Ex2-11

Chapter 2 Question 11

Question: Show that $$\int_a^b fd\phi$$ exists if and only if given $$\varepsilon>0$$, there exists $$\delta>0$$ such that $$|R_\Gamma-R_{\Gamma'}|<\varepsilon$$ if $$|\Gamma|,|\Gamma'|<\delta$$.

Proof: First assume $$\int_a^b fd\phi$$ exists, then for partition $$\Gamma=\{x_0,x_1,\ldots,x_n\}$$ of $$[a,b]$$ and $$n$$points $$\xi_1,\ldots,\xi_n$$ satisfying $$x_{i-1}\leq \xi_i\leq x_i$$ for all $$i=1,\ldots,n$$, the equality below holds$$\lim_{|\Gamma|\to 0}R_\Gamma=\int_a^b fd\phi,$$

where $$R_\Gamma$$is the Riemann-Stieltjes sum for $$\Gamma$$. Given $$\varepsilon>0$$ we may choose $$\delta>0$$ so that if $$|\Gamma|<\delta$$then$$\left|\int_a^b fd\phi-R_\Gamma\right|<\dfrac\varepsilon2\quad$$

Therefore, by triangle inequality, for $$|\Gamma|, |\Gamma'|<\delta$$ we have

$$|R_\Gamma-R_{\Gamma'}|\leq \left|\int_a^b fd\phi-R_\Gamma\right|+\left|\int_a^b fd\phi-R_{\Gamma'}\right|<\varepsilon.$$

To prove the converse, assume that given $$\varepsilon>0$$, there exists $$\delta>0$$ such that $$|R_\Gamma-R_{\Gamma'}|<\varepsilon$$ if $$|\Gamma|,|\Gamma'|<\delta$$. We define

$$U=\limsup_{|\Gamma|\to 0}R_\Gamma\quad\text{ and }\quad L=\liminf_{|\Gamma|\to 0}R_\Gamma.$$

We claim both $$U,L$$ are finite. We will only prove that $$U$$is finite, since the proof of the finiteness of $$L$$ is similar. If $$U$$is not finite, then it is either $$+\infty$$ or $$-\infty$$. If $$U$$ is $$+\infty$$, we can construct a sequence of partitions$$\{\Gamma_n\}$$ (with some intermediate choices of points $$\xi_i$$) such that

$$|\Gamma_n|<\frac1n\quad\text{ and }\quad R_{\Gamma_n}>R_{\Gamma_{n-1}}+1\quad\text{ for }n\geq 2.$$

However, we know from assumption that there is $$\delta>0$$ such that $$|R_{\Gamma}-R_{\Gamma'}|<1$$ for $$|\Gamma|,|\Gamma'|<\delta$$, this is contradictory once we choose $$\Gamma=\Gamma_n, \Gamma'=\Gamma_{n+1}$$ with $$\frac1n<\delta$$. If on the other hand, $$U=-\infty$$, then given a real number $$A$$ there is $$\delta>0$$ such that $$|\Gamma|<\delta$$ implies $$R_\Gamma<A$$. Therefore, we can choose another sequence of partitions $$\{\Gamma_n'\}$$ such that $$|\Gamma_n'|<\frac1n \quad\text{ and }\quad R_{\Gamma_n'}0$$ such that

$$|R_\Gamma-R_{\Gamma'}|<\varepsilon\quad\forall |\Gamma|,|\Gamma'|<\delta.$$

By definitions of $$\limsup$$and $$\liminf$$, we can choose partitions with $$|\Gamma|,|\Gamma'|<\delta$$ such that $$R_\GammaU-\varepsilon$$. However, this lead us to

$$R_{\Gamma'}>U-\varepsilon=L+2\varepsilon>R_\Gamma+\varepsilon\implies |R_{\Gamma'}-R_{\Gamma}|>\varepsilon,$$a contradiction.