficient structure for many practical applications where countable additivity isn't necessary.
Cauchy Sequence
A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. In a metric space, a sequence \((x_n)\) is called a Cauchy sequence if for every positive number \(\epsilon\), there is a positive integer \(N\) such that for all integers \(m, n > N\), the distance \(d(x_m, x_n) < \epsilon\).

In measure theory, Cauchy sequences in metric spaces defined via measure play an important role, particularly in assessing convergence. Part f of the original exercise demonstrates the use of Cauchy sequences within a metric space \((\hat{\mathrm{S}}, d)\) derived from a sigma-ring. The completeness property ensures that every Cauchy sequence in this space approaches a well-defined limit within the space, thus creating a robust environment to explore convergence properties of functions and sets defined with respect to a measure.
Symmetric Difference
The symmetric difference between two sets \(A\) and \(B\), denoted \(A \triangle B\), consists of elements that are in either set but not in both. Mathematically, it can be expressed as \((A \setminus B) \cup (B \setminus A)\). Symmetric differences arise naturally in measure theory when considering the equivalence of sets.This operation is crucial for defining the equivalence relation \(A \sim B\), as seen in the original exercise. When two sets have a symmetric difference of measure zero, they are equivalent because the elements that differentiate them have no significant measure.
This aspect of set interaction is integral to many proofs and concepts in measure theory, providing a foundational operation used frequently to relate sets under a measure by illustrating their differences in a structured and measurable way.

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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 sei \(X \neq \emptyset\) a) Es sei \(\mu: \mathfrak{P}(X) \rightarrow \mathbb{R}\) ein Inhalt mit \(\mu(X)=1\) und \(\mu(A) \in\\{0,1\\}\) f眉r alle \(A \subset X ; \mathfrak{U}:=\) \(\\{A \subset X: \mu(A)=1\\} .\) Zeigen Sie: (i) \(\emptyset \notin \mathfrak{U}\) (ii) \(A \in \mathscr{U}, A \subset B \subset X \Longrightarrow B \in \mathfrak{U} ;\) (iii) \(A, B \in \mathfrak{U} \Longrightarrow A \cap B \in \mathfrak{U}\) (iv) \(A \subset X \Longrightarrow A \in \mathfrak{U}\) oder \(A^{c} \in \mathfrak{U}\). (Eine Teilmenge \(\mathfrak{U} \neq \emptyset\) von \(\mathfrak{P}(X)\) mit den Eigenschaften (i), (ii), (iii) hei脽t ein Filter auf \(X\); gilt zus盲tzlich (iv), so hei脽t \(\mathfrak{U}\) ein Ultrafilter.) b) Ist \(\mathfrak{U}\) ein Ultrafilter auf \(X\), so ist \(\mu: \mathfrak{P}(X) \rightarrow \mathbb{R}, \mu(A):=1\) f眉r \(A \in \mathfrak{U}, \mu(A):=0\) f眉r \(A^{c} \in \mathfrak{U}\), ein Inhalt. \(\mu\) ist genau dann ein Ma脽, wenn f眉r jede Folge \(\left(A_{n}\right)_{n \geq 1}\) von Mengen aus U gilt \(\bigcap_{n=1}^{\infty} A_{n} \neq \emptyset\) c) Auf jeder unendlichen Menge gibt es einen nicht-trivialen Inhalt, der auf ganz \(\mathfrak{P}(X)\) definiert ist, nur die Werte 0 und 1 annimmt und auf allen endlichen Mengen verschwindet. (Hinweis: Zu jedem Filter \(\mathfrak{F}\) auf \(X\) gibt es einen Ultrafilter \(\mathfrak{U} \supset \mathfrak{F}\), s. z.B. SCHUBERT \([1], \mathrm{S}\). 49.)

Es seien \(\mu: \mathfrak{H} \rightarrow \overline{\mathbb{R}}\) und \(\nu: \mathfrak{K} \rightarrow \overline{\mathbb{R}}\) zwei Inhalte auf den Halbringen \(\mathfrak{H}, \mathfrak{K}\) 眉ber \(X\) bzw. \(Y\), und \(\rho: \mathfrak{H} * \mathfrak{K} \rightarrow \overline{\mathbb{R}}\) (s. Lemma I.5.3) sei definiert durch \(\rho(A \times B):=\mu(A) \cdot \nu(B) \quad(A \in\) \mathfrak{H} , ~ \(B \in \mathfrak{K}\) ); dabei wird das Produkt auf der rechten Seite definiert durch \(a \cdot \infty:=\infty\) f眉r \(0

Ist \(A \subset X\) und \(h_{\alpha}(A)<\infty, \beta>\alpha\), so gilt \(h_{\beta}(A)=0 .\) Es gibt also ein eindeutig bestimmtes \(\delta(A) \geq 0\), so da脽 \(h_{\alpha}(A)=0\) f眉r \(\alpha>\delta(A)\) und \(h_{\alpha}(A)=\infty\) f眉r \(\alpha<\delta(A)\); dieses \(\delta(A)\) hei脽t die Hausdorff-Dimension von \(A .\) a) F眉r jedes \(A \subset \mathbb{R}^{p}\) gilt \(\delta(A) \leq p\). b) F眉r jedes \(A \subset \mathbb{R}^{p}\) mit \(\stackrel{\circ}{A} \neq \emptyset\) gilt \(\delta(A)=p\). c) F眉r jede einfache rektifizierbare Kurve \(\gamma\) ist \(\delta([\gamma])=1\). (Es gibt jedoch stetige Funktionen \(f:[0,1] \rightarrow \mathbb{R}\), deren Graph die Hausdorff-Dimension 2 hat; s. P. WINGREN: Concerning a real-valued continuous function on the interval \([0,1]\) with graph of Hausdorff dimension 2, L'Enseignement Math., II. Ser., 41, 103-110 (1995) und Y.-Y. LIU: A function whose graph is of dimension 1 and has locally an infinite one- dimensional Hausdorff measure, C.R. Acad. Sci., Paris, Ser. I 332, 19-23 (2001).) d) F眉r \(A_{n} \subset X(n \in \mathbb{N})\) ist \(\delta\left(\bigcup_{n=1}^{\infty} A_{n}\right)=\sup \left\\{\delta\left(A_{n}\right): n \in \mathbb{N}\right\\} .\) e) F眉r jede abz盲hlbare Menge \(A \subset X\) ist \(\delta(A)=0\). f) Ist \(A \subset \mathbb{R}^{p}, \delta(A)=0\), so gilt \(\lambda^{p}(A)=0\). g) F眉r das Cantorsche Diskontinuum \(C \subset[0,1]\) gilt \(\delta(C)=\log 2 / \log 3\). h) Zu jedem \(\alpha \in] 0,1\left[\right.\) existiert eine Menge \(A \subset[0,1]\) mit \(0

Jede offene Teilmenge von \(\mathbb{R}\) ist disjunkte Vereinigung abz盲hlbar vieler offener Intervalle.
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