Riemann-Stieltjes Integral with Step Integrator

== Theorem ==
Let $a < c < b$ be real numbers.
Let $f$ be a real function that is bounded on $\closedint a b$.
Let $\alpha$ be a real function on $\closedint a b$ such that:
:$\forall x \in \hointr a c: \map \alpha x = \map \alpha a$
:$\forall x \in \hointl c b: \map \alpha x = \map \alpha b$
Suppose that:
:Either $f$ is left-continuous at $c$, or $\map \alpha c = \map \alpha a$
:Either $f$ is right-continuous at $c$, or $\map \alpha c = \map \alpha b$
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha = \map f c \paren {\map \alpha b - \map \alpha a}$
== Proof ==
Let $\epsilon > 0$ be arbitrary.
The construction of $P_\epsilon$ will depend on which of $f$ and $\alpha$ are continuous on which sides at $c$.
=== Case $f$ Left- and Right-Continuous ===
Suppose $f$ is both left- and right-continuous at $c$.
Define $\epsilon' := \dfrac \epsilon {\size {\map \alpha c - \map \alpha a} + \size {\map \alpha b - \map \alpha c} + 1} > 0$.
By Continuous at Point iff Left-Continuous and Right-Continuous:
:$f$ is continuous at $c$
Then, by definition, there is some $\delta > 0$ such that:
:$\forall x: \size {x - c} < \delta \implies \size {\map f x - \map f c} < \epsilon'$
By Existence of Subdivision with Small Norm, there exists some subdivision $P'_\epsilon$ of $\closedint a b$ such that:
:$\norm P < \delta$
where $\norm P$ denotes the norm of $P$.
Define $P_\epsilon := P'_\epsilon \cup \set c$.
Let $P = \set {x_0, \dotsc, x_n}$ be finer than $P_\epsilon$.
Then, $c \in P$, so for some $r \in \set {1, \dotsc, n - 1}$:
:$x_r = c$
By Norm of Refinement is no Greater than Norm of Subdivision:
:$\norm P < \delta$
We have:
=== Case $f$ is Left-Continuous and $\map \alpha c = \map \alpha b$ ===
Suppose $f$ is left-continuous at $c$, and $\map \alpha c = \map \alpha b$.
Define $\epsilon' := \dfrac \epsilon {\size {\map \alpha c - \map \alpha a} + 1} > 0$.
By definition of left-continuous, there is some $\delta > 0$ such that:
:$\forall x: c - \delta < x \le c \implies \size {\map f x - \map f c} < \epsilon'$
By Existence of Subdivision with Small Norm, there exists some subdivision $P'_\epsilon$ of $\closedint a b$ such that:
:$\norm P < \delta$
where $\norm P$ denotes the norm of $P$.
Define $P_\epsilon := P'_\epsilon \cup \set c$.
Let $P = \set {x_0, \dotsc, x_n}$ be finer than $P_\epsilon$.
Then, $c \in P$, so for some $r \in \set {1, \dotsc, n - 1}$:
:$x_r = c$
By Norm of Refinement is no Greater than Norm of Subdivision:
:$\norm P < \delta$
We have:
=== Case $f$ is Right-Continuous and $\map \alpha c = \map \alpha a$ ===
This case is symmetrical to the previous one.
Hence, the details are omitted whenever they are identical to those above.
Let $\epsilon' := \dfrac \epsilon {\size {\map \alpha b - \map \alpha c} + 1}$.
By definition of right-continuous, choose $\delta > 0$ such that:
:$\forall x: c \le x < c + \delta \implies \size {\map f x - \map f c} < \epsilon'$
Select $P'_\epsilon$ as above, such that:
:$\norm {P'_\epsilon} < \delta$
Define $P_\epsilon = P'_\epsilon \cup \set c$ as above.
Let $P$ be finer than $P_\epsilon$.
Then, $\norm P < \delta$.
Let $x_r = c$ as above.
Thus:
=== Case $\map \alpha a = \map \alpha c = \map \alpha b$ ===
Suppose $\map \alpha c = \map \alpha a$ and $\map \alpha c = \map \alpha b$.
It follows immediately that:
:$\forall x \in \closedint a b: \map \alpha x = \map \alpha c$
Therefore:
:$\forall x, y \in \closedint a b: \map \alpha x - \map \alpha y = \map \alpha c - \map \alpha c = 0$
Define $P_\epsilon = \set {a, b}$.
Let $P = \set {x_0, \dotsc, x_n}$ be a subdivision of $\closedint a b$ that is finer than $P_\epsilon$.
Then:
In every case, we have constructed a subdivision $P_\epsilon$ of $\closedint a b$ such that, for every $P$ finer than $P_\epsilon$:
:$\size {\map S {P, f, \alpha} - \map f c \paren {\map \alpha b - \map \alpha a}} < \epsilon$
Therefore, by definition of the Riemann-Stieltjes integral:
:$\ds \int_a^b f \rd \alpha = \map f c \paren {\map \alpha b - \map \alpha a}$
== Sources ==
* : Chapter $7$ The Riemann-Stieltjes Integral: $\S 7.8$: Theorem $7.9$
Category:Riemann-Stieltjes Integral