diff --git a/.gitignore b/.gitignore new file mode 100644 index 00000000..a4ee2280 --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +content/Excalidraw/**/*.md diff --git a/content/Definitions/Functions/Infimum.md b/content/Definitions/Functions/Infimum.md new file mode 100644 index 00000000..cb680bd0 --- /dev/null +++ b/content/Definitions/Functions/Infimum.md @@ -0,0 +1,7 @@ +# Definition +Say we have $A_{m} = \{ x_{n}\, | \, n \geq m \}$ +Then $\inf A =$ greatest lower bound of $A$. +> [!note] What is the "lower bound"? +> $c \lt a,\ \forall a \in A$ + +The infimum is denoted by $\inf$. \ No newline at end of file diff --git a/content/Definitions/Functions/Supremum.md b/content/Definitions/Functions/Supremum.md new file mode 100644 index 00000000..61002baa --- /dev/null +++ b/content/Definitions/Functions/Supremum.md @@ -0,0 +1,5 @@ +# Definition +Say we have $A_{m} = \{ x_{n}\, | \, n \geq m \}$ +Then $\sup A$ is the least upper bound of $A$. + +The **supremum** is denoted by $\sup$. \ No newline at end of file diff --git a/content/Definitions/Measure Theory/Lebesgue's Dominated Convergence Theorem.md b/content/Definitions/Measure Theory/Lebesgue's Dominated Convergence Theorem.md new file mode 100644 index 00000000..a8c3d89f --- /dev/null +++ b/content/Definitions/Measure Theory/Lebesgue's Dominated Convergence Theorem.md @@ -0,0 +1,10 @@ +# Definition +Let $g$ be a real function on $X$. + +Define $g^{+} = \max \{ g, 0 \}$, $g^{-} = -\min \{ g, 0 \}$. + +Then $g = g^{+} - g^{-}$ and $g^{\pm} \geq 0$. + +> [!example]- +> ![[Drawing 2025-03-06 11.57.37.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 11.57.37.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 11.57.37.excalidraw.light.svg|light exported image]]%% diff --git a/content/Definitions/Measure Theory/Lebesgue's Monotone Convergence Theorem.md b/content/Definitions/Measure Theory/Lebesgue's Monotone Convergence Theorem.md new file mode 100644 index 00000000..973d37e9 --- /dev/null +++ b/content/Definitions/Measure Theory/Lebesgue's Monotone Convergence Theorem.md @@ -0,0 +1,4 @@ +# Definition +Say $X$ has a [[Measure|measure]] $\mu$, and let $f_{n} : X \to [0, \infty]$ be [[Measurable|measurable]] and $f_{1} \leq f_{2} \leq f_{3} \leq \dots$. + +Then $\int f_{m} \, d\mu \to \int \lim_{ n \to \infty } f_{n} \, d\mu$ as $m \to \infty$. \ No newline at end of file diff --git a/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.png b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.png new file mode 100644 index 00000000..17a6e08f Binary files /dev/null and b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.png differ diff --git a/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.svg b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.svg new file mode 100644 index 00000000..ea2cab6a --- /dev/null +++ b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.dark.svg @@ -0,0 +1,2 @@ + \ No newline at end of file diff --git a/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.png b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.png new file mode 100644 index 00000000..8a7c6684 Binary files /dev/null and b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.png differ diff --git a/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.svg b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.svg new file mode 100644 index 00000000..5ce481cf --- /dev/null +++ b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 11.57.37.excalidraw.light.svg @@ -0,0 +1,2 @@ + \ No newline at end of file diff --git a/content/Excalidraw/Lecture 15/Drawing 2025-03-06 12.14.05.excalidraw.dark.png b/content/Excalidraw/Lecture 15/Drawing 2025-03-06 12.14.05.excalidraw.dark.png new file mode 100644 index 00000000..f437f007 Binary 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$f^{-1}([0, a\rangle) = \cap_{n=1}^{\infty} \underbrace{f_{n}^{-1}([0, a \rangle > $\implies f(x) < a$ Let $b \equiv \lim_{ n \to \infty } \int f_{n} \, d\mu \leq \int f \, d\mu$ as $f_{n} \leq f$. -Let $0 \leq s \leq f$, $s$ [[Measure|measure]] simple, and $c \in \langle 0, 1 \rangle$. +Let $0 \leq s \leq f$, $s$ [[Measurable|measurable]] [[Simple Function|simple function]], and $c \in \langle 0, 1 \rangle$. Let $A_{n} = \{ x \in X \, | \, c \times s(x) \leq f_{n}(x) \} = (\underbrace{f_{n} - cs}_{measurable function})^{-1}(\underbrace{[0, \infty]}_{open})$. Then $A_{1} \subset A_{2} \subset A_{3} \subset \dots$ [[Measurable|measurable]], and $\cup_{n} A_{n} \overbrace{=}^{\text{(*)}} X$ @@ -48,4 +48,65 @@ Then $A_{1} \subset A_{2} \subset A_{3} \subset \dots$ [[Measurable|measurable]] > > 1. $A \mapsto \int_{A} s \, d\mu$ [[Measure|measure]] ($s = 1 \implies \int_{A} s \, d\mu = \mu(A)$) > > 2. For any measure $\nu$ and $A_{1} \subset A_{2} \subset \dots$ [[Measurable|measurable]] $\implies \nu(\cup A_{n}) = \lim_{ n \to \infty } \nu(A_{n})$ -QED. \ No newline at end of file +QED. +# Corollary - Fatou's Lemma +Have [[Measure|measure]] $\mu$ on $X$, and $f_{n} : X \to [0, \infty]$ [[Measurable|measurable]]. Then $\int \lim_{ n \to \infty } \inf f_{n} \, d\mu \le \lim_{ n \to \infty } \inf \int f_{n} \, d\mu$ +> [!info] What is $\lim\inf$? +> Definition of [[Infimum|infimum]] (it is basically the opposite of a [[Supremum|supremum]]). +> +> $\{ x_{n} \} \subset [0, \infty]$ +> $\lim_{ n \to \infty }\inf x_{n} = \sup_{m}\inf_{n \geq m} x_{n}$ +> +> $\inf_{n \geq m} = y_{m} \leq y_{m+1} \leq \dots$ + +## Proof +Use [[Lebesgue's Monotone Convergence Theorem]] on $g_{m} = \inf_{n \geq m} f_{n}$. +$g_{1} \leq g_{2} \leq \dots$ are [[Measurable|measurable]] functions. +QED +# Lebesgue's Dominated Convergence Theorem +(Also defined [[Lebesgue's Dominated Convergence Theorem|here]], it's the same thing) + +Let $g$ be a real function on $X$. + +Define $g^{+} = \max \{ g, 0 \}$, $g^{-} = -\min \{ g, 0 \}$. + +Then $g = g^{+} - g^{-}$ and $g^{\pm} \geq 0$. +> [!example]- +> ![[Drawing 2025-03-06 11.57.37.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 11.57.37.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 11.57.37.excalidraw.light.svg|light exported image]]%% +# Definition +Given [[Measure|measure]] $\mu$ on $X$. +Define $L'(\mu) = \left\{ f : X \to \mathbb{C}\ \text{measurable and}\ \int |f| \, d\mu \lt \infty \right\}$. + +Define integral for $f \in L'(\mu)$ by $\int f \, d\mu \equiv \int (\mathrm{Re}f)^{+} \, d\mu - \int (\mathrm{Re}f)^{-} \, d\mu + i \int (\mathrm{Im} f)^{+} \, d\mu - i \int (\mathrm{Im} f)^{-} \, d\mu$. + +Use $f = \mathrm{Re} f + i \mathrm{Im} f = (\mathrm{Re} f)^{+} - (\mathrm{Re} f)^{-} + i((\mathrm{Im} f)^{+} - (\mathrm{Im} f)^{-})$. + +The integral definition makes sense as each integral on the RHS is finite. +($(\mathrm{Re} f)^{+} \leq |f|$) +## Lemma +Given [[Measure|measure]] $f : X \to [0, \infty]$. + +Then $\exists$ [[Measurable|measurable]] [[Simple Function|simple functions]] $s_{n}$ such that +1. $0 \leq s_{1} \leq s_{2} \leq \dots \leq f$ +2. $\lim_{ n \to \infty } s_{n} = f$ [[Pointwise|pointwise]] +### Proof +Define $h_{n} : [0, \infty] \to [0, \infty \rangle$ by +![[Drawing 2025-03-06 12.14.05.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 12.14.05.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 12.14.05.excalidraw.light.svg|light exported image]]%% +![[Drawing 2025-03-06 12.16.07.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 12.16.07.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 12.16.07.excalidraw.light.svg|light exported image]]%% +Continue like this. +![[Drawing 2025-03-06 12.23.12.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 12.23.12.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 12.23.12.excalidraw.light.svg|light exported image]]%% + +Have $0\leq h_{1} \leq h_{2} \leq \dots \leq h_{n} \to l\ \text{as}\ n \to \infty$. + +![[Drawing 2025-03-06 12.24.31.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 12.24.31.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 12.24.31.excalidraw.light.svg|light exported image]]%% + +Set $s_{n} = h_{n} \circ f$. + +![[Drawing 2025-03-06 12.25.26.excalidraw.dark.svg]] +%%[[Drawing 2025-03-06 12.25.26.excalidraw.md|🖋 Edit in Excalidraw]], and the [[Drawing 2025-03-06 12.25.26.excalidraw.light.svg|light exported image]]%% +