Lecture Notes

User Tools

Site Tools

You are here: Course Notes » Math 105: A second course to real analysis (2022 Spring) » Students Area » Selena Li
math105-s22:s:selenali:start

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision 		Previous revision
math105-s22:s:selenali:start [2022/03/04 04:14]
selenajli [Lebesgue Measure Theory Summary] 		math105-s22:s:selenali:start [2026/02/21 14:41] (current)
Line 35: Line 35:
  
 {{ :math105-s22:s:selenali:math105hw6.pdf |Homework 6}} {{ :math105-s22:s:selenali:math105hw6.pdf |Homework 6}}
 +
 +{{ :math105-s22:s:selenali:math105hw7_2.pdf |Homework 7}}
 +
 +{{ :math105-s22:s:selenali:math105hw8_2.pdf |Homework 8}}
 +
 +{{ :math105-s22:s:selenali:math105hw9.pdf |Homework 9}}
 +
 +{{ :math105-s22:s:selenali:math105hw10.pdf |Homework 10}}
 +
 +{{ :math105-s22:s:selenali:math105hw11.pdf |Homework 11}}
 +
 +{{ :math105-s22:s:selenali:math105hw12.pdf |Homework 12}}
 +
 +===== Final Essay =====
 +Here are some notes I wrote on probability measure spaces for my final essay. 
 +
 +{{ :math105-s22:s:selenali:Math105FinalEsaay.pdf |Probability Measure Spaces}}
 +
 ===== Other Notes/Thoughts ===== ===== Other Notes/Thoughts =====
 ==== Uncountable Coverings ==== ==== Uncountable Coverings ====
Line 110: Line 128:
  
 To prove this theorem, we use Fatou's Lemma on the sequences $F + f_n$ and $F - f_n$, both of which are non-negative from the condition. This allows us to lower bound $\int_{\Omega} f$ by the limsup of $\int_{\Omega} f_n$ and upper bound it by the liminf, which shows the desired result.  To prove this theorem, we use Fatou's Lemma on the sequences $F + f_n$ and $F - f_n$, both of which are non-negative from the condition. This allows us to lower bound $\int_{\Omega} f$ by the limsup of $\int_{\Omega} f_n$ and upper bound it by the liminf, which shows the desired result. 
 +
 +==== Littlewood's Three Principles ====
 +Littlewood's Three Principles summarize three main ideas in real analysis. The first principle says that every measurable set can be written as the union of finitely many intervals, up to a set of measure $\epsilon$. The second principle says that every measurable function is continuous on it's domain minus a set of measure $\epsilon$. The third principle says that a pointwise convergent sequence of measurable functions converges uniformly up to an $\epsilon$-set. 
 +
 +These three principles share a common theme that removing an arbitrarily small set can lead to nice properties, or that measurable functions/sets "nearly" have these properties. 
math105-s22/s/selenali/start.1646367240.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

Page Tools

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki