Ex2-13

Solution to Chapter 2, Exercise 13

Proof of Theorem 2.16 (iii)

We want to prove that if $$\int_a^b f~d\phi_1$$and $$\int_a^b f~d\phi_2$$ 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 $$\{\xi_i\}_{i=1}^m$$are 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

$$ \int_a^b f~(d\phi_1+d\phi_2)=\int_a^b f~d\phi_1+\int_a^b f~d\phi_2. $$