Riemann-Steiltjes Integral on R

Subjects: Real Analysis
Links: Functions of Bounded Variation on R, Riemann Integral in R

Let $\alpha :I=[a,b]\to \mathbb{R}$ be monotonically increasing. Given a partition $P$ of $I$, then:

The set of all function Reimann-Stieltjes Integrable with respect to $\alpha$ along $I$ is denoted as ${\mathcal{R}}_I(\alpha )$

There’s also the option of $\alpha$ to be of bounded variation, which is a generalization of monotonicity.

Reimann Sums

Let $f:I\to \mathbb{R}$, a Reimann-Stieltjes sum is defined as follows, given $\dot{\mathcal{P}}\in {\dot{\mathrm{\wp }}}_I$:

Integrability

Let $f:I\to \mathbb{R}$, be Reimann-Stieltjes Integrable iff:

$L$ is called the Riemann-Stieltjes integral of $f$ with respect to $\alpha$ along $I$, denoted as:

Darboux Sums

Given $f:I\to \mathbb{R},$ bounded the lower and upper sum of a parition $P$ with respect to $\alpha$ are defined as follows:

Similarly, there’s a lower integral and upper integral with respect to $\alpha$:

Lemma: If ${\mathcal{P}}^{\prime }$ is a refinement of a partition $\mathcal{P}$, and has the corresponding relation with the lower and upper sums:

Theorem: $L(f,\alpha )\le U(f,\alpha )$

Integrability

$f:I\to \mathbb{R}$ is Reimann-Stieltjes integrable iff:

Darboux-Cauchy Criterion $f:I\to \mathbb{R}$ is Reimann-Stieltjes integrable $(f\in {\mathcal{R}}_I(\alpha ))$ iff:

Theorem:

1. Given an $\epsilon >0$, and there’s a $\mathcal{P}$ such that satisfies the Darboux-Cauchy Criterion, any refinement of $\mathcal{P}$ also satisfies it.

2. Given an $\epsilon >0$, and there’s a $\mathcal{P}$ such that satisfies the Darboux-Cauchy Criterion, any $t_i,s_i\in [x_i,x_{i-1}]$ then:

   $$\sum _{i=i}^n|f(s_i)-f(t_i)|\mathrm{\Delta }{\alpha }_i<\epsilon$$

3. If $f\in {\mathcal{R}}_I(\alpha )$, and $(2)$ is satisfied, then:

Integral Properties

Additivity Theorem

1. If $f\in {\mathcal{R}}_{[a,b]}(\alpha )$, and $c\in (a,b)$, then $f\in {\mathcal{R}}_{[a,c]}(\alpha )$ and $f\in {\mathcal{R}}_{[c,b]}(\alpha )$, and:

   $${\int }_a^{b}fd\alpha ={\int }_a^{c}fd\alpha +{\int }_c^{b}fd\alpha$$

General Integrability Theorems

1. If$f$ $\in {\mathcal{C}}^0[a,b]$, then $f\in {\mathcal{R}}_I(\alpha )$.
2. If $f$ is monotonic on $[a,b]$ and $\alpha$ is continuous on $[a,b],$ then $f\in {\mathcal{R}}_I(\alpha )$.
3. Let $f$ be bounded on $[a,b]$, and only has a finite number of discontinuities on $[a,b]$, and $\alpha$ is continuous at all discontinuities of $f$, then $f\in {\mathcal{R}}_I(\alpha )$.
4. Let $f\in {\mathcal{R}}_I(\alpha )$, there’s $m,M\in \mathbb{R}[m\le f\le M]$ on $I$, $\varphi \in {\mathcal{C}}^0([m,M])$ then $\varphi \circ f\in {\mathcal{R}}_I(\alpha )$ .

Algebraic Properties

1. If $f_1,f_2\in {\mathcal{R}}_I(\alpha )$, then $f_1+f_2\in {\mathcal{R}}_I(\alpha )$, and:

   $${\int }_I{f}_1+f_{2}d\alpha ={\int }_I{f}_{1}d\alpha +{\int }_I{f}_{2}d\alpha$$

2. If $f\in {\mathcal{R}}_I(\alpha )$, and $c\in \mathbb{R}$, then $cf\in {\mathcal{R}}_I(\alpha )$, and:

   $${\int }_{I}cfd\alpha =c{\int }_{I}fd\alpha$$

3. If $f_1\le f_2$ on $I$, then:

   $${\int }_I{f}_{1}d\alpha \le {\int }_I{f}_{2}d\alpha$$

4. If $f\in {\mathcal{R}}_I(\alpha )$ y $|f|\le M$ on $[a,b]$, then:

   $$\left|{\int }_{I}fd\alpha \right|\le M(\alpha (b)-\alpha (a))$$

5. If $f\in {\mathcal{R}}_I({\alpha }_1)$ and $f\in {\mathcal{R}}_I({\alpha }_2)$, then $f\in {\mathcal{R}}_I({\alpha }_1+{\alpha }_2)$ and:

   $${\int }_{I}fd({\alpha }_1+{\alpha }_2)={\int }_{I}fd{\alpha }_1+{\int }_{I}fd{\alpha }_2$$

6. If $f\in {\mathcal{R}}_{[a,b]}(\alpha )$, and $c\in {\mathbb{R}}^{+}$, then $f\in {\mathcal{R}}_{[a,b]}(c\alpha )$ and:

   $${\int }_{I}fd(c\alpha )=c{\int }_{I}fd\alpha$$

7. If $f,g\in {\mathcal{R}}_I(\alpha )$, then $fg\in {\mathcal{R}}_I(\alpha )$.

8. If $f\in {\mathcal{R}}_I(\alpha )$, then $|f|\in {\mathcal{R}}_I(\alpha )$ and:

   $$\left|{\int }_{I}fd\alpha \right|\le {\int }_I|f|d\alpha$$

Unit Step Function

The unit step function: $u:\mathbb{R}\to \{0,1\}$, where:

Theorem: Let $s\in (a,b)$, if $f$ is bounded on $[a,b]$, and $f$ is continuous on $s$ and $\alpha (x)=u(x-s)$, then:

Theorem: Let $\mathrm{\forall }n\in \mathbb{N}(c_n\ge 0)$, and that $\sum c_n$ converge, $(s_n{)}_{n\in \mathbb{N}}$ be a sequence of different points of $[a,b]$ and:

Let $f$ be continuous on $[a,b]$. Then,

Miscellaneous

Theorem: Let $\alpha$ be monotonically increasing and ${\alpha }^{\prime }\in {\mathcal{R}}_I$. Let $f$ be bounded function on $[a,b]$. If $f\in {\mathcal{R}}_I(\alpha )$ iff $f{\alpha }^{\prime }\in {\mathcal{R}}_I$, and:

Change of variables: Let $\phi$ be strictly increasing continuous function that maps $[A,B]$ onto $[a,b]$. Let $\alpha$ be monotonically increasing on $[a,b]$ and $f\in {\mathcal{R}}_I(\alpha )$. Let’s define $\beta$ and $g$ on $[A,B]$ as:

Then $g\in {\mathcal{R}}_{[A,B]}(\beta )$, and