## Properties of the Riemann Integral

Theorem 6.1.9 Let $f:[a,b]\to\mathbb{R}$ be a bounded real valued Riemann integrable function with $Range f \subset [c,d]$. Let $\varphi:[c,d]\to\mathbb{R}$ ...

Theorem 6.1.9 Let $f:[a,b]\to\mathbb{R}$ be a bounded real valued Riemann integrable function with $Range f \subset [c,d]$. Let $\varphi:[c,d]\to\mathbb{R}$ ...

Definition (upper sum, lower sum) Let $[a,b]$ with $a<b$ be a closed and bounded interval in $\mathbb{R}$. By a partition of $\mathscr{P}$ of $[a,b]$ we m...

Theorem 4.2.13 (Intermediate Value Theorem for derivative) Let $f:I\to\mathbb{R}$ be differentiable function on the interval $I=[a,b]$. Then given $a,b\in I$...

Definition 5.2.1 Let $E\subset \mathbb{R}$ be a set and let $f:E\to\mathbb{R}$.

Definition 5.1.1 Let $I\subset \mathbb{R}$ be an interval and let $f:I\rightarrow\mathbb{R}$ be a function. Fix a $p\in I$. The derivative of $f$ at $p$ is ...

Definition 4.4.1 $E\subset \mathbb{R}, f:E\to\mathbb{R}, p$ is a limit point of $E\cap (p,\infty)$. $f$ has a right limit at $p$ if there exists a $L\in\math...

Definition 4.3.1 Let $E$ be a subset of $\mathbb{R}$ and let $f:E\to\mathbb{R}$ be a function. The function $f$ is uniformly continuous on $E$ if $\forall ...

Lemma 1

Definition 4.2.1 $f:E\to \mathbb{R}$ is continuous at $p\in E$, if $\forall \epsilon >0, \exists \delta >0$ such that $|x-p|<\delta \Rightarrow |f(...

4.1.1 Definition $E \subset \mathbb{R}, f: E \rightarrow \mathbb{R}$: a function, $p \in E^\prime$. $f$ has a limit at p if there exists $L\in \mathbb{R}$ su...

Definition of Cantor Set For each $P_i$ is non empty compact set and $P_0 \supset P_1 \supset P_2 \cdots$. Define $P$ as follows: \(\begin{align} P := \cap_{...

Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded

Theorem 3.1.13 $U \subset \mathbb{R}$ is open $\Rightarrow$ \(\exists \{I_n\}\): a finite or countable family of pairwise disjoint union of open intervals su...

Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...

Definition Let \(\{p_n\}_{n=1}^\infty\) be a sequence in $\mathbb{R}$. The sequence is a Cauchy sequence if $\forall \epsilon >0, \exists N\in \mathbb{N}$...

Definition 2.4.1 Let \(\{p_n\}\) be a sequence and let \(\{n_k\}\) be strictly increasing sequence, i.e. $n_1 < n_2 <n_3 < \cdots$. We call \(\{p_{n...

Definition 2.3.1 A sequence \(\{a_n\}\) is said to be (a) monotone increasing if $a_n \leq a_{n+1}$ for all $n\in\mathbb{N}$ (b) monotone decreasing if $a_...

Theorem 2.1.3 (Triangular inequality) For $\forall x,y\in \mathbb{R}, |x+y|\leq |x|+|y|$

Definition 1.7.2 For each positive integer $n\in \mathbb{N}$, let \(\mathbb{N}_n:=\{1,2,\ldots,n \}\). If $A$ is a set, we say (a) $A$ is finite if $A\sim \...

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

Distribution Function Let $X$ be a random variable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$. The distribution function (cumulative distribution functi...

Measurable function Let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces where $E,F,$ are sets and $\mathcal{E}$ and $\mathcal{F}$ are $\sigma...

The sample space $\Omega$ is the set of all possible outcomes of an experiment. Points in $\omega \in \Omega$ are outcomes or realizations.

Definition (Closed) A subset $A$ of a topological space $X$ is said to be closed if the set $X-A$ is open.

Definition (Product Topology on $X \times Y$) Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be topological spaces. The product space topology on $X\times...

Definition (Topology) A topology on a set $X$ is a collection $\mathfrak{T}$ having the following properties: (1) $\emptyset$ and $X$ are in $\mathfrak{T}$ (...

Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl...

Notation Let \(\mathfrak{B}=\{\mathbf{v}_1, \ldots, \mathbf{v}_n \} \text{ is a basis for } V.\) Define a function \([\cdot]_{\mathfrak{B}}:V \rightarr...

Definition 1 $\cdot: G\times G \to G$ is binary operation where we write $x\cdot y =xy$ for $x,y\in G$. $(G,\cdot)$ is group if it satisfies the followings ...

Definition 1 $\cdot: G\times G \to G$ is binary operation where we write $x\cdot y =xy$ for $x,y\in G$. $(G,\cdot)$ is group if it satisfies the followings ...

Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...