162 lines
5.9 KiB
Markdown
162 lines
5.9 KiB
Markdown
# Math4121 Lecture 9
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Exam next week.
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Transition to new book.
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## Continue on Chapter 6
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### Integrable Functions
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#### Theorem 6.11
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Suppose $f\in \mathscr{R}(\alpha)$ on $[a, b]$, $m\leq f(x)\leq M$ for all $x\in [a, b]$, and $\phi$ is continuous on $[m, M]$, and let $h(x)=\phi(f(x))$ on $[a, b]$. Then $h\in \mathscr{R}(\alpha)$ on $[a, b]$.
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Proof:
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Since $\phi$ is uniformly continuous on $[m, M]$, for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $s, t\in [m, M]$ and $|s-t| < \delta$, then $|\phi(s)-\phi(t)| < \epsilon$.
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Since $f\in \mathscr{R}(\alpha)$ on $[a, b]$, we can find a partition $P=\{x_0, x_1, \cdots, x_n\}$ of $[a, b]$ such that $U(f, P, \alpha)-L(f, P, \alpha) < \epsilon \delta$.
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Set $M_i=\sup \{f(x): x\in [x_{i-1}, x_i]\}$ and $m_i=\inf \{f(x): x\in [x_{i-1}, x_i]\}$. $M_i^*=\sup \{h(x): x\in [x_{i-1}, x_i]\}$ and $m_i^*=\inf \{h(x): x\in [x_{i-1}, x_i]\}$.
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We call a index $i$ good if $M_i-m_i < \delta$.
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If $i$ is good, then $\forall x, y\in [x_{i-1}, x_i]$, $|f(x)-f(y)| < \delta$ and by uniform continuity of $\phi$, $|\phi(f(x))-\phi(f(y))| < \epsilon$.
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Therefore, $|M_i^*-m_i^*| < \epsilon$.
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If $i$ is bad, then $M_i-m_i\geq \delta$.
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Notice that
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$$
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\begin{aligned}
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\delta\sum_{i\in\text{bad}} \Delta \alpha_i &\leq \sum_{i\in\text{bad}} (M_i-m_i) \Delta \alpha_i \\
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&\leq \sum_{i=1}^n (M_i-m_i) \Delta \alpha_i \\
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&\leq U(f, P, \alpha)-L(f, P, \alpha) \\
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&< \epsilon\delta
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\end{aligned}
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$$
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Therefore, $\sum_{i=1}^n (M_i^*-m_i^*) \Delta \alpha_i < \epsilon$.
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So,
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$$
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\begin{aligned}
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U(P,h,\alpha)-L(P,h,\alpha) &= \sum_{i=1}^n (M_i^*-m_i^*) \Delta \alpha_i \\
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&\leq \sum_{i\in\text{good}} \epsilon \Delta \alpha_i + \sum_{i\in\text{bad}}2 \sup \{|h(x)-h(y)|: x, y\in [x_{i-1}, x_i]\} \Delta \alpha_i \\
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&\leq \epsilon [\alpha(b)-\alpha(a)] + 2\epsilon \sup \{|h(x)-h(y)|: x, y\in [a, b]\}\\
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\end{aligned}
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$$
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Since $\epsilon$ is arbitrary, $h\in \mathscr{R}(\alpha)$ on $[a, b]$.
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QED
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### Properties of Integrable Functions
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#### Theorem 6.12
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Let $f,g\in \mathscr{R}(\alpha)$ on $[a, b]$.
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(a) $f+g\in \mathscr{R}(\alpha)$ on $[a, b]$, $\int_a^b (f+g)d\alpha = \int_a^b f d\alpha + \int_a^b g d\alpha$. (Linearity of the integral)
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(aa) If $c\in \mathbb{R}$, then $cf\in \mathscr{R}(\alpha)$ on $[a, b]$, and $\int_a^b cf d\alpha = c\int_a^b f d\alpha$.
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(b) If $f(x)\leq g(x),\forall x\in [a, b]$, then $\int_a^b f d\alpha \leq \int_a^b g d\alpha$.
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(c) $c\in [a, b]$, then $\int_a^c f d\alpha + \int_c^b f d\alpha = \int_a^b f d\alpha$.
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(d) If $|f(x)|\leq M$, then $|\int_a^b f d\alpha| \leq M(\alpha(b)-\alpha(a))$.
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(e) If $f\in \mathscr{R}(\beta)$ then $f\in \mathscr{R}(\alpha+\beta)$ and $\int_a^b f d(\alpha+\beta) = \int_a^b f d\alpha + \int_a^b f d\beta$.
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Proof:
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**Property (aa), (b), (e) holds for Riemann Sums themselves.**
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**For (a)**, Set $h(x)=f(x)+g(x)$. Then $h\in \mathscr{R}(\alpha)$ on $[a, b]$ and we will show $\int_a^b h d\alpha \leq \int_a^b f d\alpha + \int_a^b g d\alpha$.
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Since $f,g\in \mathscr{R}(\alpha)$ on $[a, b]$, for any $\epsilon > 0$, there exists a partition $P_1,P_2$ of $[a, b]$ such that $U(f,P_1,\alpha)-L(f,P_1,\alpha) < \epsilon$ and $U(g,P_2,\alpha)-L(g,P_2,\alpha) < \epsilon$.
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Let $P=P_1\cup P_2$. Then $U(P,f,\alpha)\leq U(P_1,f,\alpha)< \int_a^b f d\alpha + \epsilon$ and $U(P,g,\alpha)\leq U(P_2,g,\alpha)< \int_a^b g d\alpha + \epsilon$.
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So $U(P,h,\alpha)\leq U(P,f,\alpha)+U(P,g,\alpha)\leq \int_a^b f d\alpha + \int_a^b g d\alpha + 2\epsilon$.
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Since $\epsilon$ is arbitrary, $\int_a^b h d\alpha \leq \int_a^b f d\alpha + \int_a^b g d\alpha$.
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$$
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\sup cf(x) = c\sup f(x)\quad \forall c\in \mathbb{R}
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$$
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$$
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U(P,cf, \alpha) = cU(P,f,\alpha)
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$$
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**For (b)**, notice that if $f(x)\leq g(x)$, then $\sup f(x)\leq \sup g(x)$, $U(P,f,\alpha)\leq U(P,g,\alpha)$. and $L(P,f,\alpha)\leq L(P,g,\alpha)$.
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**For (c)**, if $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and if $a<c<b$, then $f\in \mathscr{R}$ on $[a,c]$ and $[c,b]$, and
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$$
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\int_a^c f d\alpha + \int_c^b f d\alpha=\int_a^b f d\alpha
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$$
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For every partition $P=\{x_0,x_1,\cdots,x_n\}$ of $[a,b]$, we have a refinement $P^*=P\cup\{c\}$ of $[a,b]$. Let $P_1=\{x_0,x_1,\cdots,x_j,c\}$ and $P_2=\{c,x_{j+1},\cdots,x_n\}$ be the partitions of $[a,c]$ and $[c,b]$ respectively. So
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$$
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\begin{aligned}
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U(P^*,f,\alpha)&=\sum_{i=0}^{n}M_i(x_i-x_{i+1})\\
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&=M_c(c-x_j)+\sum_{i=0}^{j-1}M_i(x_i-x_{i+1})+\sum_{i=j+1}^{n}M_i(x_i-x_{i+1})\\
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&=U(P_1,f,\alpha)+U(P_2,f,\alpha)
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\end{aligned}
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$$
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and
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$$
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\begin{aligned}
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L(P^*,f,\alpha)&=\sum_{i=0}^{n}m_i(x_i-x_{i+1})\\
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&=m_c(x_j-c)+\sum_{i=0}^{j-1}m_i(x_i-x_{i+1})+\sum_{i=j+1}^{n-1}m_i(x_i-x_{i+1})\\
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&=L(P_1,f,\alpha)+L(P_2,f,\alpha)
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\end{aligned}
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$$
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Since $P^*$ is a refinement of $P$, by \textbf{Theorem 6.4}, we have $U(P^*,f,\alpha)\leq U(P,f,\alpha)$ and $L(P^*,f,\alpha)\geq L(P,f,\alpha)$.
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So $\int_a^c f d\alpha+\int_c^b f d\alpha\leq U(P^*,f,\alpha)\leq U(P,f,\alpha)=\int_a^b f d\alpha$.
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Similarly, we have $\int_a^c f d\alpha+\int_c^b f d\alpha\geq L(P^*,f,\alpha)\geq L(P,f,\alpha)=\int_a^b f d\alpha$.
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Therefore, $\int_a^c f d\alpha+\int_c^b f d\alpha=\int_a^b f d\alpha$.
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**For (d)**, if $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and if $|f(x)| \leq M$ on $[a,b]$, then
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$$
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\left|\int_a^b f d\alpha\right| \leq M(\alpha(b)-\alpha(a))
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$$
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Since $|f(x)|\leq M$ on $[a,b]$, $\forall x\in [a,b]$, we have $f(x)\in [-M,M]$ on $[a,b]$. So $\sup|f(x)|\leq M$ and $\inf|f(x)|\leq M$. Since $L(P,f,\alpha)\leq \int_a^b f d\alpha\leq U(P,f,\alpha)$, we have
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So
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$$
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\begin{aligned}
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\left|\int_a^b f d\alpha\right|&\leq \max\left\{|L(P,f,\alpha)|,|U(P,f,\alpha)|\right\}\\
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&=\max\left\{\sum_{i=0}^{n-1}|M_i|\Delta x_i,\sum_{i=0}^{n-1}|m_i|\Delta x_i\right\}\\
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&\leq \sum_{i=0}^{n-1}\max\{|M_i|,|m_i|\}\Delta x_i\\
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&\leq \sum_{i=0}^{n-1}M\Delta x_i\\
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&=M(\alpha(b)-\alpha(a))
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\end{aligned}
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$$
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Therefore, $\left|\int_a^b f d\alpha\right| \leq M(\alpha(b)-\alpha(a))$.
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**For (e)**, notice that
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$$
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\begin{aligned}
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\Delta (\alpha+\beta)_i &= \alpha(x_i)-\alpha(x_{i-1})+\beta(x_i)-\beta(x_{i-1}) \\
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&= \Delta \alpha_i + \Delta \beta_i
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\end{aligned}
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$$
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QED |