Riemann-Stieltjes Integrals with Step Functions as Integrators Review

Riemann-Stieltjes Integrals with Step Functions as Integrators Review

We will now review some of the recent material that we have covered regarding Riemann-Stieltjes integrals with step functions as integrators.

• On the Step Functions page we said that a function $\alpha$ is a Step Function on the interval $[a, b]$ if there exists a partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ such that $\alpha$ is constant on each open subinterval $(x_{k-1}, x_k)$ for $k \in \{ 1, 2, ..., n \}$.
• Furthermore, we said that the Jump at $x_k$ for $k \in \{ 0, 1, 2, ..., n \}$ is defined to be $\alpha(x_k^+) - \alpha(x_k^-)$ where $\alpha(x_k^+) = \lim_{x \to x_k^+} \alpha(x)$ and $\alpha(x_k^-) = \lim_{x \to x_k^-} \alpha (x)$. If $k = 0$, then the jump at $x_0$ is $\alpha(x_0^+) - \alpha(x_0)$ and if $k = n$ then the jump at $x_n$ is $\alpha(x_n) - \alpha(x_n^-)$.
• On the Riemann-Stieltjes Integrals with Single Discontinuity Step Functions as Integrators page we saw that if $c \in (a, b)$ and if $\alpha$ is a single discontinuity step function $\alpha (x) = \left\{\begin{matrix} p & \mathrm{for \: all} \: x \in [a, c) \\ q & \mathrm{for} \: x = c\\ r & \mathrm{for \: all} \: x \in (c, b] \end{matrix}\right.$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ if not both $f$ and $g$ are left discontinuous at $c$ and not both $f$ and $g$ are right discontinuous at $c$. Furthermore, we have that:
(1)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = f(c)[\alpha(c^+) - \alpha(c^-)] \end{align}
• On the Riemann-Stieltjes Integrals with Multiple Discontinuity Step Functions as Integrators page we generalized the result above. We saw that if $\alpha$ is a step function on $[a, b]$ with discontinuities at $x_1, x_2, ..., x_n$, then at each $x_k$ if $f$ and $\alpha$ are not both left discontinuous at $x_k$ and $f$ and $\alpha$ are not both right discontinuous at $x_k$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and furthermore:
(2)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \sum_{k=1}^{n} f(x_k)[\alpha(x_k^+) - \alpha(x_k^-)] \end{align}
• If for some $x_k = a$ for some $k \in \{1, 2, ..., n \}$ then we denote $\alpha(x_k^+) - \alpha(x_k^-) = \alpha(a^+) - \alpha(a)$ and if $x_k = b$ for some $k \in \{1, 2, ..., n \}$ then we denote $\alpha(x_k^+) - \alpha(x_k^-) = \alpha(b) - \alpha(b^-)$.
• On The Greatest Integer Function page we defined a special type of function known as The Greatest Integer Function which is denoted as $[x]$ and is defined for all $x \in \mathbb{R}$ as the greatest integer less than or equal $x$ and hence satisfies the following inequality:
(3)
\begin{align} \quad [x] \leq x < x + 1 \end{align}
• For example, $[3.14] = 3$ and $[-2.3] = -3$. We noted if $\alpha (x) = [x]$ on $[a, b]$ then $\alpha$ is a step function with discontinuities at the integers in $[a, b]$.
• On the Riemann-Stieltjes Integrals with the Greatest Integer Function as an Integrator we saw that the if $(a_k)_{k=1}^{n} = (a_1, a_2, ..., a_n)$ is a finite sequence of real numbers then the sum of the sequence can be expressed as a Riemann-Stieltjes integral whose integrator is the greatest integer function. If we define $f(x) = \left\{\begin{matrix} 0 & \mathrm{if} \: x = 0\\ a_1 & \mathrm{if} \: 0 < x \leq 1 \\ a_2 & \mathrm{if} \: 1 < x \leq 2 \\ \vdots \\ a_n & \mathrm{if} \: n-1 < x \leq n \end{matrix}\right.$ we saw that then:
(4)
\begin{align} \quad \sum_{k=1}^{n} a_k = \int_0^n f(x) \: d [x] \end{align}
• This works because the function $f$ is left continuous at all integers $x$ in $[0, n]$ and $[x]$ is right continuous at all integers in $[0, n]$, the jumps of $[x]$ are $1$, and the values of the discontinuities are the values of the sequence $(a_k)_{k=1}^{n}$.