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--- math105-s22:s:selenali:start [2022/02/17 05:21]
selenajli [Homework]
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 {{ :math105-s22:s:selenali:math105lec7.pdf |Lecture 7}}
+{{ :math105-s22:s:selenali:math105lec8.pdf |Lecture 8}}
+{{ :math105-s22:s:selenali:math105lec9.pdf |Lecture 9}}
+{{ :math105-s22:s:selenali:math105lec10.pdf |Lecture 10}}
 ===== Homework =====
 {{ :math105-s22:s:selenali:105hw1.pdf |Homework 1}}
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 {{ :math105-s22:s:selenali:math105hw3_2.pdf |Homework 3}}
-{{ :math105-s22:s:selenali:math105hw4.pdf |Homework 4}}
+{{ :math105-s22:s:selenali:math105hw4_2.pdf |Homework 4}}
+{{ :math105-s22:s:selenali:math105hw5_2.pdf |Homework 5}}
+{{ :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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 One difference between the Riemann and Lebesgue integrals is that the Riemann integral is only defined for bounded functions on bounded domains. Since the Lebesgue integral is not subject to these restrictions, it opens up the possibility of integrating over a wider range of functions. A classic example is the function $\chi_{\mathbb{Q}}$, which takes on the value $1$ at all the rationals and $0$ at all the irrationals. Thus, we see that the Lebesgue integral is much more general than the Riemann integral.
+==== Lebesgue Measure Theory Summary ====
+Tao's approach to Lebesgue measure theory starts by defining a $\textbf{simple function}$, which takes on finitely many possible values. We first define how to integrate a simple function, and then use that to define how to integrate measurable functions.
+Simply put, the integral of a simple function is defined to be the product of each possible value with the measure of the subset of the region of integration that corresponds to the possible value. Then, to integrate non-negative measurable functions, we define the integral of a measurable function f to be the supremum of the integral of non-negative simple functions that are less than or equal to f.
+Some of the key results for Lebesgue integration of non-negative functions include:
+$\textbf{Lebesgue Monotone Convergence Theorem}$: For a sequence $(f_n)$ of functions from $\Omega \rightarrow \mathbb{R}$, where $\Omega \subseteq \mathbb{R}^n$ is measurable and $\forall x\in \Omega$,
+$$0 \leq f_1(x) \leq f_2(x) \leq ...$$
+we have that
+$$0 \leq \int_{\Omega} f_1(x) \leq \int_{\Omega} f_2(x) \leq ...$$
+and
+$$\int_{\Omega} \sup{f_n} = \sup{\int_{\Omega} f_n}$$
+The proof of the inequality in the $\geq$ direction is simple. For the opposite direction, we first use the definition of Lebesgue integration to look at simple functions less than or equal to $\sup{f_n}$. Then, we use the trick where we show the inequality holds after cutting by a factor of an arbitrarily small epsilon. Finally, after cutting our simple function $s$ to $(1-\epsilon)s$, we integrate over $E_n \subseteq \Omega$, where $E_n$ is the set of values where $f_n$ is greater than $(1-\epsilon)s$. Observing that $\int_{E_n}(1-\epsilon)s \leq \int_{\Omega} f_n$ and taking the supremum completes the proof. In summary, the key idea used in this proof is that to show an inequality, we can cut by arbitrarily small amounts in multiple places, and taking the as $\epsilon$ tends to zero shows the result.
+$\textbf{Interchange of Integration and Addition}$: $\int_{\Omega} (f+g) = \int_{\Omega} f + \int_{\Omega} g$
+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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