(PP 1.3) Measure theory: Measures

(0:00) Sigma-algebra generated by a collection.
(4:12) Examples of sigma-algebras.
(7:20) Definition of a measure.
(9:48) Definition of a probability measure.


A playlist of the Probability Primer series is available here:
http://www.youtube.com/view_play_list?p=17567A1A3F5DB5E4

You can skip the measure theory (Section 1) if you're not interested in the rigorous underpinnings. If you choose to do this, you should start with "(PP 1.S) Measure theory: Summary" at:
http://www.youtube.com/watch?v=upISxgopsFU&p=17567A1A3F5DB5E4

Category:

Education

Tags:

• measure theory
• probability
• math

[alkalait]

Just to clarify with the "openness" of a set: For any point in this set, I can always find another "nearby" point and that point will be also be a member of this open set. Consider the set (0,1); take the point 0.999, then I can find another "nearby" point, e.g. 0.9999, that is in the set. Take the same point 0.9999, find another in its neighbourhood, 0.9999999999, also in the set. My set is "open": I can always find more neighbouring points in the set.

The set [0,1] is not open.

[mathematicalmonk]

@alkalait In Euclidean space R^n, a set U is open if for any point x in U, there is some epsilon>0 such that if |x-y|<epsilon then y is in U.

[mathematicalmonk]

@alkalait For example, in R, the set (0,1) is open since for any x such that 0<x<1, we can choose epsilon small enough that any point between x-epsilon and x+epsilon is in the set. On the other hand, [0,1] is not open since for example when x=1, x+epsilon/2 is not in the set for any positive epsilon.

[gchime]

Thanks for the videos Monk, they are really good.

You always assume (tasitly) that OMEGA is not empty, right?

[mathematicalmonk]

@gchime Thanks for the comment! I might have implicitly assumed that someplace... If so, I should have mentioned it. Did you find something that doesn't work without assuming Omega is nonempty? Let me know and I'll annotate it.

[sponsoredwalk1]

Good video btw, cleared up some confusion I got when

I attempted the real analysis videos on uccs . edu on

this topic. People might like to check them out to complement

to these. Will let you know what I thought of everything

when I get to the end, do you have anything else (like

topology, differential geometry, differential forms, tensors,

real analysis, Abstract Algebra *hint hint*) planned?

[mathematicalmonk]

@sponsoredwalk1 Thanks for the comment. Do let me know what feedback you have. I'm planning this series on probability, then machine learning, then maybe information theory. After that, I'd love to make more. (But we will see how it goes!)

[mathematicalmonk]

You nailed it --- the Sal Khan method. (grins)

Regarding collections and sets, let me clarify. I use "collection" to refer to a set of sets, mostly because it's more natural than saying "set of sets". But this would be fine anyway since we're not talking about the (nonexistent) "set of all sets". (Regardless, I'm glad to have sparked your curiosity.)

I agree, it is enormously fun, and if you think so you should definitely become a mathematician!

[nibauAntunes]

another cool video!

[SuperGZK]

This is just too damn much fun. I need to twist your arms into making more videos on set theory, measure theory, etc. :) Not that I am not interested in probability. I am. You are going to make a mathematician out of me.

I was familiar with Russell's paradox, just not with how one works around it. At least now I am off to the Wikipedia articles on ZFC and NBG, Thank you so much.

BTW Monk, you have the Sal Khan method down. I dare say, perhaps a bit better since you don't stutter. Awesome.