Reduction of Riemann-Stieltjes Integral to Identity Integrator

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Theorem

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

Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$.

Also suppose that $\alpha$ is continuously differentiable on $\closedint a b$.

Then, let $\alpha'$ be the derivative of $\alpha$ on $\closedint a b$.


Let $g : \closedint a b \to \R$ be defined as:

$\map g x = \map f x \map {\alpha'} x$

Let $\iota$ be the identity mapping on $\closedint a b$.


Then, $g$ is Riemann-Stieltjes integrable with respect to $\iota$ on $\closedint a b$ and:

$\ds \int_a^b g \rd \iota = \int_a^b f \rd \alpha$


Proof

As $f$ is bounded on $\closedint a b$, let $M \in \R$ such that:

$\forall x \in \closedint a b: \size {\map f x} \le M$

Without loss of generality, let $M > 0$.


Let $\epsilon > 0$ be arbitrary.

By Continuous Function on Closed Real Interval is Uniformly Continuous:

$\alpha'$ is uniformly continuous on $\closedint a b$

Therefore, there exists some $\delta_\epsilon$ such that, for all $x, y \in \closedint a b$ such that $\size {x - y} < \delta$:

$\map {\alpha'} x - \map {\alpha'} y < \dfrac \epsilon {2 M \paren {b - a}}$

By Existence of Subdivision with Small Norm, let $P'_\epsilon$ be a subdivision of $\closedint a b$ such that:

$\norm {P'_\epsilon} < \delta_\epsilon$

where $\norm {P'_\epsilon}$ denotes the norm of $P'_\epsilon$.

By definition of Riemann-Stieltjes integral, there is a subdivision $P' '_\epsilon$ such that:

For any $P$ finer than $P' '_\epsilon$, $\ds \size {\map S {P, f, \alpha} - \int_a^b f \rd \alpha} < \frac \epsilon 2$

Define $P_\epsilon := P'_\epsilon \cup P' '_\epsilon$.

Then, $P_\epsilon$ is a subdivision of $\closedint a b$ that is finer than both $P'_\epsilon$ and $P' '_\epsilon$.


Let $P = \set {x_0, x_1, \dotsc, x_{n - 1}, x_n}$ be an arbitrary subdivision of $\closedint a b$ that is finer than $P_\epsilon$.

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

For every $k \in \set {1, 2, \dotsc, x_{n - 1}, x_n}$, it follows from the Mean Value Theorem that:

$\paren 1 \quad \map {\alpha'} {v_k} = \dfrac {\map \alpha {x_k} - \map \alpha {x_{k - 1}}} {x_k - x_{k - 1}}$

for some $v_k \in \openint {x_{k - 1}} {x_k}$.


Let $t_1, t_2, \dotsc, t_{n - 1}, t_n$ be fixed such that:

$t_k \in \closedint {x_{k - 1}} {x_k}$

Let $\map S {P, f, \alpha}$ denote the Riemann-Stieltjes sum of $f$ with respect to $\alpha$ for $P$, using $\paren {t_k}_k$.

Let $\map S {P, g, \iota}$ denote the Riemann-Stieltjes sum of $g$ with respect to $\iota$ for $P$, using $\paren {t_k}_k$.

For every $k \in \set {1, \dotsc, n}$, we have:

\(\text {(2)}: \quad\) \(\ds \size {v_k - t_k}\) \(\le\) \(\ds x_k - x_{k - 1}\) as $t_k, v_k \in \closedint {x_{k - 1} } {x_k}$
\(\ds \) \(\le\) \(\ds \norm P\) Definition of Norm of Subdivision
\(\ds \) \(\le\) \(\ds \norm {P'_\epsilon}\) Norm of Refinement is no Greater than Norm of Subdivision
\(\ds \) \(<\) \(\ds \delta_\epsilon\) Definition of $P'_\epsilon$

Therefore:

\(\ds \map S {P, f, \alpha} - \map S {P, g, \iota}\) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } - \sum_{k \mathop = 1}^n \map g {t_k} \paren {\map \iota {x_k} - \map \iota {x_{k - 1} } }\) Definition of Riemann-Stieltjes Sum
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \map {\alpha'} {v_k} \paren {x_k - x_{k - 1} } - \sum_{k \mathop = 1}^n \map f {t_k} \map {\alpha'} {t_k} \paren {x_k - x_{k - 1} }\) Definitions of $g$ and $\iota$, and $\paren 1$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map {\alpha'} {v_k} - \map {\alpha'} {t_k} } \paren {x_k - x_{k - 1} }\) Linear Combination of Indexed Summations
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \size {\map S {P, f, \alpha} - \map S {P, g, \iota} }\) \(=\) \(\ds \size {\sum_{k \mathop = 1}^n \map f {t_k} \paren {\map {\alpha'} {v_k} - \map {\alpha'} {t_k} } \paren {x_k - x_{k - 1} } }\)
\(\ds \) \(\le\) \(\ds \sum_{k \mathop = 1}^n \size {\map f {t_k} } \size {\map {\alpha'} {v_k} - \map {\alpha'} {t_k} } \paren {x_k - x_{k - 1} }\) Triangle Inequality for Real Numbers, Absolute Value Function is Completely Multiplicative, and Definition of Finite Subdivision
\(\ds \) \(\le\) \(\ds \sum_{k \mathop = 1}^n M \size {\map {\alpha'} {v_k} - \map {\alpha'} {t_k} } \paren {x_k - x_{k - 1} }\) Definition of $M$
\(\ds \) \(<\) \(\ds \sum_{k \mathop = 1}^n M \frac \epsilon {2 M \paren {b - a} } \paren {x_k - x_{k - 1} }\) Definition of $\delta_\epsilon$, as $\size {v_k - t_k} < \delta_\epsilon$ by $\paren 2$
\(\ds \) \(=\) \(\ds \frac \epsilon {2 \paren {b - a} } \sum_{k \mathop = 1}^n \paren {x_k - x_{k - 1} }\) Indexed Summation of Multiple of Mapping
\(\ds \) \(=\) \(\ds \frac \epsilon {2 \paren {b - a} } \paren {x_n - x_0}\) Telescoping Series
\(\ds \) \(=\) \(\ds \frac \epsilon {2 \paren {b - a} } \paren {b - a}\) Definition of Finite Subdivision
\(\ds \) \(=\) \(\ds \frac \epsilon 2\)

Finally:

\(\ds \size {\map S {P, g, \iota} - \int_a^b f \rd \alpha}\) \(=\) \(\ds \size {\map S {P, g, \iota} - \map S {P, f, \alpha} + \map S {P, f, \alpha} - \int_a^b f \rd \alpha}\)
\(\ds \) \(\le\) \(\ds \size {\map S {P, f, \alpha} - \map S {P, g, \iota} } + \size {\map S {P, f, \alpha} - \int_a^b f \rd \alpha}\) Triangle Inequality for Real Numbers, Absolute Value of Negative
\(\ds \) \(<\) \(\ds \frac \epsilon 2 + \size {\map S {P, f, \alpha} - \int_a^b f \rd \alpha}\) $\paren 3$
\(\ds \) \(<\) \(\ds \frac \epsilon 2 + \frac \epsilon 2\) Definition of $P' '_\epsilon$, since $P$ is finer than $P' '_\epsilon$
\(\ds \) \(=\) \(\ds \epsilon\)

As $t_1, \dotsc, t_n$, $P$, and $\epsilon$ were all arbitrary in their respective domains, it follows from the definition of the Riemann-Stieltjes integral that:

$\ds \int_a^b g \rd \iota = \int_a^b f \rd \alpha$

$\blacksquare$


Sources

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