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math105-s22:s:selenali:start

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math105-s22:s:selenali:start [2022/03/10 06:14]
selenajli [Homework] 		math105-s22:s:selenali:start [2026/02/21 14:41] (current)
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 {{ :math105-s22:s:selenali:math105hw6.pdf |Homework 6}} {{ :math105-s22:s:selenali:math105hw6.pdf |Homework 6}}
  
-{{ :math105-s22:s:selenali:math105hw7.pdf |Homework 7}}+{{ :math105-s22:s:selenali:math105hw7_2.pdf |Homework 7}} 
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 +{{ :math105-s22:s:selenali:math105hw8_2.pdf |Homework 8}} 
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 +{{ :math105-s22:s:selenali:math105hw9.pdf |Homework 9}} 
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 +{{ :math105-s22:s:selenali:math105hw10.pdf |Homework 10}} 
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 +{{ :math105-s22:s:selenali:math105hw11.pdf |Homework 11}} 
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 +{{ :math105-s22:s:selenali:math105hw12.pdf |Homework 12}} 
 + 
 +===== Final Essay ===== 
 +Here are some notes I wrote on probability measure spaces for my final essay.  
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 +{{ :math105-s22:s:selenali:Math105FinalEsaay.pdf |Probability Measure Spaces}} 
 ===== Other Notes/Thoughts ===== ===== Other Notes/Thoughts =====
 ==== Uncountable Coverings ==== ==== Uncountable Coverings ====
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 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. 
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 +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.1646892842.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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