Created the Real Analysis Theorems and Definitions packet
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\chapter{The Generalized Riemann Integral}
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\subimport{./}{definition-and-main-properties.tex}
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\subimport{./}{improper-and-lebesgue-integrals.tex}
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\subimport{./}{infinite-intervals.tex}
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\subimport{./}{convergence-theorems.tex}
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\section{Convergence Theorems}
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\begin{theorem}[\textbf{Uniform Convergence Theorem}]
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Let $(f_k)$ be a sequence in $\mathcal{R}^*[a,b]$ and suppose that $(f_k)$ converges \textbf{uniformly} on $[a,b]$ to $f$. Then $f \in \mathcal{R}^*[a,b]$ and
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\[\displaystyle\int_{a}^{b}f=\lim\limits_{k \to \infty}\displaystyle\int_{a}^{b}f_k\]
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holds.
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\end{theorem}
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\begin{definition}
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A sequence $(f_k)$ in $\mathcal{R}^*(I)$ is said to be \textbf{equi-integrable} if for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $I$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine partition of $I$ and $k\in\N$, then $\left|S(f_k;\dot{\mathcal{P}})-\displaystyle\int_If_k\right|<\varepsilon$.
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\end{definition}
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\begin{theorem}[\textbf{Equi-integrability Theorem}]
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If $(f_k) \in \mathcal{R}^*(I)$ is equi-integrable on $I$ and if $f(x)=\lim f_k(x)$ for all $x \in I$, then $f \in \mathcal{R}^*(I)$ and
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\[\displaystyle\int_If=\lim\limits_{k \to \infty}\displaystyle\int_If_k\]
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\end{theorem}
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\begin{definition}
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We say that a sequence of functions on an interval $I \subseteq \R$ is \textbf{monotone increasing} if it satisfies $f_1(x) \leq f_2(x) \leq \dots \leq f_k(x) \leq f_{k+1}(x) \leq \dots$ for all $k \in \N$, $x \in I$. It is said to be \textbf{monotone decreasing} if it satisfies the opposite string of inequalities, and to be \textbf{monotone} if it is either monotone increasing or decreasing.
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\end{definition}
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\begin{theorem}[\textbf{Monotone Convergence Theorem}]
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Let $(f_k)$ be a monotone sequence of functions in $\mathcal{R}^*(I)$ such that $f(x)=\lim f_k(x)$ almost everywhere on $I$. Then $f \in \mathcal{R}^*(I)$ if and only if the sequence of integrals $\left(\int_I f_k\right)$ is bounded in $\R$, in which case
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\[\int_I f = \lim\limits_{k \to \infty} \int_I f_k.\]
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\end{theorem}
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\begin{theorem}[\textbf{Dominated Convergence Theorem}]
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Let $(f_n)$ be a sequence in $\mathcal{R}^*(I)$ and let $f(x)=\lim f_k(x)$ almost everywhere on $I$. If there exist functions $\alpha, \omega$ in $\mathcal{R}^*(I)$ such that
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\[\alpha(x)\leq f_k(x)\leq\omega(x)\ \ \text{ for almost every }\ \ x \in I\]
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then $f \in \mathcal{R}^*(I)$ and
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\[\displaystyle\int_If=\lim\limits_{k \to \infty}\displaystyle\int_I f_k.\]
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Moreover, if $\alpha$ and $\omega$ belong to $\mathcal{L}(I)$, then $f_k$ and $f$ belong to $\mathcal{L}(I)$ and
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\[||f_k-f||=\displaystyle\int_I|f_k-f|\to 0\]
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\end{theorem}
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\begin{definition}
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A function $f:[a,b]\to\R$ is said to be \textbf{(Lebesgue) measurable} if there exists a sequence $(s_k)$ of step functions on $[a,b]$ such that
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\[f(x)=\lim\limits_{k \to \infty} s_k(x)\ \ \text{ for almost every }\ \ x \in [a,b]\]
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We denote the collection of measurable functions on $[a,b]$ by $\mathcal{M}[a,b]$.
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\end{definition}
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\begin{theorem}
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Let $f$ and $g$ belong to $\mathcal{M}[a,b]$ and let $c \in \R$.
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\begin{enumerate}
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\item Then the functions $cf, |f|,f+g,f-g,$ and $f\cdot g$ also belong to $\mathcal{M}[a,b]$.
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\item If $\varphi:\R \to \R$ is continuous, then the composition $\varphi \circ f \in \mathcal{M}[a,b]$.
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\item If $(f_n)$ is a sequence in $\mathcal{M}[a,b]$ and $f(x)=\lim f_n(x)$ almost everywhere on $I$, then $f \in \mathcal{M}[a,b]$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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A function $f:[a,b]\to\R$ is in $\mathcal{M}[a,b]$ if and only if there exists a sequence $(g_k)$ of continuous functions such that
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\[f(x)=\lim\limits_{k \to \infty} g(x)\ \ \text{ for almost every }\ \ x \in [a,b]\]
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\end{theorem}
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\begin{theorem}[\textbf{Measurability Theorem}]
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If $f \in \mathcal{R}^*[a,b]$, then $f \in \mathcal{M}[a,b]$
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\end{theorem}
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\begin{theorem}[\textbf{Integrability Theorem}]
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Let $f\in\mathcal{M}[a,b]$. Then $f \in \mathcal{R}^*[a,b]$ if and only if there exist functions $\alpha, \omega \in \mathcal{R}^*[a,b]$ such that
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\[\alpha(x)\leq f(x)\leq \omega(x)\ \ \text{ for almost every }\ \ x \in [a,b]\]
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Moreover, if either $\alpha$ or $\omega$ belongs to $\mathcal{L}[a,b]$, then $f \in \mathcal{L}[a,b]$.
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\end{theorem}
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\section{Definition and Main Properties}
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In \textit{Definition 5.2.2}, we defined a \textbf{gauge} on $[a,b]$ to be a strictly positive function $\delta:[a,b] \to (0,\infty)$. Further, a tagged partition $\dot{\mathcal{P}}:=\{(I_i,t_i)\}_{i=1}^n$ of $[a,b]$, where $I_i:=[x_{i-1},x_i]$, is said to be \textbf{$\delta$-fine} in the case
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\[t_i \in I_i \subseteq [t_i-\delta(t_i),t_i+\delta(t_i)]\ \ \text{for}\ \ i=1,\dots,n\]
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\begin{definition}
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A function $f:[a,b] \to \R$ is said to be \textbf{\textit{generalized} Riemann integrable} on $[a,b]$ if there exists a number $L\in\R$ such that for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine partition of $[a,b]$, then
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\[|S(f;\dot{\mathcal{P}})-L|<\varepsilon\]
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The collection of all generalized Riemann integrable functions will usually be denoted by $\mathcal{R}^*[a,b]$.
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\\\\ It will be shown that if $f \in \mathcal{R}^*[a,b]$, then the number $L$ is uniquely determined; it will be called the \textbf{generalized Riemann integral of $f$} over $[a,b]$. It will also be shown that if $f \in \mathcal{R}[a,b]$, then $f \in \mathcal{R}^*[a,b]$ and the value of the two integrals is the same. Therefore, it will not cause any ambiguity if we also denote the generalized Riemann integral of $f \in \mathcal{R}^*[a,b]$ by the symbols
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\[\displaystyle\int_{a}^{b}f\ \ \text{ or }\ \ \displaystyle\int_{a}^{b}f(x)dx\]
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\end{definition}
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\begin{theorem}[\textbf{Uniqueness Theorem}]
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If $f \in \mathcal{R}^*[a,b]$, then the value of the integral is uniquely determined.
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\end{theorem}
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\begin{theorem}[\textbf{Consistency Theorem}]
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If $f \in \mathcal{R}[a,b]$ with integral $L$, then also $f \in \mathcal{R}^*[a,b]$ with integral $L$.
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\end{theorem}
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\begin{theorem}
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Suppose that $f$ and $g$ are in $\mathcal{R}^*[a,b]$. Then:
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\begin{enumerate}
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\item If $k \in \R$, the function $kf$ is in $\mathcal{R}^*[a,b]$ and
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\[\displaystyle\int_{a}^{b}kf=k\displaystyle\int_{a}^{b}f\]
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\item The function $f+g$ is in $\mathcal{R}^*[a,b]$ and
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\[\displaystyle\int_{a}^{b}(f+g)=\displaystyle\int_{a}^{b}f+\displaystyle\int_{a}^{b}g\]
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\item If $f(x)\leq g(x)$ for all $x \in [a,b]$, then
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\[\displaystyle\int_{a}^{b}f \leq \displaystyle\int_{a}^{b}g\]
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Cauchy Criterion}]
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A function $f:[a,b] \to \R$ belongs to $\mathcal{R}^*[a,b]$ if and only if for every $\varepsilon >0$ there exists a gauge $\eta_\varepsilon$ on $[a,b]$ such that if $\dot{\mathcal{P}}$ and $\dot{\mathcal{Q}}$ are any partitions of $[a,b]$ that are $\eta_\varepsilon$-fine, then
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\[|S(f;\dot{\mathcal{P}})-S(f;\dot{\mathcal{Q}})|<\varepsilon\]
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\end{theorem}
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\begin{theorem}[\textbf{Squeeze Theorem}]
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Let $f:[a,b] \to \R$. Then $f \in \mathcal{R}^*[a,b]$ if and only if for every $\varepsilon>0$ there exist functions $\alpha_\varepsilon$ and $\omega_\varepsilon$ in $\mathcal{R}^*[a,b]$ with
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\[\alpha_\varepsilon(x) \leq f(x) \leq \omega_\varepsilon(x)\ \forall\ x \in [a,b]\]
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and such that
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\[\displaystyle\int_{a}^{b}(\omega_\varepsilon-\alpha_\varepsilon) \leq \varepsilon\]
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\end{theorem}
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\begin{theorem}[\textbf{Additivity Theorem}]
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Let $f:[a,b] \to \R$ and let $c \in (a,b)$. Then $f \in \mathcal{R}^*[a,b]$ if and only if its restrictions to $[a,c]$ and $[c,b]$ are both generalized Riemann integrable. In this case
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\[\displaystyle\int_{a}^{b}f=\displaystyle\int_{a}^{c}f+\displaystyle\int_{c}^{b}f\]
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\end{theorem}
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\begin{theorem}[\textbf{The Fundamental Theorem of Calculus (First Form)}]
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Suppose there exists a \textbf{countable} set $E$ in $[a,b]$, and functions $f,F:[a,b] \to \R$ such that:
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\begin{enumerate}
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\item $F$ is continuous on $[a,b]$.
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\item $F'(x)=f(x)$ for all $x \in [a,b]\setminus E$.
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\\Then $f$ belongs to $\mathcal{R}^*[a,b]$ and
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\[\displaystyle\int_{a}^{b}f=F(b)-F(a)\]
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Fundamental Theorem of Calculus (Second Form)}]
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Let $f$ belong to $\mathcal{R}^*[a,b]$ and let $F$ be the indefinite integral of $f$. Then we have:
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\begin{enumerate}
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\item $F$ is continuous on $[a,b]$.
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\item There exists a null set $Z$ such that if $x \in [a,b]\setminus Z$, then $F$ is differentiable at $x$ and $F'(x)=f(x)$.
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\item If $f$ is continuous at $c \in [a,b]$, then $F'(c)=f(c)$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Substitution Theorem}]
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\begin{enumerate}
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\item[]
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\item Let $I:=[a,b]$ and $J:=[\alpha, \beta]$, and let $F:I \to \R$ and $\varphi:J \to \R$ be continuous functions with $\varphi(J)\subseteq I$.
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\item Suppose there exist sets $E_f \subset I$ and $E_\varphi\subset J$ such that $f(x)=F'(x)$ for $x \in I\setminus E_f$, that $\varphi'(t)$ exists for $t \in J\setminus E_\varphi$, and that $E:=\varphi^{-1}(E_f)\cup E_\varphi$ is countable.
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\item Set $f(x):=0$ for $x \in E_f$ and $\varphi'(t):=0$ for $t \in E_\varphi$. We conclude that $f \in \mathcal{R}^*(\varphi(J))$, that $(f \circ \varphi)\cdot \varphi^t \in \mathcal{R}^*(J)$ and that
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\[\displaystyle\int_{\alpha}^{\beta}(f \circ \varphi)\cdot\varphi^t=F\circ\varphi\left.\right|_\alpha^\beta=\displaystyle\int_{\varphi(\alpha)}^{\varphi(\beta)}f\]
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Multiplication Theorem}]
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If $f \in \mathcal{R}^*[a,b]$ and if $g$ is a monotone function on $[a,b]$, then the product $f \cdot g$ belongs to $\mathcal{R}^*[a,b]$.
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\end{theorem}
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\begin{theorem}[\textbf{Integration by Parts Theorem}]
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Let $F$ and $G$ be differentiable on $[a,b]$. Then $F'G$ belongs to $\mathcal{R}^*[a,b]$ if and only if $FG'$ belongs to $\mathcal{R}^*[a,b]$. In this case we have
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\[\displaystyle\int_{a}^{b}F'G=FG\left.\right|_a^b-\displaystyle\int_{a}^{b}FG'\]
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\end{theorem}
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\begin{theorem}[\textbf{Taylor's Theorem}]
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Suppose that $f,f',f'',\dots,f^{(n)}$ and $f^{(n+1)}$ exist on $[a,b]$. Then we have
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\[f(b)=f(a)+\frac{f'(a)}{1!}(b-a)+\dots+\frac{f^{(n)}(a)}{n!}(b-a)^n+R_n\]
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where the remainder is given by
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\[R_n=\frac{1}{n!}\displaystyle\int_{a}^{b}f^{(n+1)}(t)\cdot(b-t)^n dt\]
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\end{theorem}
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\section{Improper and Lebesgue Integrals}
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\begin{theorem}[\textbf{Hake's Theorem}]
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If $f:[a,b] \to \R$, then $f \in \mathcal{R}^*[a,b]$ if and only if for every $\gamma \in (a,b)$ the restriction of $f$ to $[a,\gamma]$ belongs to $\mathcal{R}^*[a,\gamma]$ and
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\[\lim\limits_{\gamma \to b^-}\displaystyle\int_{a}^{\gamma}f=A\in\R\]
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In this case $\displaystyle\int_{a}^{b}f=A$.
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\end{theorem}
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\begin{definition}
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A function $f \in \mathcal{R}^*[a,b]$ such that $|f| \in \mathcal{R}^*[a,b]$ is said to be \textbf{Lebesgue integrable} on $[a,b]$. The collection of all Lebesgue integrable functions on $[a,b]$ is denoted by $\mathcal{L}[a,b]$.
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\end{definition}
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\begin{theorem}[\textbf{Comparison Test}]
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If $f,\omega \in \mathcal{R}^*[a,b]$ and $|f(x)| \leq \omega(x)$ for all $x \in [a,b]$, then $f \in \mathcal{L}[a,b]$ and
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\[\abs{\displaystyle\int_{a}^{b}f}\leq\displaystyle\int_{a}^{b}|f| \leq\displaystyle\int_{a}^{b}\omega\]
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\end{theorem}
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\begin{theorem}
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If $f,g \in \mathcal{L}[a,b]$ and if $c \in \R$, then $cf$ and $f+g$ also belong to $\mathcal{L}[a,b]$. Moreover
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\[\displaystyle\int_{a}^{b}cf=c\displaystyle\int_{a}^{b}f\ \ \text{ and }\ \ \displaystyle\int_{a}^{b}|f+g|\leq\displaystyle\int_{a}^{b}|f|+\displaystyle\int_{a}^{b}|g|\]
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\end{theorem}
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\begin{theorem}
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If $f \in \mathcal{R}^*[a,b]$, the following assertions are equivalent:
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\begin{enumerate}
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\item $f \in \mathcal{L}[a,b]$.
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\item There exists $\omega \in \mathcal{L}[a,b]$ such that $f(x)\leq\omega(x)$ for all $x \in [a,b]$.
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\item There exists $\alpha \in \mathcal{L}[a,b]$ such that $\alpha(x) \leq f(x)$ for all $x \in [a,b]$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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If $f,g \in \mathcal{L}[a,b]$, then the functions $\max \{f,g\}$ and $\min \{f,g\}$ also belong to $\mathcal{L}[a,b]$.
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\end{theorem}
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\begin{theorem}
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Suppose that $f,g,\alpha,$ and $\omega$ belong to $\mathcal{R}^*[a,b]$. If
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\[f\leq\omega,\ g\leq\omega\ \ \text{ or if }\ \ \alpha\leq f,\ \alpha \leq g,\]
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then $\max \{f,g\}$ and $\min\{f,g\}$ also belong to $\mathcal{R}^*[a,b]$.
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\end{theorem}
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\begin{definition}
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If $f \in \mathcal{L}[a,b]$, we define the \textbf{seminorm} of $f$ to be
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\[||f||:=\displaystyle\int_{a}^{b}|f|\]
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If $f,g \in \mathcal{L}[a,b]$, we define the \textbf{distance between $f$ and $g$} to be
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\[\text{dist}(f,g):=||f-g||=\displaystyle\int_{a}^{b}|f-g|\]
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\end{definition}
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\begin{theorem}
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The seminorm function satisfies:
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\begin{enumerate}
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\item $||f||\geq 0$ for all $f \in \mathcal{L}[a,b]$.
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\item If $f(x)=0$ for $x \in [a,b]$, then $||f||=0$.
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\item If $f \in \mathcal{L}[a,b]$ and $c \in \R$, then $||cf||=|c|\cdot||f||$.
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\item If $f,g \in \mathcal{L}[a,b]$, then $||f+g||\leq||f||+||g||$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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The distance function satisfies:
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\begin{enumerate}
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\item $\dist(f,g)\geq 0$ for all $f,g \in \mathcal{L}[a,b]$.
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\item If $f(x)=g(x)$ for $x \in [a,b]$, then $\dist(f,g)=0$.
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\item $\dist(f,g)=\dist(g,f)$ for all $f,g \in \mathcal{L}[a,b]$.
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\item $\dist(f,h)\leq\dist(f,g)+\dist(g,h)$ for all $f,g,h \in \mathcal{L}[a,b]$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Completeness Theorem}]
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A sequence $(f_n)$ of functions in $\mathcal{L}[a,b]$ converges to a function $f \in \mathcal{L}[a,b]$ if and only if it has the property that for every $\varepsilon>0$ there exists $H(\varepsilon)$ such that if $m,n\geq H(\varepsilon)$, then
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\[||f_m-f_n||=\dist(f_m,f_n)<\varepsilon\]
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\end{theorem}
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\section{Infinite Intervals}
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\begin{definition}
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\begin{enumerate}
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\item[]
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\item A function $f:[a,\infty) \to \R$ is said to be \textbf{generalized Riemann integrable} if there exists $A \in \R$ such that for every $\varepsilon>0$ there exists a gauge $\delta_\varepsilon$ on $[a,\infty]$ such that if $\dot{\mathcal{P}}$ is any $\delta_\varepsilon$-fine tagged subpartition of $[a,\infty)$, then $|S(f;\dot{\mathcal{P}})-A|\leq \varepsilon$. In this case, we write $f \in \mathcal{R}^*[a,\infty)$ and
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\[\displaystyle\int_{a}^{b}f:=A\]
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\item A function $f:[a,\infty) \to \R$ is said to be \textbf{Lebesgue integrable} if both $f$ and $|f|$ belong to $\mathcal{R}^*[a,\infty)$. In this case we write $f \in \mathcal{L}[a,\infty)$.
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\end{enumerate}
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\end{definition}
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\begin{theorem}[\textbf{Hake's Theorem}]
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If $f:[a,\infty) \to \R$, then $f \in \mathcal{R}^*[a,\infty)$ if and only if for every $\gamma \in (a,\infty)$ the restriction of $f$ to $[a,\gamma]$ belongs to $\mathcal{R}^*[a,\gamma]$ and
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\[\lim\limits_{\gamma\to\infty}\displaystyle\int_{a}^{\gamma}f=A\in\R\]
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In this case $\displaystyle\int_{a}^{\infty}f=A$.
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\end{theorem}
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\begin{theorem}[\textbf{Fundamental Theorem}]
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Suppose that $E$ is a countable subset of $[a,\infty)$ and that $f,F:[a,\infty)\to\R$ are such that:
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\begin{enumerate}
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\item $F$ is continuous on $[a,\infty)$ and $\lim\limits_{x \to \infty}F(x)$ exists.
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\item $F'(x)=f(x)$ for all $x \in (a,\infty),\ x \notin E$.
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\\Then $f$ belongs to $\mathcal{R}^*[a,\infty)$ and
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\[\displaystyle\int_{a}^{\infty}f=\lim\limits_{x \to \infty}F(x)-F(a).\]
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Hake's Theorem}]
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If $h:(-\infty,\infty)\to\R$, then $h \in \mathcal{R}^*(-\infty,\infty)$ if and only if for every $\beta < \gamma$ in $(-\infty, \infty)$, the restriction of $h$ to $[\beta,\gamma]$ is in $\mathcal{R}^*[\beta,\gamma]$ and
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\[\lim\limits_{\substack{\beta \to -\infty \\ \gamma \to +\infty}}\displaystyle\int_{\beta}^{\gamma}h=C\in\R\]
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In this case $\displaystyle\int_{-\infty}^{\infty}h=C$.
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\end{theorem}
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\begin{theorem}[\textbf{Fundamental Theorem}]
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Suppose that $E$ is a countable subset of $(-\infty, \infty)$ and that $h,H:(-\infty,\infty)\to\R$ satisfy:
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\begin{enumerate}
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\item $H$ is continuous on $(-\infty, \infty)$ and the limits $\lim\limits_{x \to \pm \infty}H(x)$ exist.
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\item $H'(x)=h(x)$ for all $x \in (-\infty, \infty),\ x \notin E$.
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\end{enumerate}
|
||||
Then $h$ belongs to $\mathcal{R}^*(-\infty,\infty)$ and
|
||||
\[\displaystyle\int_{-\infty}^{\infty}h=\lim\limits_{x \to \infty} H(x)-\lim\limits_{\gamma \to -\infty}H(\gamma)\]
|
||||
\end{theorem}
|
||||
Reference in New Issue
Block a user