Mass und Integral - Martin Schweizer - 2017

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Raffael, 23.08.2017, 11:15-11:30

Wir haben begonnen mit der Vervollständigung eines Massraumes. Da wollte er alle Details hören; wie dass die Mengen darin zusammengesetzt sind, was der Unterschied zu einem normalen Massraum etc ist. Ich war etwas unsicher, da ich diesen Teil nicht sehr genau gelernt hatte. Er stocherte ziemlich lange darauf herum. Nachher musste ich drei Aussagen des Implikationsschemas (Theorem I.2.6) aufzählen und die Beweisidee dafür geben. Wir haben dann noch ziemlich lange über Fatous Lemma gesprochen. Ich fand die Prüfung sehr unangenehm, da er mir nie konkrete Fragen stellte, sondern mir immer nur die Richtung vorgab und ich dann etwas erzählen sollte. Wenn ich dann etwas gesagt habe, meinte er oft "ja ich habe eigentlich etwas anderes im Kopf". Woher ich wissen sollte, was genau er im Kopf hatte, ist mir schleierhaft. Er ist zwar ziemlich freundlich, das hat es aber dann auch nicht wirklich besser gemacht. Ich denke es ist sehr wichtig, die Zusammenhänge zwischen den Sätzen und den Kapiteln zu sehen, darauf war bei mir die Prüfung ausgerichtet.

Giulia, 23.08.2017, 9:15-9:30

Professor Schweizer asked me if it was okay to start 5 mins earlier so I went in at 9:10. He started asking me very generally what I could tell about linear forms. After saying the first properties, I couldn't remember the exact definition of \(\tau\)-continuity so he decided to move on and asked me an example of linear form. We started talking about J(f)=\(\int f\,\mathbb{d}\mu \) and if I could remember any theorem about convergent linear form (I guess he was thinking about Theorem 1.7 and so on). Then he decided to change argument and he started asking me in which cases we can say that an integral is zero (f=0 \(\mu\) -a.e.) , definition of \(\mu\) -a.e, proof of f=0 \(\mu\) -a.e. => \(\int f\,\mathbb{d}\mu \)=0 (first for step functions, then for general measurable functions).

The assistent didn't say a word, Prof. Schweizer is very calm, he tries to help out in the reasonings and had no problem in changing argument when he saw I really didn't know some answers.

Vanessa, 23.08.2017, 14:00-14:15

Prof. Schweizer asked me at first the definition of transition kernel and where we use it; I proved theorem III 2.6. Then he asked to speak about contents and pre-measures: in particular the connections between them. At last I had to define the integral, we spoke about convergence theorems and I had to prove Prop. 2.7 of chapter 2.

Simone, 23.08.2017, 14:15-14:30

First definition of outer measure and were we use it, then Daniell-Stone and then when we have the condition on a space to prove that a linear form sigma-continuous, so state Dini and the Prop. 1.7. Then convergence theorems, and conditions on the space such that we can apply certain convergence theorem, generalized DCT. He wants that you construct from his conditions the theorems and then construct other connections between different structures (Linearforms, measures, convergence, etc...).

Philippe, 23.08.2017, 13:15-13:30

Schweizer first told me what I could say about Fubini. I told him that we needed first to show existence of a measure on a product space. I stated the theorem(2.5) and we went trough the proof. He asked me almost everything in the proof and I needed to proof extra things like, if we have a set A in the product sigma field then Aw1 is in the A2 sigma field. Then I had to write down the corresponding integral on the product space and explain why Fubini holds. Then we spoke about measures and he asked me when a content is a measure and I stated theorem 2.5. He wanted also to know where did we use these propreties. (Daniell-Kolmogorov, Ionescu-Tulcea)

Daniel, 23.08.2017, 16:15-16:30

• Chapter 3
  • Definition of product sigma-field
  • Proof of Proposition 2.1
  • Statement and proof of Theorem 2.7

• Chapter 2
  • Statement and proof of Theorem 1.17
  • Statement of Radon Nykod. and proof of "2) --> 1)"

Cyrill, 24.08.2017, 11:15-11:30

Die Prüfung hat mit 65 Minuten Verspätung begonnen. Herr Schweizer ist sehr nett und hilft, wenn man nicht weiterkommt. Zuerst Radon-Nikodym: Aussage, dann Gegenbeispiel für \( \mu \) nicht sigma-endlich (Lebesgue-Mass << Zählmass). Beweis: Wie ist der grobe Ablauf des Beweises? Dann mehr über den zweiten Schritt (konstruktion von \( g \in L_0^+\) mit \(\mu_g \leq \nu_1\). Dann "von Hand" beweisen , dass für \( f \in L^1\) \(\mu_f \) ein signiertes Mass ist (Achtung: Beppo-Levi funktioniert nicht, man muss mit dominierter Konvergenz herumbasteln).

Emilie, 24.08.2017, 13:00-13:15

We started 15 minutes later.

He asked me to speak about convergence so I stated Beppo Levi and the generalized MCT. I had to prove it, which I did, but he asked for a lot of details. Then he asked me if I knew a result which links L1 convergence and another form of convergence, so I stated generalized Lebesgue. He asked me the definition of the different form of convergences and the definition of uniform integrability. I then had to prove that L1 convergence implies stochastical convergence but I couldn't remember exactly Markov inequality. He then asked me to prove that if we have a sequence of measurable functions which converge mu almost everywhere to a function f, then f is measurable as well.

Samuel, 24.08.2017, 13:15-13:30

We started 20 minutes later than expected. Professor Schweizer started by asking what I could tell about the Lebesgue dominated convergence theorem. I stated the theorem and then gave a proof using implicitly Fatou's Lemma, so at the end of the proof he asked me to prove that as well (only the case \( f_n > g := 0 \) ). As last question he asked me how do we construct the Lebesgue pre-measure and how do we prove that we are actually dealing with a pre-measure.

Daniele, 24.08.2017, 8:30-8:15

• We started about 15 minutes later. Schweizer asked me what could I say about the space Linf, (Definition of linf, of the inf-norm)

Then I had to proof the Fischer-Riesz Theorem for p=inf ( in between he asked some 'basic' question about mesurabiliy, and I proved the fact that the product of 2 msb. function is still measurable).

• Then we moved to convergence...he just gave me an increasing seq. of measurables function, and he asked me what coud I say about the integral ( I state Thm 3.1) . Then he wanted an example of the fact that a limit of function in L1 is not necessarily in L1. I answer with fn= indicator f. of [-n,n] with lebesgue measure.

• Last topic was Linear form, he asked me about an example of Riesz space of function , and an example of Stone Vector Lattice.

Then he asked me about the Dini's Theorem , and last I had to give an example of a linear form on the space L^p ( the integral of course)

Lauro, 24.08.2017, 14:00-14:15

Was ist eine Linearform? Definition von positiv, sigma- und taustetig. Welche Implikation gibt es hier? Was ist ein Beispiel für eine sigmastetige Linearform? Wann ist eine positive Linearform sigmastetig? Aussage von Dini.

Was ist ein Mass? Was sind drei Beispiele für Masse? Was für Eigenschaften hat das Lebesguemass? Hier hab ich translationsinvariant gesagt, was ich dann beweisen musste (er hat Tipps gegeben). Was ist die Definition von sigma-endlich? Welche äquivalente Definition gibt es? (etwas aus den Serien, wusste ich nicht).

Marc, 24.08.2017, 16:00-16:15

• He started the exam by asking me what I know about the Hahn decomposition incl. the definition of a signed measure. So I had to state the theorem and prove it afterwards.
• Then he asked me where we used it and how we used it there (Radon-Nikodým). So I had to state Radon-Nikodým and show the uniqueness of the function \(f\) in this theorem.
• Finally, he wanted to know what it means for a function \( f \) to be in \(\mathcal{L}^p\) and then I had to prove \((\int \ |f|^p\ )^{1/p}\,\mathbb{d}\mu \ =0\) \(\iff\) \( f=0 \ \mu-a.e.\).

Davide, 24.08.2017, 16:15-16:30

Fatou's lemma with proof. Monotone and dominated convergence theorems. Discussion about the measurability of the limit mu-ae of a sequence of measurable functions.

Starting with a sequence of measurable functions, prove sup,liminf,lim,... are measurable.

Definition of a transition kernel. Example of non-trivial transition kernel (I_A * mu1 + I_A(complement) * mu2 with mu1, mu2 finite measures and A in the sigma-field).

Robin, 24.08.2017, 17:00-17:15

Die Prüfung begann mit ca. 50 Minuten Verspätung. Als Erstes wollte er die Definition von stochastischer Konvergenz wissen. Dann fragte er nach einer äquivalenten Charakterisierung zu stochastischer Konvergenz; diese wusste ich jedoch nicht und ich konnte dann auswählen, ob ich mich weiter mit dieser Frage beschäftigen möchte oder lieber etwas anderes hätte. Ich habe mich dann für eine neue Frage entschieden und musste zeigen, dass Konvergenz in \(L^p\) die stochastische Konvergenz impliziert. Ich habe dann leider zuerst auf die \(L^1\)-Konvergenz zurückgegriffen, konnte es dann aber auch für beliebige p zeigen.

Da ich die \(L^1\)-Konvergenz angesprochen habe, wurde ich dann gefragt unter welchen Bedingungen \(L^p\)-Konvergenz \(L^1\)-Konvergenz impliziert. Ich habe dann mit der Hölder-Ungleichung etwas herumgebastelt.

Als nächstes kam die allgemeine Frage, was ich über \(L^p\) wisse. Ich habe geantwortet, dass der Raum normiert und vollständig ist. Dann wurde ich noch nach der Norm gefragt und ob ich die erste Normeigenschaft beweisen könne.

Janosch, 25.08.2017, 7:30-7:45

We started on time, but ended 5 minutes late.

What is Lp? Talked about \(L^p\) and \(\mathcal{L}^p\). What is \(\mu\)-a.e.? \(\mathcal{L}^1, \mathcal{L}^2\): Which one is contained in the other? \(f\) measurable and \(f=g\ \mu\)-a.e: Is g measurable? \(f_n\) measurable and \(f_n\to f\): Is \(f\) measurable? Proof? Then followed a question directing me to convergence results. So I told him about the various convergence theorems. He stopped me at Fatou and wanted to see a proof.

Maic, 25.08.2017, 11:45-12:00

We started with 30 minutes delay

Topo Spaces: Daniell-Stone, Dini, Riesz He asked me what I could tell him about topo spaces. When I tried to give Definitions first, he said we assume we know that. So I just stated the theorems and he decided which parts he wanted to discuss more (Def F-open sets, sigma-cont). So we first had Daniell-Stone, where he asked me if I can tell him more about linear forms, so I gave him Riesz. There he asked for an example I barely remember. Then he wanted to see the relation between sigma-cont and teta-cont. He clearly wanted Prop 1.7. Unfortunately I didn't know it, so I tried to talk about Dini and that it would lead to his result.

Conv: Monoton, Fatou, DCT He asked me what I can tell him about the convergence of functions concerning integrals. It was a really open question, so I decided to start with the generalized monoton conv thm, he asked me if I know also other conv thm, so I gave him Fatou and DCT of lebesgue. He didn't want to hear any proof but rather asked me if I can give him an example for Fatou, so that the relation is strict. I gave him fn=n*I(0,1/n) assuming we have leb measure. At last, he asked me some questions about this example.

Impression: Schweizer is really kind and tries to help if you struggle. But I disliked that he wanted to hear the examples, because I didn't prepare for this and in the test situation I wasn't allways flexible enough to construct them as he desired.

Milos, 25.08.2017, 11:00-11:15

• Satz von Egorov mit Beweis
• Satz 1.2.6 (nur Aussage)
• Definition Prämass, Lebesgueprämass
• Beweis, dass Lebesgueprämass sigma-additiv ist.
• Definition \(L^p\)-Räume.
• Zusammenhänge zwischen \(L^p\)- und \(L^q\)-Räumen
• Gegenbeispiel, dass \(L^p\) Teilmenge von \(L^1\) für das Lebesguemass.
• Beweis, Korollar 2.1.15

Anonym, 22.01.2018

• Lp Raueme und Eigenschafen
• Lp Norm
• Normeigenschaften
• I(f)=0 iff f=0 a. e. Beweisen
• Minkowski Ungleichung plus Beweis
• Radon Nikodym: Aussage und Beweis 2->1 mit f in L1 (Dominierte Konvergenz statt monotone Konvergenz)
• f messbar iff ex f_n folge messbarer Funktionen fn->f beweisen beide richtungen