Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part

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Theorem

Let $a < c < b$ be real numbers.

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 $\closedint a b$, and also on one of the two intervals $\closedint a c$ and $\closedint c b$.


Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on the other interval as well, and:

$\ds \int_a^c f \rd \alpha + \int_c^b f \rd \alpha = \int_a^b f \rd \alpha$


Proof

Let $\closedint p q$ be the interval from among $\closedint a c$ and $\closedint c b$ on which we know $f$ is Riemann-Stieltjes integrable with respect to $\alpha$.

Let $\closedint u v$ be the other interval.

Then, we want to prove that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint u v$, and:

$\ds \int_u^v f \rd \alpha = \int_a^b f \rd \alpha - \int_p^q f \rd \alpha$


Let $\epsilon > 0$ be arbitrary.

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

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

Define $P_\epsilon := \paren {P'_\epsilon \cup {u, v}} \cap \closedint u v$.

Then, $P_\epsilon$ is a subdivision of $\closedint u v$.


Let $P = \set {x_0, \dotsc, x_n}$ be a subdivision of $\closedint u v$ that is finer than $P_\epsilon$.

Define:

$P' = \set {y_0, \dotsc, y_m} := P \cup P'_\epsilon \cup P' '_\epsilon$
$P' ' = \set {z_0, \dotsc, z_r} := P' \cap \closedint p q$

Since $P'_\epsilon \subseteq P' \subseteq \closedint a b$:

$P'$ is a subdivision of $\closedint a b$ that is finer than $P'_\epsilon$

Since $P' '_\epsilon \subseteq P' ' \subseteq \closedint p q$:

$P' '$ is a subdivision of $\closedint p q$ that is finer than $P' '_\epsilon$

We have:

\(\ds \map S {P, f, \alpha}\) \(=\) \(\ds \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }\) Definition of Riemann-Stieltjes Sum
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {t'_k} \paren {\map \alpha {y_k} - \map \alpha {y_{k - 1} } } - \sum_{k \mathop = 1}^r \map f {z_k} \paren {\map \alpha {z_k} - \map \alpha {z_{k - 1} } }\) where $t'_k = \begin{cases} t_j & : y_k = x_j \\ z_j & : y_k = z_j \end{cases}\quad$ and $j \ge 1$
\(\ds \) \(=\) \(\ds \map S {P', f, \alpha} - \map S {P' ', f, \alpha}\) Definition of Riemann-Stieltjes Sum

Therefore:

\(\ds \size {\map S {P, f, \alpha} - \paren {\int_a^b f \rd \alpha - \int_p^q f \rd \alpha} }\) \(=\) \(\ds \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha + \int_p^q f \rd \alpha - \map S {P' ', f, \alpha} }\) Above
\(\ds \) \(\le\) \(\ds \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha} + \size {\int_p^q f \rd \alpha - \map S {P' ', f, \alpha} }\) Triangle Inequality for Real Numbers
\(\ds \) \(=\) \(\ds \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha} + \size {\map S {P' ', f, \alpha} - \int_p^q f \rd \alpha}\) Absolute Value of Negative
\(\ds \) \(<\) \(\ds \frac \epsilon 2 + \frac \epsilon 2\) Definitions of $P'_\epsilon$ and $P' '_\epsilon$
\(\ds \) \(=\) \(\ds \epsilon\)

As $P$ finer than $P_\epsilon$ and $\epsilon > 0$ were arbitrary, it follows from the definition of the Riemann-Stieltjes integral that:

$\ds \int_u^v f \rd \alpha = \int_a^b f \rd \alpha - \int_p^q f \rd \alpha$

$\blacksquare$


Sources

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