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

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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 $\closedint a c$, and also on $\closedint c b$.


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

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


Proof

Let $\epsilon > 0$ be arbitrary.

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

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

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

Since:

  • $P_\epsilon \subseteq \closedint a c \cup \closedint c b = \closedint a b$
  • $a \in P'_\epsilon \subseteq P_\epsilon$
  • $b \in P' '_\epsilon \subseteq P_\epsilon$

it follows that $P_\epsilon$ is a subdivision of $\closedint a b$.


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

Since $c \in P'_\epsilon \subseteq P_\epsilon \subseteq P$, there must be some $m \in {1, 2, \dotsc, n - 2, n - 1}$ such that:

$x_m = c$

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

$y_\ell = x_{\ell + m}$

Then, define:

$P' := P \cap \closedint a c = \set {x_0, x_1, \dotsc, x_{m - 1}, x_m}$
$P' ' := P \cap \closedint c b = \set {y_0, y_1, \dotsc, y_{n - m - 1}, y_{n - m}}$

It follows that:

$P'$ is a subdivision of $\closedint a c$ that is finer than $P'_\epsilon$
$P' '$ is a subdivision of $\closedint c b$ 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
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{k \mathop = m + 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }\) Indexed Summation over Adjacent Intervals
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{\ell \mathop = 1}^{n - m} \map f {t_{\ell + m} } \paren {\map \alpha {x_{\ell + m} } - \map \alpha {x_{\ell + m - 1} } }\) Indexed Summation over Translated Interval
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{\ell \mathop = 1}^{n - m} \map f {t_{\ell + m} } \paren {\map \alpha {y_\ell} - \map \alpha {y_{\ell - 1} } }\) Definition of $y_\ell$
\(\ds \) \(=\) \(\ds \map S {P', f, \alpha} + \map S {P' ', f, \alpha}\) Definition of Riemann-Stieltjes Sum, as $y_{\ell - 1} = x_{\ell + m - 1} \le t_{\ell + m} \le x_{\ell + m} = y_\ell$

Therefore:

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


As $P \supseteq P_\epsilon$ and $\epsilon > 0$ were arbitrary, by the definition of the Riemann-Stieltjes integral:

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

$\blacksquare$


Sources

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