Lecture 7: Sigma Algebras

MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez
View the complete course: ocw.mit.edu/co...
TH-cam Playlist: • Lecture 7: Sigma Algebras
Last time, we introduced outer measures, which have most properties we want for a measure. We define Lebesgue measurable sets and ultimately the Lebesgue measure. We also define sigma-algebras, including the important Borel sigma-algebra.
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[@aaronrobertcattell8859]

An abbreviation is a shortening of a word or a phrase. An acronym is an abbreviation that forms a word. An initialism is an abbreviation that uses the first letter of each word in the phrase (thus, some but not all initialisms are acronyms).

[@mastershooter64]

Thanks

[@SSNewberry]

This mainly about definitions and the intricacies of proving what seems to be obvious but is not. Extremely fine attack.

[@cristianismopuroesimples5302]

Excelente, quem estuda análise bayesiana, é imprescindível aprender sigma-algebra

[@coreconceptclasses7494]

Plz upload complex analysis.

[@kingarth0r]

Alpha algebras are afraid of sigma algebras

[@fierydino9402]

Thank you.

[@abramcz]

Whats with the disclaimer at the start about this being recorded by a robotic camera? Has MIT of all places has run out of AV techs to record a lecture properly?

[@carl3260]

100:28 is it simpler to say E \in A => E^c \in A, so R = E u E^c \in A, so ø = R^c \in A ? (i.e. more direct from initial axioms)

[@sunritroykarmakar4406]

Instead of the compactness argument at 21:00 why cant we consider (a + e/2 , b- e/2) is a subset of I so the lower bound on the outer measure of I is b-a - epsilon

[@komorebireed4714]

I got the same confusion. For a finite interval, regardless of whether it is closed, open, or half-closed, it seems sufficient to prove that m^*(I) = \ell(I) by [a+e/2, b-e/2] \sub I \sub [a-e/2, b-e/2].

[@tuongnguyen9391]

I thought this was some measure theory stuff :))) but some how it goes into functional analysis

[@BigFish-ii8zd]

This is pure measure theory, not functional analysis. I'm not sure why you are saying this "goes into functional analysis".

[@frankchen4229]

​@@BigFish-ii8zdffs....Open the description slot....

[@freeeagle6074]

Everything in maths is functional.

[@joenissan]

Exactly what is this used for? When would this be used in life?

[@TheTacticalDood]

This is building up for the Lebesgue integral which is used everywhere. Proability theory is a big application for instance.