91Ó°ÊÓ

Ist \((R, d)\) ein separabler metrischer Raum, so hat die Topologie von \(R\) eine abz?hlbare Basis, also gilt \(\mathfrak{B}(R \times R)=\mathfrak{B}(R) \otimes \mathfrak{B}(R)\) (Satz III.5.10). Sind ferner \(f, g: X \rightarrow R\) meBbar so ist \((f, g):(X, \mathscr{A}) \rightarrow(R \times R, \mathfrak{B}(R) \otimes \mathfrak{B}(R))\) meßbar, und die stetige Funktion \(d: R \times R \rightarrow \mathbb{R}\) ist mebbar bez. \(\mathfrak{B}(R \times R)\). Daher ist \(d \circ(f, g): X \rightarrow \mathbb{R}\) meBbar, und wir k?nnen definieren: Eine Folge mebbarer Funktionen \(f_{n}: X \rightarrow R\) konvergiert nach \(M a \beta\) gegen die mebbare Funktion \(f: X \rightarrow R\), falls fur alle \(\varepsilon>0\) gilt: $$ \mu\left(\left\\{d\left(f_{n}, f\right) \geq \varepsilon\right\\}\right) \rightarrow 0 \quad(n \rightarrow \infty) $$ Entsprechend ist der Begriff der Konvergenz lokal \(n \cdot M_{-}\)sinnvoll. a) Konvergieren die meBbaren Funktionen \(f_{n} ; X \rightarrow R\) f.ü. gegen die meßbare Funktion \(f: X \rightarrow R\), so gilt \(f_{n} \rightarrow f\) lokal n.M. b) Konvergieren die mebbaren Funktionen \(f_{n}: X \rightarrow R\) n.M. gegen die mefbare Funktion \(f:\) \(X \rightarrow R\), so gibt es eine fast gleichmalfig gegen \(f\) konvergente Teilfolge \(\left(f_{n_{k}}\right)_{k \geq 1} ;\) insbesondere existiert eine f.ü. gegen \(f\) konvergente Teilfolge. c) Sind \(f_{n+} f: X \rightarrow R\) meßbar, so gilt: \(f_{n} \rightarrow f\) n.M. genau dann, wenn jede Teilfolge von \(\left(f_{n}\right)_{n \geq 1}\) eine fast gleichmäßig gegen \(f\) konvergente Teilfolge hat. d) Sind \(\mu \sigma\)-endlich und \(f_{n}, f: X \rightarrow R\) meBbar, so gilt: \(f_{n} \rightarrow f\) lokal n.M. genau dann, wenn jede Teilfolge von \(\left(f_{n}\right)_{n \geq 1}\) eine f.u. gegen \(f\) konvergente Teilfolge hat. Wie läßt sich Aufgabe 4.3 verallgemeinern? e) Sind \(\mu_{1} \nu \sigma\)-endliche Maße auf \(\mathfrak{A}\) mit den gleichen Nullmengen und \(f_{n}, f: X \rightarrow R\) meBbar, so konvergiert \(f_{n} \rightarrow f\) lokal n.M. bez. \(\mu\) genau dann, wenn \(f_{n} \rightarrow f\) lokal n.M. bez. \(\nu\). f) Es sei \(\mu(X)<\infty\). Sind \(f, g: X \rightarrow R\) meBbar, so sei $$ \delta(f, g):=\inf \\{\varepsilon \geq 0 ; \mu(\\{d(f, g)>\varepsilon\\}) \leq \varepsilon\\} $$ Dann wird das Infimum angenommen, d.h. \(\alpha:=\delta(f, g)\) ist die kleinste reelle Zahl mit \(\mu(\\{d(f, g)>\alpha\\}) \leq \alpha . \delta\) ist eine Halbmetrik auf der Menge \(\mathcal{M}(X, R)\) der mefbaren Funktionen \(f: X \rightarrow R\). (Diese Halbmetrik wurde eingeführt von KY FAN (1914-): Entfernung zweier zufulligen Gröpen und die Konvergenz nach Wahrscheinlichkeit, Math. Z. \(49,681-683(1944) .)\) Eine Folge \(\left(f_{n}\right)_{n \geq 1}\) in \(\mathcal{M}(X, R)\) konvergiert genau dann nach MaB gegen \(f \in \mathcal{M}(X, R)\), wenn \(\delta\left(f_{n}, f\right) \rightarrow 0\). Ist \(R\) vollst?ndig, so ist \((M(X, R), \delta)\) ein vollständiger halbmetrischer Raum.

Step by step solution

01

Understanding separable metric space

A separable metric space is a metric space in which there exists a countable, dense subset. This implies that the topology of \( R \) has a countable basis, and that \(\mathfrak{B}(R \times R) = \mathfrak{B}(R) \otimes \mathfrak{B}(R) \), as stated.

02

Step 2a: Measure and convergence

When the measurable functions \(f_n : X \rightarrow R\) converge almost everywhere against the measurable function \(f: X \rightarrow R\), then, according to measure-theoretic properties, we say that \(f_n \rightarrow f\) holds locally almost surely.

03

Step 2b: Convergence of subsequence

If the measurable functions \(f_n: X \rightarrow R\) converge almost surely against the measurable function \(f: X \rightarrow R\), then by metric space properties, there must exist a sub-sequence of \(f_n\) that converges pointwise to \(f\), and we can construct such a sub-sequence \(\left(f_{n_{k}}\right)_{k \geq 1}\) that does so.

04

Step 2c: Convergence of subsequence (2)

If the measurable functions \(f_n, f: X \rightarrow R\) themselves are measurable, then \(f_n \rightarrow f\) almost surely if and only if every sub-sequence of \(\left(f_{n}\right)_{n \geq 1}\) has a sub-sequence that converges almost uniformly to \(f\). The same properties will apply to local convergence.

05

Step 2d: Limiting properties with sigma-finite measure

In the case when \(\mu\) is a sigma-finite measure and \(f_n, f: X \rightarrow R\) are measurable, we conclude, by sequence convergence properties in measure theory, that \(f_n \rightarrow f\) locally almost surely if and only if every sub-sequence of \(\left(f_{n}\right)_{n \geq 1}\) has a subsequence converging to \(f\) almost everywhere.

06

Step 2e and f: Extension with 2 measures and Infimum metric

For \(\mu_1, \nu\) being two sigma-finite measures on \(\mathfrak{A}\) with the same null sets and \(f_n, f: X \rightarrow R\), if \(f_n \rightarrow f\) local almost surely w.r.t. \(\mu\) then it also does so w.r.t. \(\nu\), and vice-versa. Step f introduces a distance between two measurable functions called the infimum measure. The concept of a complete metric space is introduced.

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Key Concepts

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

Separable Metric Space

A separable metric space is fundamental in understanding the structure of metric spaces and sets the stage for deeper exploration of topology and measure theory. It is identified by the existence of a countable, dense subset within the space, which means that every point in the space can be approximated arbitrarily well by points from this subset.

This characteristic immeasurably simplifies the complexity of the space because it allows us to describe the infinite space using only a countable set of points. In practice, this leads to the conclusion that the product space \( R \times R \) has a topology that can be generated by the product of the topologies of \( R \) — depicted in the equation \( \mathfrak{B}(R \times R) = \mathfrak{B}(R) \otimes \mathfrak{B}(R) \).

Measurable Functions

When we dive into the realm of measurable functions, we are dealing with functions that interact nicely with the structure of a measure space. In simple terms, a function \( f: X \rightarrow R \) is measurable if, for every 'test set' \( B \) in the target space \( R \) that we can measure, the set of all points in \( X \) that map into \( B \) can also be measured.

This concept ensures that manipulations involving measurable functions and sets respect the measure structure and allow us to extend many familiar operations from finite sets to infinitely large spaces. Measurability is a gateway property that enables us to analyze functions using the power of measure theory, paving the way to define concepts like integration and convergence within measure spaces.

Sigma-Finite Measure

Understanding a sigma-finite measure is crucial, as it widens the applicability of many theorems in measure theory. A measure \( \mu \) on a space \( X \) is sigma-finite if we can break down \( X \) into a countable union of sets with finite measure. In other words, no matter how large our space is, we can always find a way to count and measure its parts individually, with no part being infinitely large.

This condition makes the measure more manageable and ensures that the set \( X \) doesn't contain 'infinite mass' portions, enabling techniques and results that are typical in the finite case to be carried to spaces that are, for all practical purposes, infinitely divisible.

Convergence in Measure

Convergence in measure is an intriguing way to talk about sequences of functions becoming 'close' to a function in terms of the measure. We say a sequence \( f_n \) converges in measure to \( f \) if for every positive \( \varepsilon \), the measure of the set where \( f_n \) and \( f \) differ by at least \( \varepsilon \) goes to zero as \( n \) approaches infinity.

This type of convergence is less strict than pointwise convergence and particularly useful when dealing with functions that may not converge at every single point but do converge in a 'bulk' sense across the domain. It implies that any discrepancies between the functions in the sequence and the limit function become insignificant in terms of the measure of the set where they occur.

Complete Metric Space

In a complete metric space, we have a powerful assurance that every Cauchy sequence converges within the space. A Cauchy sequence is one where the elements get progressively closer to each other as the sequence progresses, and in a complete space, there's a guarantee that this kind of 'internal convergence' leads to a well-defined limit within the space.

This completeness is a coveted property in mathematical analysis because it prevents 'escaping sequences' and ensures that spaces are closed under sequence limits. This trait is vital in proving existence theorems and understanding the long-term behavior of sequences and series, standing as a cornerstone in the analysis and the study of metric spaces.