Mass und Integral - 2020

Dave, 24.08., 15:00 - 15:20

First, I had to prove that every metric measure on \( \mathbb{R}^n \) is a Borel measure. Prof. Da Lio only wanted to hear the proof and wasn't interested in additional things around the statement (e.g. didn't want to hear why we only have to consider closed sets). Then I just copied the proof from the lecture notes and when I came to the step, where one has to show that \( \sum_{l=0}^\infty \mu(R_{l}) < \infty \), she stopped me and moved on.

The second question was about what types of convergence there are and their relations. I mentioned convergence \(\mu\)-a.e. and convergence in measure and proved (again, just copying the script word by word) that convergence \(\mu\)-a.e. implies convergence in measure. To my surprise I didn't have to show the converse direction (conv. in measure yields subsequence with conv. \(\mu\)-a.e.) and she ended the exam there.

During the exam, Prof. Da Lio only spoke when she posed her questions, or when I asked her if I have to explain something more explicit, to which the answer was always no. Otherwise she just sat there and didn't show any reaction. As long as you just present things like they are in the script, she seems to be happy with it.

Jonathan, 26.08., 15:00 - 15:20

I was let in about two minutes early. The atmosphere was calm and only disturbed by my nervous questions if I should/could use my own pen. Prof Da Lio merely said she didn't know how this works, but that I could remove my mask. After, I showed my Legi to the assistant and recieved some paper, the exam started, without my question about pens being answered (I used the one present).

The first question was that I had to prove that \( L^p \)-spaces are separaple. I replied that this was only true for \(p<\infty\). Then I proceeded to proof this statement. Professor Da Lio only sighed in approval a few times. I did not know if this meant that I could stop the proof here. I did not ask and continued eagerly. Freddie Mercury singing in my head: "Don't stop me nooooowww..."

After this was finished. Prof. Da Lio asked me why I excluded \(p=\infty\). I replied that \(L^\infty\) was not separable. The fact that neither she or her assistant got up in cheers for this answer, lead me to believe that I should prove this statement, too. I did, fully expecting that this was only a side-note, start with the essentials for a counter-example. As she did not stop me, I proved the counter-example properly. Talked about \(f_a:=\chi_{[0,a]}\) for \( a \in [0,1]\). How they were mutually a distance of one apart in the \(L^\infty\text{-norm}\). Then I talked about the fact that any countable set \(E \subseteq L^\infty([0,1],\mathcal{L}^1)\) and any balls with radius \(\frac{1}{2}\) for an element \(f \in E\) could cover at most one of these \(f_a\) but there were more than countably many of these \(f_a\). Finally, Da Lio said that this would suffice.

Then she asked if I could prove, she stared at the cieling lost in thoughts, that the convolution of \( f\in L_{loc}^1 \) and \( g\in C^0_c\) was \(C^0\). The voice in my head that was singing songs from Queen before, started swearing. I forgot that proof. In an effort to conceal my panic, I wrote down here question again and in a blind effort, I started to write down \(|f* g(x+\delta)-f* g (x)|=...\leq \int |f(y)||g(x+\delta-y)-g(x-y)|dx\). But then I got stuck. I defined quite chaotically \(A:= \{ y\in \mathbb{R}^n : y+x,y+x+\delta'\in \text{supp}(f)\text{ for } \delta'<1\}\). My idea, I had prepared at home was to say that \(1>\delta>0\) such that \(|x-y|<\delta \implies |g(y)-g(x)|<\epsilon' \). Then I would get from my previous calculations \(|f* g(x+\delta)-f* g (x)|\leq \epsilon' \int_A |f(y)|dy\). But I could not formulate this properly. Instead I just started to sweat profusely and started mumbling. Prof. Da Lio finally stopped this sad act and said that I needn't to worry, I was doing fine. Because, I had nothing smart to say, I thanked her. Out of pity for this mumbling student in front of her, she said that Lebesgue might help. I had no idea how that would help, but naturally I replied: "Of, course!" As I did not proceed with writing down anything, she called my bluff and said that it would be easy if \( f \) was in \(L^1\). My answer was promptly: "Yes, right!" But this time, I could actually deliver. I said that we need only to consider \(f_{|A}\). And therefore we had a supremum in \(L^1\) and could use Lebesgue and take the limit inside of the integral. A feeling of relief overcame the room. Finally, Da Lio asked her assistant if the exam could end here, he graciously agreed.

After what felt like an eternity, I handed the papers I was writting on to the assistant. After this Odyssey of emotions and an exam that felt at parts like a greek drama, I left the room feeling years older. A quick look at my watch: Only fifteen minutes had passed: the exam actually had ended early.

Carmen, 24.08., 10:40 - 11:00

You should wait outside and they will pick you up. I did not know that and I made the mistake of knocking at the door while another student was inside. I think they did not like that.

When I entered she did not ask my name or anything, she just began with the questions. She asked me to state Fatou´s Lemma and to prove it. She gave me the option to prove it with Beppo Levi´s or directly. I choose it to prove it directly, which is much longer, but I had this prove more in mind. I did some mistake in the proof because it contains a lot of index and I mixed them. She try to correct me this mistakes by asking “does it make sense to define it like this?”. I corrected it and moved on. Then she asked me why it is necessary that the functions are nonnegative. I could not answered that and she went to the next question. The she asked me the Caratheódory Criterion for measurability, and she told me to prove that the set containing alls measurable sets is a sigma-Algebra. I prove the first two points but then when I started proving the finite union is also inside the time was over and I left.

Overall she is not really nice during the exam, she does not smile and is really serious while answering the question. Do not get intimidated by that, she is just trying to do a poker face. I tried to make a joke and I think she did not like it, so you can learn from my mistakes and don’t do that.

Yannik, 28.08., 10:40 - 11:00

She started with asking the definition of \(L^p\) spaces and the proof that they are complete. I gave the definition and started the proof but could not remember it. I started with saying what we need to prove: every Cauchy sequence converges. But I had a blackout, I literaly could not write down anything.

She said she can give me an other proof but we stay with \(L^p\)-spaces. (... thanks) I then had to prove that \(L^p\)-spaces are normed. I mentioned the Minkowski inequality. I managed the first few steps of the prove. But I was not confident and at some point I did not make much more progress.

Then she asked me the definition of Lebesgue measure. I defined it as the Caratheodory Hahn extension of the volume of Elementary sets. I had to prove the sub-additivity of the Caratheodory Hahn extension. I explained what we exactly have to prove and then started but soon the time was over.

The exame is very stressful if you can not replicate what is written in the script. It is also not very dynamic as there is not much dialog between Da Lio and you.

p.s. still got a 3.5

Jonathan, 28.08., 13:00 - 13:20

She asked me what I could say about Minkowski's Inequality. I told her, it was the last step we needed in order to prove that the Lp-Nom is indeed a norm. I then statet and proved it. Before I was entirely finished, she interrupted me and asked whether I remembered the proof that the continuous functions with compact support were dense in Lp. I did not because I had not learned it. Then she asked me, whether I remembered the proof that Lp was separable. I did not because I had not learned it. She then moved to a different topic and asked me to prove that a metric measure is borel. I did that and she asked me, why the sets Fk, which you define in the proof, are closed (the function that assigns to a point the distance to F is continuous, so the Fk can be represented as the preimage of a closed set, namely [0,1/k], under a continuous function). After presenting about half of the proof, time was up. I asked whether I should briefly explain the idea, but she said, it was alright.

To sum it up: - Memorize as many of the proofs as you can and make sure you can present them, for example by trying to write them down. You will quickly see whether you can present them or whether you just thought you could - In the exam, don't expect Da Lio to say much because she is rather passive. I personally almost never looked at her while presenting the proofs and I did not expect to see much reaction of her. Just present as much as you can and and tell her, when you are stuck and want to move on. That way, you have a higher chance of being able to show much of what you have learned