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--- math105-s22:s:selenali:start [2022/03/03 22:44]
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: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 =====
 ==== Uncountable Coverings ====
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 Tao's proof of this theorem is easier to follow (at least in my opinion) than Pugh's. The main idea here is to construct increasing sequences of simple functions approaching $f$ and $g$, and apply results for simple functions and take the supremum.
+$\textbf{Fatou's Lemma}$: For a sequence $(f_n)$ of functions from $\Omega \rightarrow \mathbb{[0, \infty]}$, where $\Omega \subseteq \mathbb{R}^n$ is measurable,
+$$\int_{\Omega} \lim_{n \rightarrow \infty} \inf{f_n} \leq \lim_{n\rightarrow \infty} \inf{\int_{\Omega} f_n}$$
+The key idea in the proof is rewriting the liminf on the LHS as a sup, and apply the Monotone Convergence Theorem.
+Finally, for integration of measurable functions that are not necessarily non-negative, Tao introduces the idea of an $\textbf{absolutely integrable}$ function, which is a function whose absolute value is Lebesgue integrable. Also, we introduce the positive and negative part of a function $f$, defined by
+$$f^+ = \max{(f, 0)}, f^- = -\min{(f, 0)}$$
+Then, we define
+$$\int f = \int f^+ - \int f^-$$
+Many of the properties of the integral of non-negative functions can be generalized to the general Lebesgue integral.
+A key theorem involving absolutely integrable functions is:
+$\textbf{Lebesgue Dominated Convergence Theorem}$:  Let there be a sequence $(f_n)$ of functions from $\Omega \rightarrow \mathbb{R}^*$ converging pointwise, where $\Omega \subseteq \mathbb{R}^n$ is measurable. If $\exists F: \Omega \rightarrow [0, \infty]$ s.t. $|f_n| \leq F $ for each $n$,
+$$\int_{\Omega} \lim_{n\rightarrow\infty} f_n = \lim_{n\rightarrow\infty}\int_{\Omega}f_n$$
+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.

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