https://courses.pzhou.org/ 2026-04-17T11:17:25+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture-1&rev=1771684870&do=diff Lecture 1 [ note], video * Motivation for Lebesgue measure (read Tao 7.1) * What is the definition of outer-measure? * Hmm, the outer-measure of a closed box? Why is it so complicated? * Interlude, the Banach-Tarski sphere. Discussion Time: * How to prove that $\{1,2,3\}$ has zero outer-measure? text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_2&rev=1771684870&do=diff Lecture 2 [ note], video Last time we had the definition of outer measure, and we basically followed Tao-II's presentation. This time, we will go through Lemma 7.2.5 (relatively easy) and Lemma 7.2.6 (about outer measure of a box, a bit hard). Pugh gives a different proof for the outer measure of a box being what it supposed to be, namely the naive volume, and he uses Lebesgue number. I am going to follow Tao's approach, although it is longer. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_3&rev=1771684870&do=diff Lecture 3 [note] video Today we continue going over Tao's sequence of Lemma 7.4.2 - 7.4.11 Lemma 7.4.2 I will prove the easy case (1-dim), you will do the general case in HW. Let $E = (0,+\infty)$ be the open half space in $\R$. For any subset $A \In \R$, we need to prove that $$ m^*(A) = m^*(A \cap E^c) + m^*(A \cap E). $$ Let $A_+ = A \cap E$ and $A_- = A \cap E^c$, note that if $0 \in A$, then $0 \in A_-$. First, we note that $$ m^*(A) \leq m^*(A_-) + m^*(A_+)$$ by finite sub-additivi… text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_4&rev=1771684869&do=diff Lecture 4 [note], video Cor 7.4.7 If $A \In B$ are both measurable, then $B \RM A$ is measurable, and $m(B \RM A) = m(B) - m(A)$. We need to show that for any subset $E \In \R^n$... wait a second, do we really need to go by definitions again? After all these preparations, we should be able to exploit our sweat. Hint text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_5&rev=1771684870&do=diff Lecture 5 $\gdef\mcal{\mathcal{M}}$ video Welcome back to in-person instruction. I will continue type in here as a way to prepare for class. After a long toil of last two weeks, we have established the existence of measurable sets and Lebesgue measure on $\R^n$. We know open sets and closed sets are measurable, and countable operations won't take us away from measurable sets. The Lebesgue measure on measurable sets satisfies all the intuitive properties that you wish it has. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_6&rev=1771684870&do=diff Lecture 6 video Theorem 21 If $E \In \R^n, F \In \R^k$ are measurable, then $E \times F$ is measurable, with $m(E) \times m(F) = m(E \times F)$. Let's first treat some special case. If $m(E)=0$, and $m(F) = \infty$, what is $m(E \times F)$? You have seen a special case as $m ( \{ 0 \} \times \R)=0$ in $\R^2$. The general proof is similar, for each $\epsilon$, and each $n \in \N$, we can find a countable collection of boxes that covers $E \times B(0, n)$ with total volume less than $\epsilo… text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_7&rev=1771684870&do=diff Lecture 7 video We will first follow Pugh's approach, then we will cover Tao's approach in exercises. * Use undergraph of a non-negative function to define measurability and its measure. If the measure is finite, then call this function integrable. text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_8&rev=1771684869&do=diff Lecture 8 video We finished Pugh 6.6. We mainly go over $\int f+g = \int f + \int g$. text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_9&rev=1771684869&do=diff Lecture 9 We will cover Tao's 7.5 and 8.1 today. Here we will use Tao's definition of measurable set, and Lebesgue integration, which a priori is not the same as Pugh's. Tao 7.5: Measurable function Let $\Omega \In \R^m$ be measurable, and $f: \Omega \to \R^m$ be a function. If for all open sets $V \In \R^m$, we have $f^{-1}(V)$ being measurable, then $f$ is called a measurable function. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_10&rev=1771684870&do=diff Lecture 10 We did Tao 8.2. Main result is monotone convergence theorem: given a monotone increasing sequence of non-negative measurable functions $f_n$, we have $$ \int \lim f_n = \lim \int f_n$$ or equivalently $$ \int \sup f_n = \sup \int f_n$$ The $\geq $ direction is easy, the $\leq$ direction is hard, which requires 3 steps lowering of the LHS $\int \sup f_n$: text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_11&rev=1771684869&do=diff Lecture 11 Today we covered Tao 8.3, 8.4, and 8.5. Here is the video, but I made a stupid mistake regarding Fubini theorem. I made a mistake in today's presentation in 8.5. Namely, given a measurable function $f(x,y)$. First of all, for a fixed $x$, the function $f_x(y) = f(x,y)$ as a function of $y$, may not be measurable at all. For example, take a measurable subset $E \In \R^2$, it is possible that certain slice $E_x = E \cap \{x\} \times \R$, when viewed as a subset of $\R$, is non-meas… text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_12&rev=1771684869&do=diff Lecture 12 $\gdef\uint{\overline{\int}}$ $\gdef\lint{\underline{\int}}$ * Redo Fubini's Theorem (Tao 8.5.1) * Vitali Covering Lemma. Upper and Lower Lebesgue integral To deal with possibly non-integrable functions, we need to define 'upper Lebesgue integral' and 'lower Lebesgue integral', which works for non-integrable functions $$ \overline{\int} f(x) = \inf \{ \int g(x), \text{g absolutely integrable, and $g(x)>f(x)$} \} $$ similarly for lower Lebesgue integral. By monotonicity of in… text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_13&rev=1771684870&do=diff Lecture 13 $\gdef\vcal{\mathcal V}$ Last time we were in the middle of proving Vitali Covering Lemma, which informally says: “given a Vitali cover of a bounded measurable set $A$ using closed balls and given any $\epsilon>0$, we can find an almost cover of $A$ by countably many disjoint closed balls, where we waste no more than $\epsilon$ in the cover, and we miss only a null set in the cover text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_14&rev=1771684869&do=diff Lecture 14 We finish the proof of Lebesgue density theorem. video text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_15&rev=1771684870&do=diff Lecture 15 video Last time, we considered the (long and hard) Lebesgue density theorem, which says, given any Lebesgue locally integrable function $f: \R^n \to \R$, then for almost all $p$, the density $\delta(p,f)$ of $f$ at $p$ exists and equals to the value $f(p)$. text/html 2026-02-21T14:41:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_16&rev=1771684869&do=diff Lecture 16: Linear algebra * from matrix to linear transformation. (basis free philosophy) * $V \otimes W$, $Hom(V, W)$, $V^*$ * norm on vector spaces. norm of a linear transformation. * continuity of linear transformation. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_17&rev=1771684870&do=diff Lecture 17 We follow Pugh 5.2. * A function $f: \R^n \to \R^m$ is differentiable at $p \in \R^n$ if $f$ can be approximable by a constant plus a linear term plus a remainder. * If approximation exists, then the differential is unique (and has a formula) text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:lecture_18&rev=1771684870&do=diff Lecture 18 video $\gdef\pa{\partial}$ Last time we have given the definition what it means for a map $f: U \to \R^m$ where $U \In \R^n$ to be differentiable at a point $p$, and we proved a bunch of nice properties about $(Df)_p$, like linear dependence on $f$, chain rules etc. text/html 2026-02-21T14:41:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math105-s22:notes:start&rev=1771684870&do=diff Notes