Probability and Measure, Lecture 2: Carathéodory's Extension Theorem

In this video lecture, we prove Carathéodory's Extension Theorem. This theorem allows us to take a pre-measure defined on a ring or semi-ring and extend it to a measure on a sigma field. It will be a useful tool for constructing measures such as Lebesgue measure.

[aidosmaulsharif]

You are a wonderful person. I have been reading Billingsley's "Probability and Measure". Tbh, it's very enigmatic book and was struggling with it. Thank you for your clear explanations, your personal op ions on things, your reasoning. It makes measure theory so natural and simple because of you. Thank you!!!

[LDB-czwf]

I appreciate that you provide intuition for new definitions and gently introduce them.
btw, "outer measure" is spelled with just one "t" (unless it's a Canadian spelling I'm unaware of).

[haopei]

@27'45'' I think we can not let E be the entire set \Omega: as E \in \mathcal{A} and \mathcal{A} is just a ring not a field. And later @29'15'' E could be the entire of the countable union of the set of subsets {A_i}. Looking forward to hearing back from you. Thank you for the effort and quality of the lectures!

[kamalhaider]

It's interesting. Did you recommend any book in order to follow all playlist???

[benjamindavid]

I think for the inequality at 1:27:18 to hold we are assuming that the outer measure is continuous, which might still need some justification?

[AaBbCize]

At 1:14:25, why does the second equality follow from the first? While intuitive (the RHS is based upon a partition of E), its not clear to me why writing u*(E) in this manner is rigorously justified. We haven’t shown that u* is countably (or even finitely) additive, so why are we allowed break up u*(E) in this manner?

Any help would be much appreciated!

[AllanAngusADA]

At around 1:26, you say mu-star "less than" but you write "greater than." I think you meant what you said. Yes?

[micahdelaurentis]

this is quite a burden on working memory, but very good none the less!