The Additivity Theorem

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 $[a,b]$, 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 Failed to parse (unknown function "\mathscr"): {\displaystyle P_{\epsilon_1} \in \mathscr{P}[a, c]} such that for all partitions Failed to parse (unknown function "\mathscr"): {\displaystyle P' \in \mathscr{P}[a, c]} finer than $P_{\epsilon _{1}}$, ($P_{\epsilon _{1}}\subseteq P'$) and for any choice of $t_{k}$'s in each $k^{\mathrm {th} }$ subinterval we have that:

${\begin{aligned}\quad \mid S(P',f,\alpha )-A\mid <\epsilon _{1}={\frac {\epsilon }{2}}\quad (*)\end{aligned}}$

• Similarly, since $\int _{c}^{b}f(x)\;d\alpha (x)=B$ we have that for $\epsilon _{2}={\frac {\epsilon }{2}}>0$ there exists a partition Failed to parse (unknown function "\mathscr"): {\displaystyle P_{\epsilon_2} \in \mathscr{P}[c, b]} such that for all partitions Failed to parse (unknown function "\mathscr"): {\displaystyle P'' \in \mathscr{P}[c, b]} finer than $P_{\epsilon _{2}}$, $(P_{\epsilon _{2}}\subseteq P''$) and for any choice of $u_{k}$'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 \mathscr{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}}