Ex2-13

Solution to Chapter 2, Exercise 13
Theorem 2.16 (i)
If \int_a^b f~d\phiexists, then so do \int_a^b cf~d\phi and \int_a^b f~d(c\phi) for any constant c, and
\int_a^b cf~d\phi=\int_a^b f~d(c\phi)=c\int_a^b f~d\phi.
Proof:
Consider a partition \Gamma=\{a=x_0<x_1<\ldots<x_m=b\}of [a,b] and arbitrarily select intermediate points \{\zeta_i\}_{i=1}^m, where x_{i-1}\leq \zeta_i\leq x_i.
\begin{align}
R_\Gamma(cf)&=\sum_{i=1}^m cf(\zeta_i)[\phi(x_i)-\phi(x_{i-1})]\\
&=c\left(\sum_{i=1}^m f(\zeta_i)[\phi(x_i)-\phi(x_{i-1})]\right)\\
&=cR_\Gamma(f).
\end{align}Then for any sequence of partitions \{\Gamma_k\}with \lim_{k\to\infty}gap(\Gamma_k)=0, we have
\begin{align}
\lim_{k\to \infty}R_{\Gamma_k}(cf)&=\lim_{k\to \infty}cR_{\Gamma_k}(f)\\
&=c\lim_{k\to \infty}R_{\Gamma_k}(f)\\
&=c\int_a^b f~d\phi.
\end{align}Since \int_a^b f~d\phiexists, clearly c\int_a^b f~d\phi also exists for any constant c. Thus, \lim_{k\to \infty}R_{\Gamma_k}(cf)=\int_a^b cf~d\phi exists and \int_a^b cf~d\phi=c\int_a^b f~d\phi.
Similarly,
\begin{align}
R_\Gamma(f, c\phi)&=\sum_{i=1}^m f(\zeta_i)[c\phi(x_i)-c\phi(x_{i-1})]\\
&=\sum_{i=1}^m f(\zeta_i)c[\phi(x_i)-\phi(x_{i-1})]\\
&=c\left(\sum_{i=1}^m f(\zeta_i)[\phi(x_i)-\phi(x_{i-1})]\right)\\
&=cR_\Gamma(f,\phi).
\end{align}For any sequence of partitions \{\Gamma_k\}with \lim_{k\to\infty}gap(\Gamma_k)=0,
\begin{align}
\lim_{k\to \infty}R_{\Gamma_k}(f,c\phi)&=\lim_{k\to \infty}cR_{\Gamma_k}(f,~\phi)\\
&=c\lim_{k\to \infty}R_{\Gamma_k}(f,~\phi)\\
&=c\int_a^b f~d\phi.
\end{align}
Since \int_a^b f~d\phiexists, clearly c\int_a^b f~d\phi also exists for any constant c. Thus, \lim_{k\to \infty}R_{\Gamma_k}(f,c\phi)=\int_a^b f~dc\phi exists and \int_a^b f~d(c\phi)=c\int_a^b f~d\phi. This completes the proof.
Proof of Theorem 2.16 (ii)
If \int_a^b f_1 \operatorname{d}\!\phi and \int_a^b f_2 \operatorname{d}\!\phi both exist, so does \int_a^b (f_1+f_2) \operatorname{d}\!\phi, and 
\int_a^b (f_1+f_2) \operatorname{d}\!\phi = \int_a^b f_1 \operatorname{d}\!\phi + \int_a^b f_2 \operatorname{d}\!\phi. 
Let \Gamma be a partition \Gamma=\{x_0,x_1,\dots,x_n\} of [a,b], the sample point \{\xi_i\} is \xi_i \in [x_{i-1},x_i]. Then 
\begin{align} 
R_\Gamma (f_1+f_2) &= \sum_{i=1}^m [f_1(\xi_1)+f_2(\xi_i)][\phi(x_i)-\phi(x_{i-1})] \\
&= \sum_{i=1}^m f_1(\xi_1)[\phi(x_i)-\phi(x_{i-1})] + \sum_{i=1}^m f_2(\xi_i)[\phi(x_i)-\phi(x_{i-1})] \\
&= R_\Gamma(f_1)+R_\Gamma(f_2).
\end{align}
Then for any sequence of partitions \{\Gamma_k\}, \lim_{k\to\infty} gap(\Gamma_k)=0, 
\begin{align}
\lim_{k\to\infty} R_{\Gamma_k}(f_1+f_2) &= \lim_{k\to\infty} R_{\Gamma_k}(f_1) + \lim_{k\to\infty} R_{\Gamma_k}(f_2) \\
&= \int_a^b f_1 \operatorname{d}\!\phi + \int_a^b f_2 \operatorname{d}\!\phi.
\end{align}
Thus \lim_{k\to\infty}R_{\Gamma_k}(f_1+f_2) exists and so \int_a^b (f_1+f_2) \operatorname{d}\!\phi. Also, 
\int_a^b (f_1+f_2) \operatorname{d}\!\phi = \int_a^b f_1 \operatorname{d}\!\phi + \int_a^b f_2 \operatorname{d}\!\phi.
Proof of Theorem 2.16 (iii)
We want to prove that if \int_a^b f~d\phi_1and \int_a^b f~d\phi_2 exist, so does \int_a^b f~(d\phi_1+d\phi_2), and
\int_a^b f~(d\phi_1+d\phi_2)=\int_a^b f~d\phi_1+\int_a^b f~d\phi_2.
Suppose \Gamma=\{x_0,x_1,...,x_m\}is a partition of [a,b]and \{\xi_i\}_{i=1}^mare intermediate points satisfying x_{i-1} \leq \xi_i \leq x_i.
\begin{align}
R_\Gamma(\phi_1+\phi_2)&=\sum_{i=1}^m f(\xi_i)[\phi_1(x_i)+\phi_2(x_i)-\phi_1(x_{i-1})-\phi_2(x_{i-1})] \\
&=\sum_{i=1}^m f(\xi_i)[\phi_1(x_i)-\phi_1(x_{i-1})] + \sum_{i=1}^m f(\xi_i)[\phi_2(x_i)-\phi_2(x_{i-1})]\\
&=R_\Gamma(\phi_1)+R_\Gamma(\phi_2)
\end{align}
Then for any sequence of partitions \{\Gamma_k\}with \lim_{k\to\infty}gap(\Gamma_k)=0, we have
\begin{align}
\lim_{k\to\infty} R_{\Gamma_k}(\phi_1+\phi_2)&=\lim_{k\to\infty}R_{\Gamma_k}(\phi_1) + \lim_{k\to\infty}R_{\Gamma_k}(\phi_2)\\
&=\int_a^b f~d\phi_1+\int_a^b f~d\phi_2
\end{align}
This shows that \lim_{k\to\infty} R_{\Gamma_k}(\phi_1+\phi_2)exists and hence \int_a^b f~(d\phi_1+d\phi_2)exists. Also, we have
\int_a^b f~(d\phi_1+d\phi_2)=\int_a^b f~d\phi_1+\int_a^b f~d\phi_2.