# Query about a proof of Riemann integrability of step/simple functions

Let $$f:[a,b]\to\Bbb{R}$$ be a step function is of the form $$f=a_11_{[a,t_1]}+\sum\limits_{i=2}^n a_{i} 1_{(t_{i-1},t_{i}]}$$ i.e. in simple words $$f(x)=a_1$$ for all $$x\in [a,t_1]$$ and $$f(x)=a_{i}$$ for all $$x\in(t_{i-1},t_{i}]\ \forall i\ge 2$$.

I write $$E_1=[a,t_1],E_{i}=(t_{i-1},t_i]\ \forall i\ge 2$$, then $$f=\sum\limits_{i=1}^n a_{i} 1_{E_{i}}$$ Define $$\mathcal{CI}(f)=\sum\limits_{i=1}^n a_{i}(t_i-t_{i-1})$$ We have to prove that for all $$\epsilon >0$$, there exists $$\delta>0$$ such that $$\left|\mathcal{CI}(f)-\sum\limits_{j=1}^k f(\xi_j)(s_j-s_{j-1})\right|<\epsilon$$ for any portion $$\mathcal{P}=\{a=s_0<\cdots with $$\lVert \mathcal{P}\rVert(:=\text{max}\{s_j-s_{j-1}|\ j=1,\ldots,n\})<\delta$$ and for any choice of $$\xi_j\in(s_j,s_{j-1})$$

I have tried to prove it in the following manner-

Let $$\mathcal{P}=\{a=s_0<\cdots be a partition of $$[a,b]$$ and choose any $$\xi_j\in (s_{j-1},s_j)$$. Define a step function $$g:[a,b]\to\mathbb{R}$$ as follows- $$g=\sum\limits_{j=1}^k f(\xi_j)1_{F_j}\text{ where } F_1=[a,s_1],F_j=(s_{j-1},s_j]\ \forall j\ge 2$$

Then as per our notation $$\mathcal{CI}(g)=\sum\limits_{j=1}^k f(\xi_j)(s_j-s_{j-1})$$

We have to approximate $$|\mathcal{CI}(f)-\mathcal{CI}(g)|$$. I have tried in the following manner-

$$\mathcal{CI}(f)=\sum\limits_{i=1}^n a_i\lambda(E_i)$$

$$=\sum\limits_{i=1}^n a_i\lambda(E_i\cap [a,b])$$

$$=\sum\limits_{i=1}^n a_i\lambda(E_i\cap\bigsqcup\limits_{j=1}^k F_j)$$

$$=\sum\limits_{i=1}^n\sum\limits_{j=1}^k a_i\lambda(E_i\cap F_j)$$

Similarly, $$\mathcal{CI}(g)=\sum\limits_{i=1}^n\sum\limits_{j=1}^k f(\xi_j)\lambda(E_i\cap F_j)$$

Then $$|\mathcal{CI}(f)-\mathcal{CI}(g)|=|\sum\limits_{i,j} x_i-f(\xi_j)\lambda(E_i\cap F_j)|\le \sum\limits_{i,j}(|x_i|+|f(\xi_j)|)|\lVert \mathcal{P}\rVert\le 2nk \lVert f\rVert_{\infty} \lVert P\rVert$$

Where $$\lVert f\rVert_\infty=\text{sup}|f|$$.

Here $$n,\lVert f\rVert_\infty$$ is fixed, but $$k$$ could vary. So I am unable to give approximation $$|\mathcal{CI}(f)-\mathcal{CI}(g)|<\epsilon$$.

There is a hint provided in the book that is-

Show that, $$\left|\mathcal{CI}(f)-\sum\limits_{j=1}^k f(\xi_j)(s_j-s_{j-1})\right|<(n-1)\lVert P\rVert \underset{i}{\text{max}}{|a_i|}$$

But I'm unable to prove that. Can anyone help me in this regard? Thanks for help in advance.

Form the partition $$Q$$ with points $$a = x_0 < x_1 < \ldots < x_m = b$$ where

$$\{x_0,x_1, \ldots x_m\} = \{t_0,t_1, \ldots, t_n\}\cup\{s_0,s_1, \ldots, s_k\}$$

Note that $$Q$$ has at most $$n-1$$ more points and subintervals than the partition $$P = (s_0,s_1,\ldots, s_k)$$. The maximum difference in the number of points occurs when $$\{t_1, \ldots, t_{n-1}\}\cap\{s_1, \ldots, s_{k-1}\}= \emptyset.$$

We can write $$\mathcal{CI}(f)$$ and $$\mathcal{CI}(g)$$ as sums of the form

$$\mathcal{CI}(f) = \sum_{p=1}^m\alpha_p(x_p-x_{p-1}),\quad\mathcal{CI}(g) = \sum_{p=1}^m\beta_p(x_p-x_{p-1}),$$

where $$\alpha_p, \beta_p \in \{a_0,a_1,\ldots, a_n\}$$ and $$\alpha_p$$ and $$\beta_p$$ are different on at most $$n-1$$ new subintervals introduced by forming partition $$Q$$. We must have $$|\alpha_p - \beta_p| \leqslant 2M$$ where $$M = \max(|a_0|,|a_1|,\ldots,|a_n|)$$, and, thus

$$\left|\mathcal{CI}(f)- \mathcal{CI}(g) \right|< (n-1)\cdot \|Q\|\cdot 2M < (n-1) \cdot \|P\|\cdot 2M$$

Choose a partition $$P$$ such that $$\delta = \|P\| < \frac{\epsilon}{2(n-1)M}$$ to finish.