Difference between revisions of "The Additivity Theorem"

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<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>
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<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>
 
<ul>
 
<ul>
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<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>
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<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>
 
</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>
 
<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>
 
<ul>
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<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>
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<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>
 
</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>
 
<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>
 
<ul>
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<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>
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<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>
 
</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>
 
<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>
 
<ul>
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<li>Hence <span class="math-inline"><math>\int_a^b f(x) \: d \alpha (x)</math></span> exists and:</li>
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<li>Hence <span class="math-inline"><math>\int_a^b f(x) \; d \alpha (x)</math></span> exists and:</li>
 
</ul>
 
</ul>
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<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
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<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>
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==Licensing==
 +
Content obtained and/or adapted from:
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* [http://mathonline.wikidot.com/riemann-stieltjes-integrability-on-subintervals Riemann-Stieltjes Integrability on Subintervals, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 15:57, 9 November 2021

The Additivity Theorem for Riemann Integrable Functions: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } be a real-valued function on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] } , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\in (a,b) } . Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } is Riemann integrable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] } if and only if it is also Riemann integrable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,c] } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [c,b] } . In this case,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f }
  • Proof: Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^c f(x) \; d \alpha (x) = A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_c^b f(x) \; d \alpha (x) = B} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B \in \mathbb{R}} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} be given.
    • Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^c f(x) \; d \alpha (x) = A} we have that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_1 = \frac{\epsilon}{2} > 0} there exists a partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon_1} \in \mathcal{P}[a, c]} such that for all partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P' \in \mathcal{P}[a, c]} finer than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon_1}} , (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon_1} \subseteq P'} ) and for any choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_k} 's in each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mathrm{th}}} subinterval we have that:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mid S(P', f, \alpha) - A \mid < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}}
    • Similarly, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_c^b f(x) \; d \alpha (x) = B} we have that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_2 = \frac{\epsilon}{2} > 0} there exists a partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon_2} \in \mathcal{P}[c, b]} such that for all partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'' \in \mathcal{P}[c, b]} finer than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon_2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (P_{\epsilon_2} \subseteq P''} ) and for any choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_k} 's in each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mathrm{th}}} subinterval we have that:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mid S(P'', f, \alpha) - B \mid < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}}
    • Let . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon}} is a partition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a, b]} and for all partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \in \mathcal{P}[a, b]} finer than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon}} , (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon} \subseteq P} ) we must have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (*)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (**)} hold. Then for any choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_k} 's in each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mathrm{th}}} subinterval we have that:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}
    • Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b f(x) \; d \alpha (x)} exists and:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}


    Licensing

    Content obtained and/or adapted from:

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