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Blogs about: 245a Real Analysis

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“Introduction to measure theory” now published

My graduate text on measure theory (based on these lecture notes) is now published by the AMS as part of the Graduate Studies in Mathematics series.  (See also my own blog page for this book, which am… more →

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"Introduction to measure theory" now published — 3 comments

Terence Tao wrote 8 months ago: My graduate text on measure theory (based on these lecture notes) is now published by the AMS as par … more →

Tags: math.CA, Update, book, Measure Theory

245A, Notes 6: Outer measures, pre-measures, and product measures — 21 comments

Terence Tao wrote 1 year ago: In this course so far, we have focused primarily on one specific example of a countably additive mea … more →

Tags: math.CA, math.PR, Caratheodory extension theorem, outer measure, Hahn-Kolmogorov extension theorem, Kolmogorov extension theorem, product measure

245A: Problem solving strategies — 23 comments

Terence Tao wrote 1 year ago: This is going to be a somewhat experimental post. In class, I mentioned that when solving the type o … more →

Tags: math.CA, math.GM, tricks, Measure Theory, Problem-solving

245A, Notes 5: Differentiation theorems — 35 comments

Terence Tao wrote 1 year ago: Let be a compact interval of positive length (thus ). Recall that a function is said to be different … more →

Tags: math.CA, absolute continuity, bounded variation, Hardy-Littlewood maximal inequality, Lebesgue differentiation theorem, rising sun lemma, total variation

Covering a non-closed interval by disjoint closed intervals — 47 comments

Terence Tao wrote 1 year ago: The following question came up in my 245A class today: Is it possible to express a non-closed interv … more →

Tags: math.GN, Intervals

245A, Notes 4: Modes of convergence — 27 comments

Terence Tao wrote 1 year ago: If one has a sequence of real numbers , it is unambiguous what it means for that sequence to converg … more →

Tags: math.CA, dominated convergence theorem, Egorov's theorem, convergence in measure, almost everywhere convergence, uniform convergence, uniform integrability

245A, Notes 3: Integration on abstract measure spaces, and the convergence theorems — 39 comments

Terence Tao wrote 1 year ago: Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces … more →

Tags: math.CA, sigma-algebra, measure space, monotone convergence theorem, measurable space, Lebesgue dominated convergence theorem, Fatou's lemma, absolutely integrable functions

245A, Notes 2: The Lebesgue integral — 34 comments

Terence Tao wrote 1 year ago: In the previous notes, we defined the Lebesgue measure of a Lebesgue measurable set , and set out th … more →

Tags: math.CA, Lebesgue integration, Egorov's theorem, Littlewood's three principles, Lusin's theorem, absolute integrability

245A, Notes 1: Lebesgue measure — 67 comments

Terence Tao wrote 1 year ago: In the prologue for this course, we recalled the classical theory of Jordan measure on Euclidean spa … more →

Tags: math.CA, Axiom of Choice, lebesgue measure, outer measure

245A, prologue: The problem of measure — 49 comments

Terence Tao wrote 1 year ago: One of the most fundamental concepts in Euclidean geometry is that of the measure of a solid body in … more →

Tags: math.CA, elementary sets, Riemann Integral, Jordan measure, piecewise constant functions

Course announcement: 245A, Real analysis — 17 comments

Terence Tao wrote 1 year ago: In a few weeks (and more precisely, starting Friday, September 24), I will begin teaching Math 245A, … more →

Tags: Admin, Announcement


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