Lecture Notes

User Tools

Site Tools

You are here: Course Notes » Math 105: A second course to real analysis (2022 Spring) » Students Area » David Alcalay
math105-s22:s:david_alcalay: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:david_alcalay:start [2022/01/25 00:05]
135.180.101.5 		math105-s22:s:david_alcalay:start [2026/02/21 14:41] (current)
Line 1: Line 1:
-===David Alcalay===+====David Alcalay==== 
  
 ==About Me== ==About Me==
Line 6: Line 7:
  
 Hello! My name is David Alcalay. I'm a third year pure-math major here at UC Berkeley. I grew up in Davis, California, and transferred a year ago from community college. Analysis has always been my favorite area of mathematics, so I'm really excited to be taking Math 105. I'm really interested in Cantor sets, and also the topic of constructible numbers (think ruler-and-compass).  Hello! My name is David Alcalay. I'm a third year pure-math major here at UC Berkeley. I grew up in Davis, California, and transferred a year ago from community college. Analysis has always been my favorite area of mathematics, so I'm really excited to be taking Math 105. I'm really interested in Cantor sets, and also the topic of constructible numbers (think ruler-and-compass). 
- 
-I have bipolar disorder, type 2, a mental illness characterized by periods of high energy, called hypomanic episodes, and periods of depression. I'm very passionate about the subject, and welcome any questions about my personal experiences with the illness, navigating UC Berkeley as a student with a mental illness, or more generally. 
  
 Outside of my education, I spend most of my time reading, hiking, doing art, and playing board games. Outside of my education, I spend most of my time reading, hiking, doing art, and playing board games.
Line 21: Line 20:
  
 My dream, upon finishing my education, is to become a research mathematician, specifically researching Cantor sets. My dream, upon finishing my education, is to become a research mathematician, specifically researching Cantor sets.
 +
 +==Homework==
 +  * {{ :math105-s22:s:david_alcalay:math_105_homework_1.pdf |Homework 1 (Incomplete)}}
 +  * {{ :math105-s22:s:david_alcalay:math_105_homework_2.pdf |Homework 2 (Rough Draft)}}
 +
  
 ==Lecture Notes== ==Lecture Notes==
   * {{ :math105-s22:s:david_alcalay:1.18.22_math_105_lecture_notes.pdf |Lecture 1 - 18 January 2022}}   * {{ :math105-s22:s:david_alcalay:1.18.22_math_105_lecture_notes.pdf |Lecture 1 - 18 January 2022}}
   * {{ :math105-s22:s:david_alcalay:1.20.22_math_105_lecture_notes.pdf |Lecture 2 - 20 January 2022}}   * {{ :math105-s22:s:david_alcalay:1.20.22_math_105_lecture_notes.pdf |Lecture 2 - 20 January 2022}}
 +  * {{ :math105-s22:s:david_alcalay:1.25.22_math_105_lecture_notes.pdf |Lecture 3 - 25 January 2022}}
 +  * {{ :math105-s22:s:david_alcalay:2.8.21_math_105_lecture_notes.pdf |Lecture 7 - 8 February 2022}}
  
  
-==Musings on Course Material==+ 
 +===Musings on Course Material===
  
   * What modifications need to be made to the content covered in lecture so far so that the concepts of measure we've defined make sense in an arbitrary metric space?   * What modifications need to be made to the content covered in lecture so far so that the concepts of measure we've defined make sense in an arbitrary metric space?
Line 34: Line 41:
     * When metrics define measures, are the measures unique?     * When metrics define measures, are the measures unique?
     * How do isometric embeddings from one metric space to another affect measures that come from the metrics on the domain and codomain? (Does this question even make sense?)     * How do isometric embeddings from one metric space to another affect measures that come from the metrics on the domain and codomain? (Does this question even make sense?)
 +
  
  
Line 40: Line 48:
 The following proof is from Measure and Category, second edition, by John C. Oxtoby. The following proof is from Measure and Category, second edition, by John C. Oxtoby.
  
-**Definition:** A Bernstein set +**Definition:** A subset $B \subseteq \mathbb{R}$ is called a Bernstein set if the following holds: For each subset $F \subseteq \mathbb{R}$ such that $F$ is an uncountable closed subset of $\mathbb{R}$, the intersections $B \cap F$ and $B^c \cap F$ is nonempty. 
 + 
 +**Theorem (F. Bernstein):** There exists a Bernstein set. (pg. 23) 
 + 
 +The proof relies on the Well-Ordering Principle, and uses transfinite induction to choose points from each uncountable closed subset of the reals. 
 + 
 +**Theorem:** Bernstein sets are not Lebesgue-measurable. 
 + 
 +//proof.// Let $B$ be a Bernstein set. Let $A \subseteq B$ be a Lebesgue-measurable subset. Suppose that $F \subseteq A$ is closed. If $F$ was uncountable, then $F$ would meet both $B$ and $B^c$ since both $B$ and $B^c$ are Bernstein sets by Bernstein's theorem, and $F$ meeting $B^c$ is a contradiction. Thus $F$ is at most countable, and so $m(F) = 0$. 
 + 
 +**Lemma:** For a set $A$, we have $m^*(A) = \sup\{m(X) \mid X \subseteq A \text{ and } X \text{ is closed}\}$ (pg.15-16).  
 + 
 +The by the lemma above, we thus have $m(A) = 0$. Hence if $B$ is Lebesgue measurable, then $m(B) = 0$. But $B^c$ is also a Bernstein set, and thus $m(B^c) = 0$. But $B \cup B^c = \mathbb{R}$, and so $m(B)$ and $m(B^c)$ can't both be zero. It follows that $B$ is not Lebesgue measurable. $\square$ 
 + 
  
  
math105-s22/s/david_alcalay/start.1643069103.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