Theorem: Integration by Parts/Riemann-Stieltjes Integral

Let $f, \alpha$ be a real functions that are bounded on $\closedint a b$.
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$.
Then, $\alpha$ is Riemann-Stieltjes integrable with respect to $f$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha + \int_a^b \alpha \rd f = \map f b \map \alpha b - \map f a \map \alpha a$

Proof:
Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P_\epsilon$ be a subdivision of $\closedint a b$ such that:
:For every $P$ finer than $P_\epsilon$, $\size {\map S {P, f, \alpha} - \int_a^b f \rd \alpha} < \epsilon$
Let $P = \set {x_0, \dotsc, x_n}$ be an arbitrary subdivision finer than $P_\epsilon$.
Let $t_k \in \closedint {x_{k - 1} } {x_k}$ for all $k \in \set {1, \dotsc, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \map S {P, \alpha, f}
      | r = \sum_{k \mathop = 1}^n \map \alpha {t_k} \paren {\map f {x_k} - \map f {x_{k - 1} } }
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {t_k} - \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {t_k}
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
We also have:
{{begin-eqn}}
{{eqn | l = \map f b \map \alpha b - \map f a \map \alpha a
      | r = \map f {x_n} \map \alpha {x_n} - \map f {x_0} \map \alpha {x_0}
      | c = {{Defof|Finite Subdivision}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\map f {x_k} \map \alpha {x_k} - \map f {x_{k - 1} } \map \alpha {x_{k - 1} } }
      | c = Telescoping Series
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {x_k} - \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {x_{k - 1} }
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
Combining the above:
{{begin-eqn}}
{{eqn | l = \map f b \map \alpha b - \map f a \map \alpha a - \map S {P, \alpha, f}
      | r = \paren {\sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {x_k} - \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {x_{k - 1} } } - \paren {\sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {t_k} - \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {t_k} }
}}
{{eqn | r = \paren {\sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {x_k} - \sum_{k \mathop = 1}^n \map f {x_k} \map \alpha {t_k} } + \paren {\sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {t_k} - \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \map \alpha {x_{k - 1} } }
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {x_k} \paren {\map \alpha {x_k} - \map \alpha {t_k} } + \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \paren {\map \alpha {t_k} - \map \alpha {x_{k - 1} } }
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
Now, consider $y_\ell \in \R$ defined for all $\ell \in \set {0, 1, \dotsc, 2 n - 1, 2 n}$ as:
:$y_{2 k} = x_k$
:$y_{2 k - 1} = t_k$
Since $y_{2 k - 2} = x_{k - 1} \le t_k = y_{2 k - 1}$ and $t_{2 k - 1} = t_k \le x_k = y_{2 k}$, we have:
:$y_0 \le y_1 \le \dotso \le y_{2 n - 1} \le y_{2 n}$
As $y_0 = x_0 = a$ and $y_{2 n} = x_n = b$, it follows that:
:$P' := \set {y_0, y_1, \dotsc, y_{2 n - 1}, y_{2 n}}$
is a subdivision of $\closedint a b$.
Furthermore, since $P_\epsilon \subseteq P \subseteq P'$, it follows that $P'$ is finer than $P_\epsilon$.
Let $t'_\ell \in \closedint {y_{\ell - 1}} {y_\ell}$ be defined for all $\ell \in \set {1, \dotsc, 2 n}$ as:
:$t'_{2 k} := x_k = y_{2 k} \in \closedint {y_{2 k - 1}} {y_{2 k}}$
:$t'_{2 k - 1} := x_{k - 1} = y_{2 k - 2} \in \closedint {y_{2 k - 2}} {y_{2 k - 1}}$
Then:
{{begin-eqn}}
{{eqn | l = \map S {P', f, \alpha}
      | r = \sum_{\ell \mathop = 1}^{2 n} \map f {t'_\ell} \paren {\map \alpha {y_\ell} - \map \alpha {y_{\ell - 1} } }
      | c = {{Defof|Riemann-Stieltjes Sum}}, noting that repeated values $y_\ell$ can be ignored, since if $y_{\ell - 1} = y_\ell$, then $\map \alpha {y_\ell} - \map \alpha {y_{\ell - 1} } = 0$
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {t'_{2 k} } \paren {\map \alpha {y_{2 k} } - \map \alpha {y_{2 k - 1} } } + \sum_{k \mathop = 1}^n \map f {t'_{2 k - 1} } \paren {\map \alpha {y_{2 k - 1} } - \map \alpha {y_{2 k - 2} } }
      | c = Separating the sum into even and odd $\ell$
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {x_k} \paren {\map \alpha {x_k} - \map \alpha {t_k} } + \sum_{k \mathop = 1}^n \map f {x_{k - 1} } \paren {\map \alpha {t_k} - \map \alpha {x_{k - 1} } }
      | c = Definitions of $y_\ell$ and $t'_\ell$
}}
{{eqn | r = \map f b \map \alpha b - \map f a \map \alpha a - \map S {P, \alpha, f}
      | c = Above
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \size {\map S {P, \alpha, f} - \paren {\map f b \map \alpha b - \map f a \map \alpha a - \int_a^b f \rd \alpha} }
      | r = \size {- \paren {\map f b \map \alpha b - \map f a \map \alpha a - \map S {P, \alpha, f} - \int_a^b f \rd \alpha} }
}}
{{eqn | r = \size {\map f b \map \alpha b - \map f a \map \alpha a - \map S {P, \alpha, f} - \int_a^b f \rd \alpha}
      | c = Absolute Value of Negative
}}
{{eqn | r = \size {\map S {P', f, \alpha} - \int_a^b f \rd \alpha}
      | c = Above
}}
{{eqn | o = <
      | r = \epsilon
      | c = Definition of $P_\epsilon$, since $P'$ is finer than $P_\epsilon$
}}
{{end-eqn}}
Since $P$ finer than $P_\epsilon$ and $\epsilon > 0$ were arbitrary, by definition of the Riemann-Stieltjes integral:
:$\int_a^b \alpha \rd f = \map f b \map \alpha b - \map f a \map \alpha a - \int_a^b f \rd \alpha$
The result follows.
{{qed}}