Suppose Alpha Increases on a B Alpha is Continuous at X and F X 0
Problem 1
Suppose $\alpha$ increases on $[a, b], a \leq x_{0} \leq b, \alpha$ is continuous at $x_{0}, f\left(x_{0}\right)=1$, and $f(x)=0$ if $x \neq x_{0} .$ Prove that $f \in \Omega(\alpha)$ and that $\int f d \alpha=0 .$
Problem 2
Suppose $f \geq 0, f$ is continuous on $[a, b]$, and $\int_{a} f(x) d x=0$. Prove that $f(x)=0$ for all $x \in[a, b]$. (Compare this with Exercise 1.)
Problem 3
Define three functions $\beta_{1}, \beta_{2}, \beta_{3}$ as follows: $\beta_{j}(x)=0$ if $x<0, \beta_{j}(x)=1$ if $x>0$ for $j=1,2,3 ;$ and $\beta_{1}(0)=0, \beta_{2}(0)=1, \beta_{3}(0)=t .$ Let $f$ be a bounded function on $[-1,1]$
(a) Prove that $f \in \mathscr{R}\left(\beta_{1}\right)$ if and only if $f(0+)=f(0)$ and that then
$$
\int f d \beta_{1}=f(0)
$$
(b) State and prove a similar result for $\beta_{2}$.
(c) Prove that $f \in \mathscr{R}\left(\beta_{3}\right)$ if and only if $f$ is continuous at 0 .
(d) If $f$ is continuous at 0 prove that
$$
\int f d \beta_{1}=\int f d \beta_{2}=\int f d \beta_{3}=f(0)
$$
Problem 4
If $f(x)=0$ for all irrational $x, f(x)=1$ for all rational $x$, prove that $f \notin \mathscr{R}$ on $[a, b]$ for any $a<b$.
Problem 5
Suppose $f$ is a bounded real function on $[a, b]$, and $f^{2} \in \mathscr{\text { on }}[a, b] .$ Does it follow that $f \in \mathscr{R}$ ? Does the answer change if we assume that $f^{3} \in \mathscr{R} ?$
Problem 6
Let $P$ be the Cantor set constructed in Sec. 2.44. Let $f$ be a bounded real function on $[0,1]$ which is continuous at every point outside $P$. Prove that $f \in \mathscr{R}$ on $[0,1]$. Hint: $P$ can be covered by finitely many segments whose total length can be made as small as desired. Proceed as in Theorem $6.10 .$
Problem 7
Suppose $f$ is a real function on $(0,1]$ and $f \in \mathscr{R}$ on $[c, 1]$ for every $c>0 .$ Define
$$
\int_{0}^{1} f(x) d x=\lim _{\epsilon \rightarrow 0} \int_{e}^{1} f(x) d x
$$
if this limit exists (and is finite).
(a) If $f \in \mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.
(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.
Problem 8
Suppose $f \in \mathscr{P}$ on $[a, b]$ for every $b>a$ where $a$ is fixed. Define
$$
\int_{a}^{\infty} f(x) d x=\lim _{b \rightarrow \infty} \int_{a}^{b} f(x) d x
$$
if this limit exists (and is finite). In that case, we say that the integral on the left converges. If it also converges after $f$ has been replaced by $|f|$, it is said to converge absolutely.
Problem 9
Show that integration by parts can sometimes be applied to the "improper" integrals defined in Exercises 7 and 8 . (State appropriate hypotheses, formulate a theorem, and prove it.) For instance show that
$$
\int_{0}^{\infty} \frac{\cos x}{1+x} d x=\int_{0}^{\infty} \frac{\sin x}{(1+x)^{2}} d x .
$$
Show that one of these integrals converges absolutely, but that the other does not.
Problem 10
Let $p$ and $q$ be positive real numbers such that
$$
\frac{1}{p}+\frac{1}{q}=1
$$
Prove the following statements.
(a) If $u \geq 0$ and $v \geq 0$, then
$$
u v \leq \frac{u^{p}}{p}+\frac{v^{q}}{q}
$$
Equality holds if and only if $u^{\prime}=v^{\theta}$.
(b) If $f \in \mathscr{R}(\alpha), g \in \mathscr{R}(\alpha), f \geq 0, g \geq 0$, and
$$
\int_{a}^{b} f^{\nu} d \alpha=1=\int_{a}^{b} g^{q} d \alpha
$$
then
$$
\int_{=}^{b} f g d \alpha \leq 1
$$
(c) If $f$ and $g$ are complex functions in $\mathscr{R}(\alpha)$, then
$$
\left|\int_{a}^{b} f g d \alpha\right| \leq\left\{\int_{a}^{b}|f|^{B} d \alpha\right\}^{1 / D}\left\{\int_{a}^{b}|g|^{q} d \alpha\right\}^{1 / \varepsilon}
$$
This is Hölder's inequality. When $p=q=2$ it is usually called the Schwarz inequality. (Note that Theorem $1.35$ is a very special case of this.)
(d) Show that Hölder's inequality is also true for the "improper" integrals described in Exercises 7 and $8 .$
Problem 11
Let $\alpha$ be a fixed increasing function on $[a, b]$. For $u \in \mathscr{R}(\alpha)$, define
$$
\|u\|_{2}=\left\{\int_{0}^{b}|u|^{2} d \alpha\right\}^{1 / 2} .
$$
Suppose $f, g, h \in \mathscr{R}(\alpha)$, and prove the triangle inequality
$$
\|\boldsymbol{f}-\boldsymbol{h}\|_{2} \leq\|\boldsymbol{f}-\boldsymbol{g}\|_{\mathbf{2}}+\|g-\boldsymbol{h}\|_{2}
$$
as a consequence of the Schwarz inequality, as in the proof of Theorem $1.37$.
Problem 12
With the notations of Exercise 11, suppose $f \in \mathscr{R}(\alpha)$ and $\varepsilon>0$. Prove that there exists a continuous function $g$ on $[a, b]$ such that $\|f-g\|_{2}<\varepsilon$. Hint: Let $P=\left\{x_{0}, \ldots, x_{n}\right\}$ be a suitable partition of $[a, b]$, define
$$
g(t)=\frac{x_{i}-t}{\Delta x_{1}} f\left(x_{i-1}\right)+\frac{t-x_{i-1}}{\Delta x_{t}} f\left(x_{t}\right)
$$
if $x_{i-1} \leq t \leq x_{i}$
Problem 13
Define
$$
f(x)=\int_{x}^{x+t} \sin \left(t^{2}\right) d t
$$
(a) Prove that $|f(x)|<1 / x$ if $x>0$. Hint: Put $t^{2}=u$ and integrate by parts, to show that $f(x)$ is equal to
$$
\frac{\cos \left(x^{2}\right)}{2 x}-\frac{\cos \left[(x+1)^{2}\right]}{2(x+1)}-\int_{x^{2}}^{(x+1)^{2}} \frac{\cos u}{4 u^{3 / 2}} d u
$$
Replace $\cos u$ by $-1$.
(b) Prove that
$$
2 x f(x)=\cos \left(x^{2}\right)-\cos \left[(x+1)^{2}\right]+r(x)
$$
where $|r(x)|<c / x$ and $c$ is a constant.
(c) Find the upper and lower limits of $x f(x)$, as $x \rightarrow \infty$.
(d) Does $\int_{0}^{\infty} \sin \left(t^{2}\right) d t$ converge?
Problem 14
Deal similarly with
$$
f(x)=\int_{x}^{x+1} \sin \left(e^{t}\right) d t
$$
Show that
$$
e^{x}|f(x)|<2
$$
and that
$$
e^{x} f(x)=\cos \left(e^{x}\right)-e^{-1} \cos \left(e^{x+1}\right)+r(x)
$$
where $|r(x)|<C e^{-x}$, for some constant $C$.
Problem 15
Suppose $f$ is a real, continuously differentiable function on $[a, b], f(a)=f(b)=0$, and
$$
\int_{0}^{b} f^{2}(x) d x=1
$$
Prove that
$$
\int_{a}^{0} x f(x) f^{\prime}(x) d x=-\frac{1}{2}
$$
and that
$$
\int_{0}^{b}\left[f^{\prime}(x)\right]^{2} d x \cdot \int_{0}^{b} x^{2} f^{2}(x) d x>1
$$
Problem 16
For $1<s<\infty$, define
$$
\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{\prime}}
$$
(This is Riemann's zeta function, of great importance in the study of the distribution of prime numbers.) Prove that
(a) $\zeta(s)=s \int_{1}^{\infty} \frac{[x]}{x^{3+1}} d x$
and that
(b) $\zeta(s)=\frac{s}{s-1}-s \int_{1}^{\infty} \frac{x-[x]}{x^{s+1}} d x$
where $[x]$ denotes the greatest integer $\leq x$. Prove that the integral in (b) converges for all $s>0$. Hint: To prove (a), compute the difference between the integral over $[1, N]$ and the $N$ th partial sum of the series that defines $\zeta(s)$.
Problem 17
Suppose $\alpha$ increases monotonically on $[a, b], g$ is continuous, and $g(x)=G^{\prime}(x)$ for $a \leq x \leq b$. Prove that
$$
\int_{a}^{b} \alpha(x) g(x) d x=G(b) \alpha(b)-G(a) \alpha(a)-\int_{a}^{b} G d \alpha
$$
Hint: Take $g$ real, without loss of generality. Given $P=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$, choose $t_{1} \in\left(x_{i-1}, x_{i}\right)$ so that $g\left(t_{i}\right) \Delta x_{t}=G\left(x_{i}\right)-G\left(x_{i-1}\right) .$ Show that
$$
\sum_{i=1}^{n} \alpha\left(x_{i}\right) g\left(t_{i}\right) \Delta x_{i}=G(b) \alpha(b)-G(a) \alpha(a)-\sum_{i=1}^{n} G\left(x_{i-1}\right) \Delta \alpha_{i}
$$
Problem 18
Let $\gamma_{1}, \gamma_{2}, \gamma_{3}$ be curves in the complex plane, defined on $[0,2 \pi]$ by
$$
\gamma_{1}(t)=e^{t t}, \quad \gamma_{2}(t)=e^{201}, \quad \gamma_{3}(t)=e^{2 \pi i t \sin (1 / t)}
$$
Show that these three curves have the same range, that $\gamma_{1}$ and $\gamma_{2}$ are rectifiable, that the length of $\gamma_{1}$ is $2 \pi$, that the length of $\gamma_{2}$ is $4 \pi$, and that $\gamma_{3}$ is not rectifiable.
Problem 19
Let $\gamma_{1}$ be a curve in $R^{*}$, defined on $[a, b] ;$ let $\phi$ be a continuous $1-1$ mapping of $[c, d]$ onto $[a, b]$, such that $\phi(c)=a$; and define $\gamma_{2}(s)=\gamma_{1}(\phi(s))$. Prove that $\gamma_{2}$ is an arc, a closed curve, or a rectifiable curve if and only if the same is true of $\gamma_{1}$. Prove that $\gamma_{2}$ and $\gamma_{1}$ have the same length.
Source: https://www.numerade.com/books/chapter/the-riemann-stieltjes-integral/
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