Integration by Substitution/Riemann-Stieltjes Integral/Decreasing

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Theorem

Let $g$ be a real function that is continuous and strictly decreasing on $\closedint a b$.

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

Further suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint {\map g b} {\map g a}$.


Let $h, \beta : \closedint a b \to \R$ be defined as:

$\map h x = \map f {\map g x}$
$\map \beta x = \map \alpha {\map g x}$

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

$\ds \int_a^b h \rd \beta = - \int_{\map g b}^{\map g a} f \rd \alpha$


Proof

For any $x \in \closedint a b$, $\map g x > \map g a$ implies $x < a$ by definition of strictly decreasing, so:

$\forall x \in \closedint a b: \map g x \le \map g a$

Likewise, $\map g x > \map g b$ implies $x < b$, so:

$\forall x \in \closedint a b: \map g x \ge \map g b$

Together:

$\Img g \subseteq \closedint {\map g b} {\map g a}$

But by the Intermediate Value Theorem:

$\closedint {\map g b} {\map g a} \subseteq \Img g$

Therefore:

$\Img g = \closedint {\map g b} {\map g a}$

Then, by Inverse of Strictly Monotone Function, $g$ has an inverse $g^{-1} : \closedint {\map g b} {\map g a} \to \closedint a b$.

Furthermore, $g^{-1}$ is strictly decreasing.


Let $\epsilon > 0$ be arbitrary.

By definition of the Riemann-Stieltjes integral, there exists a subdivision $P'_\epsilon$ of $\closedint {\map g b} {\map g a}$ such that:

For each $P'$ finer than $P'_\epsilon$, $\ds \size {\map S {P', f, \alpha} - \int_{\map g b}^{\map g a} f \rd \alpha} < \epsilon$.

Suppose $P'_\epsilon = \set {y_0, y_1, \dotsc, y_{n - 1}, y_n}$.

Then, define $P_\epsilon = g^{-1} \sqbrk {P'_\epsilon}$.

Since $g^{-1}$ is strictly decreasing:

$a = \map {g^{-1}} {\map g a} = \map {g^{-1}} {y_n} < \map {g^{-1}} {y_{n - 1}} < \dotso < \map {g^{-1}} {y_1} < \map {g^{-1}} {y_0} = \map {g^{-1}} {\map g b} = b$

we have that $P_\epsilon$ is a subdivision of $\closedint a b$.


Now, let $P = \set {x_0, \dotsc, x_m}$ be a subdivision of $\closedint a b$ that is finer than $P_\epsilon$.

For each $k \in \set {0, 1, \dotsc, m - 1, m}$, define:

$x'_k = x_{m - k}$

Define $P' = \set {\map g {x'_0}, \map g {x'_1}, \dotsc, \map g {x'_{m - 1}}, \map g {x'_m}$.

Since $g$ is strictly decreasing, $P'$ is a subdivision of $\closedint {\map g b} {\map g a}$.

Furthermore, for every $k$:

$y_k = \map g {\map {g^{-1}} {y_k}} \in g \sqbrk {P_\epsilon} \subseteq g \sqbrk P = P'$

Thus, $P'$ is finer than $P'_\epsilon$.

Hence:

\(\ds \map S {P, h, \beta}\) \(=\) \(\ds \sum_{k \mathop = 1}^m \map h {t_k} \paren {\map \beta {x_k} - \map \beta {x_{k - 1} } }\) Definition of Riemann-Stieltjes Sum
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {\map g {t_k} } \paren {\map \alpha {\map g {x_k} } - \map \alpha {\map g {x_{k - 1} } } }\) Definitions of $h$ and $\beta$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {\map g {t_{m - k + 1} } } \paren {\map \alpha {\map g {x_{m - k + 1} } } - \map \alpha {\map g {x_{m - k} } } }\) Indexed Summation does not Change under Permutation, with $\map \sigma k = m - k + 1$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {\map g {t_{m - k + 1} } } \paren {\map \alpha {\map g {x'_{k - 1} } } - \map \alpha {\map g {x'_k} } }\) Definition of $x'_k$
\(\ds \) \(=\) \(\ds - \sum_{k \mathop = 1}^m \map f {\map g {t_{m - k + 1} } } \paren {\map \alpha {\map g {x'_k} } - \map \alpha {\map g {x'_{k - 1} } } }\) Indexed Summation of Multiple of Mapping
\(\ds \) \(=\) \(\ds - \map S {P', f, \alpha}\) Definition of Riemann-Stieltjes Sum, since $\map g {x'_{k - 1} } = \map g {x_{m - k + 1} } \le \map g {t_{m - k + 1} } \le \map g {x_{m - k} } = \map g {x'_k}$

Therefore:

\(\ds \size {\map S {P, h, \beta} - \paren {- \int_{\map g a}^{\map g b} f \rd \alpha} }\) \(=\) \(\ds \size {- \map S {P', f, \alpha} + \int_{\map g a}^{\map g b} f \rd \alpha}\) Above
\(\ds \) \(=\) \(\ds \size {\map S {P', f, \alpha} - \int_{\map g a}^{\map g b} f \rd \alpha}\) Absolute Value of Negative
\(\ds \) \(<\) \(\ds \epsilon\) Definition of $P'_\epsilon$


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

$\ds \int_a^b h \rd \beta = - \int_{\map g a}^{\map g b} f \rd \alpha$

$\blacksquare$


Sources

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