Linear Combination of Riemann-Stieltjes Integrals/Integrator

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Theorem

Let $f, \alpha, \beta$ be real functions that are bounded on $\closedint a b$.

Let $c_1, c_2 \in \R$.

Suppose that $f$ is Riemann-Stieltjes integrable with respect both $\alpha$ and $\beta$ on $\closedint a b$ and:

$\ds \int_a^b f \rd \alpha = A$
$\ds \int_a^b f \rd \beta = B$


Then, $f$ is Riemann-Stieltjes integrable with respect to $\gamma : \closedint a b \to \R$ defined as:

$\map \gamma x = c_1 \map \alpha x + c_2 \map \beta x$

and furthermore:

$\ds \int_a^b f \rd \gamma = c_1 A + c_2 B$


Proof

Let $\epsilon > 0$ be arbitrary.

By definition of the Riemann-Stieltjes integral, let $P'_\epsilon, P' '_\epsilon$ be subdivisions of $\closedint a b$ such that:

For every $P$ finer than $P'_\epsilon$, $\size {\map S {P, f, \alpha} - A} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$
For every $P$ finer than $P' '_\epsilon$, $\size {\map S {P, f, \beta} - B} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$

Define $P_\epsilon = P'_\epsilon \cup P' '_\epsilon$ to be a subdivision that is finer than both $P'_\epsilon$ and $P' '_\epsilon$.


Let $P = {x_0, \dotsc, x_n} \supseteq P_\epsilon$ be an arbitrary subdivision that is finer than $P_\epsilon$.

Clearly, $P$ is also finer than $P'_\epsilon$ and $P' '_\epsilon$.

Finally, fix the $t_1, \dotsc, t_k$ for the following Riemann-Stieltjes sum.


We have:

\(\ds \map S {P, f, \gamma}\) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \gamma {x_k} - \map \gamma {x_{k - 1} } }\) Definition of Riemann-Stieltjes Sum
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \paren {\paren {c_1 \map \alpha {x_k} + c_2 \map \beta {x_k} } - \paren {c_1 \map \alpha {x_{k - 1} } + c_2 \map \beta {x_{k - 1} } } }\) Definition of $\gamma$
\(\ds \) \(=\) \(\ds c_1 \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + c_2 \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \beta {x_k} - \map \beta {x_{k - 1} } }\) Linear Combination of Indexed Summations
\(\ds \) \(=\) \(\ds c_1 \map S {P, f, \alpha} + c_2 \map S {P, f, \beta}\) Definition of Riemann-Stieltjes Sum

And so:

\(\ds \size {\map S {P, f, \gamma} - \paren {c_1 A + c_2 B} }\) \(=\) \(\ds \size {\paren {c_1 \map S {P, f, \alpha} + c_2 \map S {P, f, \beta} } - \paren {c_1 A + c_2 B} }\)
\(\ds \) \(=\) \(\ds \size {c_1 \paren {\map S {P, f, \alpha} - A} + c_2 \paren {\map S {P, f, \beta} - B} }\)
\(\ds \) \(\le\) \(\ds \size {c_1} \size {\map S {P, f, \alpha} - A} + \size {c_2} \size {\map S {P, f, \beta} - B}\) Triangle Inequality for Real Numbers and Absolute Value Function is Completely Multiplicative
\(\ds \) \(\le\) \(\ds \size {c_1} \frac \epsilon {\size {c_1} + \size {c_2} + 1} + \size {c_2} \frac \epsilon {\size {c_1} + \size {c_2} + 1}\) Definitions of $P'_\epsilon$ and $P' '_\epsilon$
\(\ds \) \(=\) \(\ds \frac {\size {c_1} + \size {c_2} } {\size {c_1} + \size {c_2} + 1} \epsilon\)
\(\ds \) \(<\) \(\ds \epsilon\)


As $\epsilon > 0$ was arbitrary, by the definition of the Riemann-Stieltjes integral:

$\ds \int_a^b f \rd \gamma = c_1 A + c_2 B$

$\blacksquare$


Sources

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