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math105-s22:s:david_alcalay:start [2022/01/25 00:23]
135.180.101.5 		math105-s22:s:david_alcalay:start [2026/02/21 14:41] (current)
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 ====David Alcalay==== ====David Alcalay====
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 ==About Me== ==About Me==
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 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). 
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-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.
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 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}}
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-==Bernstein Sets and Nonmeasurability==+ 
 +===Bernstein Sets and Nonmeasurability===
  
 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 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 intersection $B \cap F$ is nonempty.+**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 set $B$ of real numbers such that $B$ and $B^c$ both meet every closed uncountable subset of the real numbers. (pg. 23)+**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. The proof relies on the Well-Ordering Principle, and uses transfinite induction to choose points from each uncountable closed subset of the reals.
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 **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).  **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.+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.1643070231.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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