philosophy of some proof(THE RIEMANN-STIETJES INTEGRAL)

6.6 Theorem Suppose $\alpha$ is a monotonically increasing function on $[a,b]$.  f $\in \mathscr{R}(\alpha)$ on $[a,b]$ if and only if for every $\epsilon>0$ there exists a partition P such that

$$ \begin{equation} U(P,f,\alpha)-L(P,f,\alpha)<\epsilon. \end{equation} $$

  This theorem provides a fundamental idea about solving questions like “how to check a function to be Riemann-integrable or not”. By definition, we have

$$ U(P,f,\alpha)-L(P,f,\alpha)=\sum\limits_{i=1}^{n}(M_i-m_i)\Delta\alpha_{i}, $$

where for a partition P,

$$ \begin{equation} \begin{aligned} M_i&=\sup f(x) \hspace{1cm}(x_{i-1}\leq x \leq x_{i}), \\ m_i&=\inf f(x) \hspace{1cm}(x_{i-1}\leq x \leq x_{i}), \\ \Delta\alpha_{i}&=\alpha(x_{i})-\alpha(x_{i-1}). \end{aligned} \end{equation} $$

  Given $\epsilon>0$. Naturally, we could imagine that if $f$ is Riemann-integrable, then $\sum\limits_{i=1}^{n}(M_i-m_i)\Delta\alpha_{i}$ should be small enough for some partition P. We hope to construct some P which meet the requirement.

  There is an idea on constructing a suitable P. For each term, we either make $(M_i-m_i)$ or $\Delta\alpha_{i}$ small. Then this P may suffice.

  The proof of the theorem below should be an example.

6.8 Theorem If f is continuous on $[a, b]$ then f $\in \mathscr{R}(\alpha)$ on $[a,b]$.

Proof Given $\epsilon>0$. Since $f$ is continuous, and $[a,b]$ is compact, we know $f$ is uniformly continuous, i.e., given $\epsilon>0$, there exist a $\delta>0$ such that $\forall x,y\in [a,b]$, if $d(x,y)<\delta$, then $d(f(x),f(y))<\epsilon$. Hence, if we choose a partition with $\Delta\alpha_{i}<\delta$, then $(M_i-m_i)<\epsilon$. We thus get

$$ \begin{equation} \begin{aligned} U(P,f,\alpha)-L(P,f,\alpha)&=\sum\limits_{i=1}^{n}(M_i-m_i)\Delta\alpha_{i} \\ &< \epsilon\sum\limits_{i=1}^{n}\Delta\alpha_{i} \\ &= \epsilon (\alpha(b)-\alpha(a)). \end{aligned} \end{equation} $$

Since $\epsilon$ is arbitrary, for each $\eta>0$, we pick $\epsilon$ which satisfy the inequality $\epsilon (\alpha(b)-\alpha(a))<\eta$ . We finally get

$$ \begin{equation} \begin{aligned} U(P,f,\alpha)-L(P,f,\alpha)<\eta, \end{aligned} \end{equation} $$

and therefore $f\in \mathscr{R(\alpha)}$.