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Source of most content: Gunhee Cho and RCA by Rudin. Almost everywhere Def. Given a M.S. $(X, F, \mu)$, we say a statement $P$ holds almost everywhere (a.e.) on $X$ if $\exists N \in F$ s.t. $\mu(N)=0$ and $P$ is true for $X\setminus N$. NB1; Give the relation to measurable fns $f,g: X \to [0, \infty]$, $f \sim g$ if $\mu( \{x \in X : f(x)\n...

Source of most content: Gunhee Cho and RCA by Rudin. Converse of Radon-Nikodym theorem \[\text{Given a M.S. } (X, F, \mu), \text{ let } f: X \to [0, \infty] \text{ be measurable function. Define } \varphi: F \to [0, \infty] (E\in F \mapsto \int_E f \mathrm{d}\mu). \nonumber\] \[\text{Then, } \varphi \text{ is a measure. Moreover, given a me...

Source of most content: Gunhee Cho and RCA by Rudin. Measure Def. Given a measurable space $(X, \mathfrak{M})$, if $\mu: \mathfrak{M} \rightarrow [0, \infty]$ is a function satisfying countable additivity, then we call $\mu$ a measure on $X$, and $(X, \mathfrak{M}, \mu)$ is called measure space (M.S.) NB1; Countable additivity means for any ...

Source of most content: Gunhee Cho and RCA by Rudin. Borel set Def. Given a T.S. $(X, T)$, we call $\mathfrak{B}=\cap_{T \subseteq \mathfrak{M}}\mathfrak{M}$, the smallest $\sigma$-algebra containing $T$ Borel $\sigma$-algebra of $X$ and the element of it is called by a Borel set. NB1; By definition, every open set and closed set are a Borel...

Source of most content: Gunhee Cho. Measurable set Def. Given a set $X$, we call $\mathfrak{M} \subseteq 2^X$ a $\sigma$-algebra in $X$ if it satisfies $X \in \mathfrak{M}$. If $A \in \mathfrak{M}$, then $A^c \in \mathfrak{M}$. If $\{ A _n \} _{n=1}^\infty \subseteq \mathfrak{M}$, then $\cup _{n=1}^\infty A _n \in \mathfrak{M}...

Source of most content: Gunhee Cho and Lecture notes by Parimal Parag. Cover and Compact Def. Given a T.S. $(X, T_X)$, $\{ A_\alpha \}_{\alpha \in J} \subset 2^X$ is a cover of $X$ if $X \subset \bigcup _{\alpha \in J} A _\alpha$. If $\{ A _\alpha \} _{\alpha \in J} \subset T_X$, then it is called open cover of $X$. Def. A T.S. $(X, T_X)$ ...

Source of most content: Gunhee Cho. Product Topology Def. Given an index set $J$, let $(A _\alpha) _{\alpha \in J}$ be a seq. of sets. Then, Generalized Cartesian product of $(A _\alpha) _{\alpha \in J}$ is $\Pi _{\alpha \in J} A _\alpha := \{ f: J \rightarrow \cup _{\alpha \in J} A _\alpha : \forall \alpha, f(\alpha) \in A _\alpha \} $. Fo...

Source of most content: Gunhee Cho and Lecture note by Matsumura Hausdorff Space Def. Given T.S. $(X, T_X)$, $\{ a_n\}_{n=1}^\infty \subseteq X$ and $a \in X$. We say $\{ a_n \}$ converges to $a$, $a_n \stackrel{n \rightarrow \infty}{\rightarrow} a$ if $\forall$ nbhd $V$ of $a$, $\exists N \in \mathbb{N}$ s.t. $\forall n \geq N$, $a_n \in V$....