The Additivity Theorem - Revision history

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Lila: Created page with "
Let $f$ be a real-valued function on the interval..."

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Created page with "<blockquote style="background: white; border: 1px solid black; padding: 1em;"> Let <math> f </math> be a real-valued function on the interval..."

New page

<blockquote style="background: white; border: 1px solid black; padding: 1em;">

Let <math> f </math> be a real-valued function on the interval <math> \[ a,b\] </math>, and let <math> c\in (a,b) </math>. Then, <math> f </math> is Riemann integrable on <math> [a,b] </math> if and only if it is also Riemann integrable on <math> [a,c] </math> and <math> [c,b] </math>. In this case, we have that

<math> </math>

</blockquote>

← Older revision		Revision as of 21:57, 9 November 2021
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	</ul>		</ul>
	<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \; d \alpha (x) = \int_a^c f(x) \; d \alpha (x) + \int_c^b f(x) \; d \alpha (x) \quad \blacksquare \end{align}</math></div>		<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \; d \alpha (x) = \int_a^c f(x) \; d \alpha (x) + \int_c^b f(x) \; d \alpha (x) \quad \blacksquare \end{align}</math></div>
		+
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [http://mathonline.wikidot.com/riemann-stieltjes-integrability-on-subintervals Riemann-Stieltjes Integrability on Subintervals, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 10: / Line 10: @@
 </ul>
 <ul>
-<li>Since <span class="math-inline"><math>\int_a^c f(x) \; d \alpha (x) = A</math></span> we have that for <span class="math-inline"><math>\epsilon_1 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_1} \in \mathscr{P}[a, c]</math></span> such that for all partitions <span class="math-inline"><math>P' \in \mathscr{P}[a, c]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_1}</math></span>, (<span class="math-inline"><math>P_{\epsilon_1} \subseteq P'</math></span>) and for any choice of <span class="math-inline"><math>t_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
+<li>Since <span class="math-inline"><math>\int_a^c f(x) \; d \alpha (x) = A</math></span> we have that for <span class="math-inline"><math>\epsilon_1 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_1} \in \mathcal{P}[a, c]</math></span> such that for all partitions <span class="math-inline"><math>P' \in \mathcal{P}[a, c]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_1}</math></span>, (<span class="math-inline"><math>P_{\epsilon_1} \subseteq P'</math></span>) and for any choice of <span class="math-inline"><math>t_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P', f, \alpha) - A \mid < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}</math></div>
 <ul>
-<li>Similarly, since <span class="math-inline"><math>\int_c^b f(x) \; d \alpha (x) = B</math></span> we have that for <span class="math-inline"><math>\epsilon_2 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_2} \in \mathscr{P}[c, b]</math></span> such that for all partitions <span class="math-inline"><math>P'' \in \mathscr{P}[c, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_2}</math></span>, <span class="math-inline"><math>(P_{\epsilon_2} \subseteq P''</math></span>) and for any choice of <span class="math-inline"><math>u_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
+<li>Similarly, since <span class="math-inline"><math>\int_c^b f(x) \; d \alpha (x) = B</math></span> we have that for <span class="math-inline"><math>\epsilon_2 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_2} \in \mathcal{P}[c, b]</math></span> such that for all partitions <span class="math-inline"><math>P'' \in \mathcal{P}[c, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_2}</math></span>, <span class="math-inline"><math>(P_{\epsilon_2} \subseteq P''</math></span>) and for any choice of <span class="math-inline"><math>u_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P'', f, \alpha) - B \mid < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}</math></div>
 <ul>
-<li>Let <span class="math-inline"><math>P_{\epsilon} = P_{\epsilon_1} \cup P_{\epsilon_2}</math></span>. Then <span class="math-inline"><math>P_{\epsilon}</math></span> is a partition of <span class="math-inline"><math>[a, b]</math></span> and for all partitions <span class="math-inline"><math>P \in \mathscr{P}[a, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon}</math></span>, (<span class="math-inline"><math>P_{\epsilon} \subseteq P</math></span>) we must have that <span class="math-inline"><math>(*)</math></span> and <span class="math-inline"><math>(**)</math></span> hold. Then for any choice of <span class="math-inline"><math>v_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
+<li>Let <span class="math-inline"><math>P_{\epsilon} = P_{\epsilon_1} \cup P_{\epsilon_2}</math></span>. Then <span class="math-inline"><math>P_{\epsilon}</math></span> is a partition of <span class="math-inline"><math>[a, b]</math></span> and for all partitions <span class="math-inline"><math>P \in \mathcal{P}[a, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon}</math></span>, (<span class="math-inline"><math>P_{\epsilon} \subseteq P</math></span>) we must have that <span class="math-inline"><math>(*)</math></span> and <span class="math-inline"><math>(**)</math></span> hold. Then for any choice of <span class="math-inline"><math>v_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P, f, \alpha) - (A + B) \mid = \mid S(P', f, \alpha) + S(P'', f, \alpha) - (A + B) \mid \leq \mid S(P', f, \alpha - A \mid + \mid S(P'', f, \alpha) - B \mid < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</math></div>

@@ Line 7: / Line 7: @@
 </blockquote>
-<li><strong>Proof</strong>: Suppose that <span class="math-inline"><math>\int_a^c f(x) \: d \alpha (x) = A</math></span> and <span class="math-inline"><math>\int_c^b f(x) \: d \alpha (x) = B</math></span> for some <span class="math-inline"><math>A, B \in \mathbb{R}</math></span>. Let <span class="math-inline"><math>\epsilon > 0</math></span> be given.</li>
+<li><strong>Proof</strong>: Suppose that <span class="math-inline"><math>\int_a^c f(x) \; d \alpha (x) = A</math></span> and <span class="math-inline"><math>\int_c^b f(x) \; d \alpha (x) = B</math></span> for some <span class="math-inline"><math>A, B \in \mathbb{R}</math></span>. Let <span class="math-inline"><math>\epsilon > 0</math></span> be given.</li>
 </ul>
 <ul>
-<li>Since <span class="math-inline"><math>\int_a^c f(x) \: d \alpha (x) = A</math></span> we have that for <span class="math-inline"><math>\epsilon_1 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_1} \in \mathscr{P}[a, c]</math></span> such that for all partitions <span class="math-inline"><math>P' \in \mathscr{P}[a, c]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_1}</math></span>, (<span class="math-inline"><math>P_{\epsilon_1} \subseteq P'</math></span>) and for any choice of <span class="math-inline"><math>t_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
+<li>Since <span class="math-inline"><math>\int_a^c f(x) \; d \alpha (x) = A</math></span> we have that for <span class="math-inline"><math>\epsilon_1 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_1} \in \mathscr{P}[a, c]</math></span> such that for all partitions <span class="math-inline"><math>P' \in \mathscr{P}[a, c]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_1}</math></span>, (<span class="math-inline"><math>P_{\epsilon_1} \subseteq P'</math></span>) and for any choice of <span class="math-inline"><math>t_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P', f, \alpha) - A \mid < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}</math></div>
 <ul>
-<li>Similarly, since <span class="math-inline"><math>\int_c^b f(x) \: d \alpha (x) = B</math></span> we have that for <span class="math-inline"><math>\epsilon_2 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_2} \in \mathscr{P}[c, b]</math></span> such that for all partitions <span class="math-inline"><math>P'' \in \mathscr{P}[c, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_2}</math></span>, <span class="math-inline"><math>(P_{\epsilon_2} \subseteq P''</math></span>) and for any choice of <span class="math-inline"><math>u_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
+<li>Similarly, since <span class="math-inline"><math>\int_c^b f(x) \; d \alpha (x) = B</math></span> we have that for <span class="math-inline"><math>\epsilon_2 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_2} \in \mathscr{P}[c, b]</math></span> such that for all partitions <span class="math-inline"><math>P'' \in \mathscr{P}[c, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_2}</math></span>, <span class="math-inline"><math>(P_{\epsilon_2} \subseteq P''</math></span>) and for any choice of <span class="math-inline"><math>u_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
 </ul>
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P'', f, \alpha) - B \mid < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}</math></div>
@@ Line 22: / Line 22: @@
 <div style="text-align: center;"><math>\begin{align} \quad \mid S(P, f, \alpha) - (A + B) \mid = \mid S(P', f, \alpha) + S(P'', f, \alpha) - (A + B) \mid \leq \mid S(P', f, \alpha - A \mid + \mid S(P'', f, \alpha) - B \mid < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</math></div>
 <ul>
-<li>Hence <span class="math-inline"><math>\int_a^b f(x) \: d \alpha (x)</math></span> exists and:</li>
+<li>Hence <span class="math-inline"><math>\int_a^b f(x) \; d \alpha (x)</math></span> exists and:</li>
 </ul>
-<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^c f(x) \: d \alpha (x) + \int_c^b f(x) \: d \alpha (x) \quad \blacksquare \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \; d \alpha (x) = \int_a^c f(x) \; d \alpha (x) + \int_c^b f(x) \; d \alpha (x) \quad \blacksquare \end{align}</math></div>

@@ Line 24: / Line 24: @@
 <li>Hence <span class="math-inline"><math>\int_a^b f(x) \: d \alpha (x)</math></span> exists and:</li>
 </ul>
-<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^c f(x) \: d \alpha (x) + \int_c^b f(x) \: d \alpha (x) \quad \blacksquare \end{al
+<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^c f(x) \: d \alpha (x) + \int_c^b f(x) \: d \alpha (x) \quad \blacksquare \end{align}</math></div>

← Older revision		Revision as of 21:53, 9 November 2021
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	</blockquote>		</blockquote>
		+
		+	<li><strong>Proof</strong>: Suppose that <span class="math-inline"><math>\int_a^c f(x) \: d \alpha (x) = A</math></span> and <span class="math-inline"><math>\int_c^b f(x) \: d \alpha (x) = B</math></span> for some <span class="math-inline"><math>A, B \in \mathbb{R}</math></span>. Let <span class="math-inline"><math>\epsilon > 0</math></span> be given.</li>
		+	</ul>
		+	<ul>
		+	<li>Since <span class="math-inline"><math>\int_a^c f(x) \: d \alpha (x) = A</math></span> we have that for <span class="math-inline"><math>\epsilon_1 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_1} \in \mathscr{P}[a, c]</math></span> such that for all partitions <span class="math-inline"><math>P' \in \mathscr{P}[a, c]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_1}</math></span>, (<span class="math-inline"><math>P_{\epsilon_1} \subseteq P'</math></span>) and for any choice of <span class="math-inline"><math>t_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\begin{align} \quad \mid S(P', f, \alpha) - A \mid < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}</math></div>
		+	<ul>
		+	<li>Similarly, since <span class="math-inline"><math>\int_c^b f(x) \: d \alpha (x) = B</math></span> we have that for <span class="math-inline"><math>\epsilon_2 = \frac{\epsilon}{2} > 0</math></span> there exists a partition <span class="math-inline"><math>P_{\epsilon_2} \in \mathscr{P}[c, b]</math></span> such that for all partitions <span class="math-inline"><math>P'' \in \mathscr{P}[c, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon_2}</math></span>, <span class="math-inline"><math>(P_{\epsilon_2} \subseteq P''</math></span>) and for any choice of <span class="math-inline"><math>u_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\begin{align} \quad \mid S(P'', f, \alpha) - B \mid < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}</math></div>
		+	<ul>
		+	<li>Let <span class="math-inline"><math>P_{\epsilon} = P_{\epsilon_1} \cup P_{\epsilon_2}</math></span>. Then <span class="math-inline"><math>P_{\epsilon}</math></span> is a partition of <span class="math-inline"><math>[a, b]</math></span> and for all partitions <span class="math-inline"><math>P \in \mathscr{P}[a, b]</math></span> finer than <span class="math-inline"><math>P_{\epsilon}</math></span>, (<span class="math-inline"><math>P_{\epsilon} \subseteq P</math></span>) we must have that <span class="math-inline"><math>()</math></span> and <span class="math-inline"><math>(*)</math></span> hold. Then for any choice of <span class="math-inline"><math>v_k</math></span>'s in each <span class="math-inline"><math>k^{\mathrm{th}}</math></span> subinterval we have that:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\begin{align} \quad \mid S(P, f, \alpha) - (A + B) \mid = \mid S(P', f, \alpha) + S(P'', f, \alpha) - (A + B) \mid \leq \mid S(P', f, \alpha - A \mid + \mid S(P'', f, \alpha) - B \mid < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</math></div>
		+	<ul>
		+	<li>Hence <span class="math-inline"><math>\int_a^b f(x) \: d \alpha (x)</math></span> exists and:</li>
		+	</ul>
		+	<div style="text-align: center;"><math>\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^c f(x) \: d \alpha (x) + \int_c^b f(x) \: d \alpha (x) \quad \blacksquare \end{al

@@ Line 3: / Line 3: @@
 '''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case,
-<math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math>
+<div style="text-align: center;"> <math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math> </div>
 </blockquote>

@@ Line 1: / Line 1: @@
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
-'''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case, we have that
+'''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case,
-<math>  </math>
+<math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math>
 </blockquote>