Difference between revisions of "The Additivity Theorem"

Revision as of 15:56, 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 $c\in (a,b)$ . Then, $f$ is Riemann integrable on $[a,b]$ if and only if it is also Riemann integrable on $[a,c]$ and $[c,b]$ . In this case,

$\int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f$

Proof: Suppose that

\int _{a}^{c}f(x)\;d\alpha (x)=A

and

\int _{c}^{b}f(x)\;d\alpha (x)=B

for some

A,B\in \mathbb {R}

. Let

\epsilon >0

be given.

Since $\int _{a}^{c}f(x)\;d\alpha (x)=A$ we have that for $\epsilon _{1}={\frac {\epsilon }{2}}>0$ there exists a partition $P_{\epsilon _{1}}\in {\mathcal {P}}[a,c]$ such that for all partitions $P'\in {\mathcal {P}}[a,c]$ finer than $P_{\epsilon _{1}}$ , ( $P_{\epsilon _{1}}\subseteq P'$ ) and for any choice of $t_{k}$ '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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{\epsilon} = P_{\epsilon_1} \cup P_{\epsilon_2}}$ . 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}}

@@ 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>

Difference between revisions of "The Additivity Theorem"

Revision as of 15:56, 9 November 2021

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