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math105-s22:s:samuels [2022/02/24 06:38]
samuel_speas 		math105-s22:s:samuels [2026/02/21 14:41] (current)
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 Here are some {{ :math105-s22:s:measure_theory_notes.pdf | notes}}, mostly from Tao's Introduction to Measure Theory. Here are some {{ :math105-s22:s:measure_theory_notes.pdf | notes}}, mostly from Tao's Introduction to Measure Theory.
 A few exercises are thrown in, mostly sketches. I focused on material not found or emphasized in Analysis II. I'll update the pdf regularly, if anyone stopping by is interested in reading them I will be typesetting the occasional solution to exercises in this book. If anyone wants to do so together, or to discuss some exercises in general at any point, that would be nice! A few exercises are thrown in, mostly sketches. I focused on material not found or emphasized in Analysis II. I'll update the pdf regularly, if anyone stopping by is interested in reading them I will be typesetting the occasional solution to exercises in this book. If anyone wants to do so together, or to discuss some exercises in general at any point, that would be nice!
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 As Terence Tao said, the Lebesgue integral can handle "noise" or "error". The difference between the two and the motivation for the Lebesgue theory is demonstrated nicely by  $[0,1]\Q^2$, the "bullet-riddled-square". The square and the set of bullets $[0,1]^2\cup\Q^2$ have Jordan inner measure zero, and Jordan outer measure 1. The indicator function of the bullets over the square is not Riemann integrable, but it is Lebesgue integrable, as the "noise" $Q^2$ poses no problems.  As Terence Tao said, the Lebesgue integral can handle "noise" or "error". The difference between the two and the motivation for the Lebesgue theory is demonstrated nicely by  $[0,1]\Q^2$, the "bullet-riddled-square". The square and the set of bullets $[0,1]^2\cup\Q^2$ have Jordan inner measure zero, and Jordan outer measure 1. The indicator function of the bullets over the square is not Riemann integrable, but it is Lebesgue integrable, as the "noise" $Q^2$ poses no problems. 
 +
 +**Littlewood's Three Principals:
 +** 
 +(i) Every (measurable) set is nearly a finite sum of intervals;
 +
 +(ii) Every (absolutely integrable) function is nearly continuous;
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 +(iii) Every (pointwise) convergent subsequence of functions is nearly uniformly convergent.
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 +These three principals are an intuitive rephrasing of some results that we have learned. We can find such a sum in (i) using a G-delta cover, and expressing at open set as a countable collection of disjoint (up to some error), or almost disjoint boxes.
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 +Littlewood's second theorem is a consequence of Egorov's theorem. We can take some closed set on which our function is continuous, and an epsilon set on which it is not. This is also essentially Lusin's theorem, mentioned above.
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 +The third principle is essentially Egorov's theorem.
  
  
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 {{ :math105-s22:s:105-hw5.pdf |homework 5}} {{ :math105-s22:s:105-hw5.pdf |homework 5}}
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 +{{ :math105-s22:s:105-hw6.pdf |homework 6}}
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 +{{ :math105-s22:s:105-hw7.pdf |homework 7 (needs revision)}}
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 +{{ :math105-s22:s:105-hw8.pdf |homework 8}}
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 +{{ :math105-s22:s:105-hw9.pdf |homework 9}}
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 +{{ :math105-s22:s:105-hw10.pdf |homework 10}}
math105-s22/s/samuels.1645684693.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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