Differences

This shows you the differences between two versions of the page.

--- math105-s22:s:selenali:start [2022/03/03 21:52]
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 ====
@@ Line 84: / Line 102: @@
 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.

Lecture Notes

User Tools

Site Tools

Differences

Page Tools