Mult. Products of R-S Integrable Functions with Increasing Integrators
Multiple Products of Riemann-Stieltjes Integrable Functions with Increasing Integrators
Recall from The Product of Riemann-Stieltjes Integrable Functions with Increasing Integrators page that if $f$ and $g$ are functions defined on $[a, b]$ and $\alpha$ is an increasing function on $[a, b]$ then if $f$ and $g$ are Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then $fg$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
The following theorem generalizes the one stated above for products of more than two functions.
| Theorem 1: Let $f_1, f_2, ..., f_n$ be functions for some $n \in \mathbb{N}$ defined on $[a, b]$ and let $\alpha$ be an increasing function. If $f_1, f_2, ..., f_n$ are all Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then the product $\prod_{i=1}^n f_i$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$. |
We will carry this proof out by induction.
- Proof: Let $S(n)$ be the statement that if the functions $f_1, f_2, ..., f_n$ are Riemann-Stieltjes (which we abbreviate as "R-S" from now on) integrable with respect to $\alpha$ then $\prod_{i=1}^{k} f_i$ is also Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
- For the base step, $S(1)$ is trivially true and $S(2)$ is true by the theorem stated above.
- Assume that for $k > 2$ then $S(k)$ is true, that is, if $f_1, f_2, ..., f_k$ are R-S integrable with respect to $\alpha$ on $[a, b]$ then $\prod_{i=1}^{k} f_i$ is also R-S integrable with respect to $\alpha$ on $[a, b]$. We want to then show that $S(k+1)$ is true, that is, if $f_1, f_2, ..., f_k, f_{k+1}$ are R-S integrable with respect to $\alpha$ on $[a, b]$ then so is $\prod_{i=1}^{k+1} f_i$.
- Note that:
\begin{align} \quad \prod_{i=1}^{k+1} f_i = \left ( \prod_{i=1}^{k} f_i \right ) \cdot f_{k+1} \end{align}
- Let $g = \prod_{i=1}^{k} f_i$. Then by assumption $g$ is R-S integrable and $f_{k+1}$ is R-S integrable so $gf_{k+1}$ is R-S integrable with respect to $\alpha$ on $[a, b]$ so $S(k+1)$ is true.
- So for all $n \in \mathbb{N}$, if $f_1, f_2, ..., f_n$ is a set of Riemann-Stieltjes integrable functions with respect to $\alpha$ on $[a, b]$ then $\prod_{i=1}^{n} f_i$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$. $\blacksquare$