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Integral and Measure: From Rather Simple to Rather Complex

Vigirdas Mackevičius

Chapter 5

Other Definitions Riemann and Stieltjes Integrals - all with Video Answers

Educators

Chapter Questions

03:05

Problem 1

Let

$$
f(x)=\left\{\begin{array}{l}
1 \text { if } x \text { is rational, } \\
0 \text { otherwise }
\end{array}\right.
$$

Prove that the function $f$ is not Riemann-integrable in the interval [0,1].

Linda Hand
Linda Hand
Numerade Educator
01:08

Problem 2

Let

$$
f(x):= \begin{cases}\sin \frac{1}{x}, & x \in(0,1] \\ 0, & x=0\end{cases}
$$

Show that $f \in R[a, b]$ but $f \notin D[a, b]$.

Aman Gupta
Aman Gupta
Numerade Educator

Problem 3

Let

$$
f(x):= \begin{cases}x^\alpha \sin \frac{1}{x^\beta}, & x \in(0,1] \\ 0, & x=0\end{cases}
$$

Prove that $f$ is a finite-variation function in the interval $[0,1]$ if and only if $\alpha>\beta$.

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02:17

Problem 4

Find the limit

$$
\lim _{|P| \rightarrow 0} \sum_i f\left(\xi_i\right) \int_{x_i}^{x_{i-1}} g(x) \mathrm{d} x
$$

for $f, g \in C[a, b]$, where $\xi$ is a tag set of a partition $P$ of the interval $[a, b]$ (see definition 5.1).

Willis James
Willis James
Numerade Educator

Problem 5

Let $f \in C[a, b]$ and $P=\left\{a=x_0<x_1<\cdots<x_k=b\right\}$. Prove that there exists a tag set $\xi$ of $P$ such that

$$
\int_a^b f(x) \mathrm{d} x=S(f ; P, \xi)
$$

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Problem 6

Give an example of a bounded convergent sequence $\left\{f_n\right\} \subset R[a, b]$ with limit $f=\lim _{n \rightarrow \infty} f_n$ that is not Riemann integrable.

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00:43

Problem 7

Find the total variations $V(f ;-1,1)$ and $V(g ;-1,1)$ of the functions $f(x):=|x|+|x+1|, x \in[-1,1]$, and $g(x)=\operatorname{sgn} x, x \in[-1,1]$. Find the integrals

$$
\int_{-1}^1\left(1+x^2\right) \mathrm{d} f(x) \quad \text { and } \quad \int_{-1}^1\left(1+x^2\right) \mathrm{d} g(x)
$$

Erika Bustos
Erika Bustos
Numerade Educator
02:31

Problem 8

Show that $f \in V[a, b]$ is an increasing function if and only if

$$
V(f ; a, b)=f(b)-f(a)
$$

Dushyant Barot
Dushyant Barot
Numerade Educator

Problem 9

Show the following properties of the variation of a function:

$$
\begin{aligned}
V(f \pm g ; a, b) & \leqslant V(f ; a, b)+V(g ; a, b) \\
V(\alpha f ; a, b) & =|\alpha| V(f ; a, b) \\
V(|f| ; a, b) & \leqslant V(f ; a, b)
\end{aligned}
$$

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Problem 10

Prove that if $f \in C[a, b]$, then $V(|f| ; a, b)=V(f ; a, b)$.

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Problem 11

Let $f, g \in V[a, b]$. Show that $f g \in V[a, b]$ and $\max \{f, g\} \in V[a, b]$ and that if, moreover, $g(x) \geqslant C>0, x \in[a, b]$, then $\frac{f}{g} \in V[a, b]$. Give an example which shows that from the condition $g(x) \neq 0, x \in[a, b]$, it does not follow that $\frac{f}{g} \in V[a, b]$.

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Problem 12

Let $f \in V[a, b]$. Denote $V(x):=V(f ; a, x)$ for $x \in[a, b]$. Prove that $V$ and $V-f$ are increasing (not strictly) functions. Therefore, every function $f \in V[a, b]$ can be written as the sum of two increasing functions.

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01:20

Problem 13

Express the function $f(x):=\sin x, x \in[0,2 \pi]$, as the sum of two increasing functions.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator

Problem 14

Show that $V[a, b] \subset D[a, b]$.

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Problem 15

Check the linearity of the Stieltjes integral:

$$
\int_a^b\left(\alpha_1 f_1(x)+\alpha_2 f_2(x)\right) \mathrm{d} g(x)=\alpha_1 \int_a^b f_1(x) \mathrm{d} g(x)+\alpha_2 \int_a^b f_2(x) \mathrm{d} g(x)
$$

provided that the integrals on the right-hand side of the equality exist.

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01:46

Problem 16

Give an example where

$$
\int_a^b f(x) \mathrm{d} g(x) \neq \int_a^c f(x) \mathrm{d} g(x)+\int_c^b f(x) \mathrm{d} g(x)
$$

(i.e. the Stieltjes integral not always possesses the additivity property). Show that in such a case, both functions $f$ and $g$ have a discontinuity at the point $c$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:59

Problem 17

Find $\int_0^1 x \mathrm{~d}([3 x]-x)$, where $[x]$ denotes the integer part of a number $x$.

Gregory Higby
Gregory Higby
Numerade Educator
01:13

Problem 18

Let $\left\{g_n\right\}$ be an increasing sequence of increasing functions converging pointwise to a function $g \in C[a, b]$. Show that
a) $g_n \rightrightarrows g$, but not necessarily $V\left(g_n-g ; a, b\right) \rightarrow 0$;
b) $\int_a^b f(x) \mathrm{d} g_n(x) \rightarrow \int_a^b f(x) \mathrm{d} g(x)$ for every $f \in C[a, b]$.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
07:51

Problem 19

An increasing function $g$ is such that

$$
\int_0^\pi \sin x \mathrm{~d} g(x)=g(\pi)-g(0)
$$

Prove that $g(x)=g(0)$ for $x \in[0, \pi / 2)$ and $g(x)=g(\pi)$ for $x \in(\pi / 2, \pi]$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 20

An increasing function $g$ is such that

$$
\int_0^2 f(x) \mathrm{d} g(x)=f(1)
$$

for every $f \in C[0,2]$. Prove that $g(x)=g(0)$ for $x \in[0,1)$ and $g(x)=g(2)$ for $x \in(1,2]$.

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Problem 21

An increasing function $g:[0,1] \rightarrow \mathbb{R}$ is such that

$$
\int_0^1 f(x) \mathrm{d} g(x)=\frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right)
$$

for every $f \in C[0,1]$. Find all such functions $g$.

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Problem 22

Find

$$
\lim _n \int_0^1 x^n \mathrm{~d} g(x) \text { and } \lim _n \int_0^1\left(x^n+e^{-n x}\right) \mathrm{d} g(x),
$$

provided that $g$ is an increasing function on $[0,1]$.

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