Ex2-30

Chapter 2, Question 30

Let $$f$$and $$\phi$$ be real-valued functions on $$[a,b]$$.

(a) If $$\int_a^b fd\phi$$ exists and $$\phi$$ is not constant on any subinterval of $$[a,b]$$, show that $$f$$ is bounded on $$[a,b]$$.

Proof: We assume $$f$$is unbounded on $$[a,b]$$, $$\phi$$ is not constant on any subinterval of $$[a,b]$$ and $$\int_a^b fd\phi$$ exists, then get a contradiction. Without loss of generality, we assume $$f$$is unbounded above, then there is a sequence $$\{x_n\}$$in $$[a,b]$$such that $$\{f(x_n)\}$$is strictly increasing and unbounded. By closedness of $$[a,b]$$ and the fact that every sequence in $$\mathbb R$$ has a monotone subsequence [see for example, Theorem 2.32 in Advanced Calculus, Fitzpatrick], we find that $$\{x_n\}$$ has a monotone subsequence $$\{x_{n_k}\}$$that converges to a point $$x\in[a,b]$$. For the sake of simplicity, we rename the subsequence as $$\{y_k\}$$and without loss of generality, assume $$x>a$$ and $$\{y_k\}$$ is an increasing sequence. For the case where $$x=a$$ and $$\{y_k\}$$ decreasing, one can repeat the proof below by changing just a few words.

From assumptions, we know $$\lim_{|\Gamma|\to 0}R_\Gamma$$exists and is equal to $$\int_a^b fd\phi$$, so there is $$\delta>0$$ such that

$$\left|\int_a^b fd\phi-R_\Gamma\right|<1\quad\text{ whenever }\Gamma\text{ is a partition such that }|\Gamma|<\delta.$$

Since $$\phi$$ is not constant on any subinterval, we can choose $$x'<x, x-x'<\delta$$ such that $$\phi(x)\neq \phi(x')$$ (this can be done, or otherwise $$\phi$$ will be constant on the interval $$(x-\delta,x]$$). Let $$\Gamma$$ be a partition satisfying $$|\Gamma|<\delta$$ that contains the two adjacent partition points $$x',x$$. Then we fix $$\xi_1,\ldots,\xi_n$$to be the inputs of $$f$$ in $$R_\Gamma$$ except the one in $$[x',x]$$, say $$\xi=\xi_i$$, which can be changed. Now we treat $$R_\Gamma=R_\Gamma(\xi)$$ as a function of $$\xi\in[x',x]$$ and investigate its behavior. Using the triangle inequality we should have

$$|R_\Gamma(\xi)-R_\Gamma(\xi')|<2\quad\text{ for all }\xi,\xi'\in[x',x].$$

We choose $$K$$ a positive integer large enough such that $$y_k\in[x',x]$$ for $$k\geq K$$. Then for $$k\geq K$$ we have

$$2>|R_\Gamma(x)-R_\Gamma(y_k)|=|f(x)-f(y_k)||\phi(x)-\phi(x')|,$$

which is a contradiction since $$\{f(y_k)\}_{k=K}^\infty$$is unbounded above and $$|\phi(x)-\phi(x')|>0$$.