The importance of measure theory for probability theory

letmeitellyou · 3 months

Hmm ok I think I see: You can think of a curve as an intersection of hypersurfaces in Rⁿ and then write a delta function for each of them and take their product?

So yah I guess delta functions do more than I gave them credit for.

letmeitellyou · 3 months

Thanks for the pointers.

For 1. I might try using a white board in the future. I considered washing the board ahead of time but I figured it would get chalky over time anyway... maybe there is some blackboard trick I am not aware of.

For 2. Yah, maybe i got greedy trying to fit too much of the board on camera (though with higher resolution it might be legible). I don't have a camera man so it was one continuous take. There might be a better way to do things...

For 3. The camera I bought was supposed to be SDHC compatible but it bugged out when I tried to use SDHC. So I ended up using a 2GB SD card which limited my options. I have sent the camera back for replacement so hopefully this should be fixed.

For 4. Yah... I dunno what causes that. I have no real AV knowledge... The lapel mic I bought was highly rated so I am hoping it is decent enough (and it cost $100).

letmeitellyou · 3 months

Yah the sigma algebra A is supposed to be curly... my handwriting is a bit bad so apologies on that.

letmeitellyou · 3 months

Measure theory is a different language so its a bit like saying "why use Python when you already know C and can get the job done in C just fine".

Some people love learning new languages for their own sake and find them beautiful (I think a lot of pure mathematicians would fall under this category).

Some people have applications in mind where the new language is really a better fit (For pure mathematicians, to prove rigorous probability theorems measure theory is invaluable. For people in finance I assume they care about sampling random functions etc. But for plenty of probability applications the undergrad notions are often the better approach)

Some people have to learn the new language for their job (Just like a job might require you to code in Python, many advanced math books require that you know measure theory to read them).

letmeitellyou · 3 months

See my comment above.

In one dimension when measure is concentrated at a point you can use a delta function but you can also have measure concentrated along a 1D curve embedded in R² (or a 2D surface embedded in R³⁾ and so you can't always get by with delta functions.

letmeitellyou · 3 months

The short answer is that you probably don't need measure theory for most calculations. And certainly I tend to use undergrad-style calculations to "get things done".

One example (alluded to in the video) where measure theory would be helpful would be something like the following:

Suppose Z_1 and Z_2 are i.i.d. Uniform(0,1) and let X=Z_1 and Y=max(Z_1,Z_2). Now consider the joint distribution of X and Y. In this case half of the "weight" of the probability lies on the line y=x and is "one-dimensional" while the other half of the "weight" lies in the triangle 1>=y>x>=0. I don't know of any way to express the p.d.f. of this probability distribution and while there may be a notion of c.d.f. in higher dimensions I think it is a bit cumbersome.

Also I find the idea of sampling a random function a compelling to study measure theory (and I assume this is why financial math requires it).

letmeitellyou · 3 months

The short answr is that you cannot define the probablity for the uniform measure on the power set (which I agree is weird).

Offhand I really don't know the details of why not but I read them once, heh. It has to do with Vitali sets, the Banach paradox, or maybe other subtle points. I'd reccomend reading the wikipedia articles (and maybe I should again too!)

letmeitellyou · 3 months

Yes this is a good point as well. Another good reason is that measure theory allows you to apply analysis to probability and rigorously prove theorems (e.g. the CLT)

letmeitellyou · 3 months

That looks like a good site, thanks!

My plan is maybe to make more "mini lectures" i.e. brief talks on concepts I am familiar with and have some insight on that I think aren't as widely known.

There are video lectures (and a lot of written lectures) on probability theory in general out there and most of what I would cover if I continued would be fairly standard (though when I taught the course there were a few points where I deviated from the book so I might touch on those).

letmeitellyou · 3 months

Hmm... from checking out the wikipedia article, it seems that Kolmogorov's axioms are another approach (perhaps an earlier approach?)

But the wikipedia article even says that "These assumptions can be summarised as: Let (Ω, F, P) be a measure space with P(Ω)=1. Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P." so I am guessing the modern, measure-theoretic definition has supplanted these axioms.

So yah as far as I can tell they are the same thing (Keep in mind that for more general measure theory you would not have the condition P(\Omega) = 1). Really the main condition that makes a measure a measure is the sigma additivity (and measures are used for lots of things e.g. mass, heat, etc..)

letmeitellyou · 3 months

I do research in probability theory so I figured I would make a video explaining what I wish someone had told me when I started graduate level probability theory: Why do we bother with measure theory?

I might make some other mini-lectures in the future on other topics so any comments on the video/format would be welcomed.

letmeitellyou

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