2
votes
2answers
68 views

Lebesgue measurable homework problem

Let $X \subseteq \mathbb{R}$. A subset $E \subseteq \mathbb{R}$ is called a hull of $X$ if $E$ is measurable $X \subseteq E$ If $F$ is any measurable set such that $X \subseteq F$, then $E$\ $F$ is ...
2
votes
2answers
78 views

Is the sequence $ \frac{1}{10^n} $ convergent?

I must prove that $ f: \Bbb{N} \to \Bbb{R}; n \to\frac{1}{10^n} $ is a convergent sequence. I thought: If $f$ is convergent then $\exists L \in \Bbb{R}(\forall \epsilon >0 (\exists m \in ...
5
votes
1answer
75 views

An integration question.

An help in the following problem: Let $f:[-1,1] \longrightarrow \mathbb{R}$ a $C^1$ function, i.e., continuously differentiable. Suppose that we have $$\int_{-1}^{1} f(x)\;dx = \pi ...
5
votes
2answers
158 views

An infinite $\sigma$-algebra contains an infinite sequence of nonempty, disjoint sets.

I am trying to solve Exercise 3 a) given here. The problem states: Let $\mathcal{M}$ be an infinite $\sigma$-algebra. Prove that $\mathcal{M}$ contains an infinite sequence of nonempty, disjoint ...
3
votes
1answer
187 views

Question from Folland real analysis 6.38

I have been staring at this for hours. I cannot figure out how to prove the following from Folland, problem 6.38. Show that: $$f \in L^p \iff \sum_{k=-\infty}^{+\infty}2^{kp}\lambda_f(2^k)<\infty$$ ...
3
votes
0answers
79 views

Lebesgue outer measure, real analysis

Let $E$ have finite outer measure. Show that $E$ is measurable if and only if for each open, bounded interval $(a,b)$, $(b-a)=m^*((a,b) \cap E)+ m^* ((a,b) \cap E^c)$. The forward direction ...
2
votes
5answers
1k views

Infinite Cartesian product of countable sets is uncountable

Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$. ...
2
votes
1answer
119 views

Prove $\inf\{x+y+z\mid x,y,z\in\mathbb{R}, 0<x<y<z\}=0$

Prove $\inf\{x+y+z\mid x,y,z\in\mathbb{R}, 0<x<y<z\}=0$ Let $A=\{x+y+z\mid x,y,z\in\mathbb{R}, 0<x<y<z\}$. Need to show 3 things: (1) $\inf A$ exists (2) 0 is a lower bound for ...
7
votes
2answers
2k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
1
vote
1answer
53 views

Integral of $\|x\|_\infty^{-n-1}$

Define $f:\mathbb{R}^n \to \mathbb{R}$ as $f(x) = \|x\|_\infty^{-n-1}$, where for $x = (x_1, x_2, \cdots, x_n)$, $\|x\|_\infty = \max\{|x_i|, i = 1,\cdots,n\}$ (the usual coordinate-wise ...
0
votes
1answer
41 views

A more rigorous proof that if $\forall n, \int_{[0,1]} f^n = C$, $f(x) = \chi_{\{f = 1\}}(x)$

Consider the following problem: Let $f \geq 0$ be measurable on $[0,1]$. If there is a constant $C$ such that for all $n \in \mathbb{N}$, $\int_{[0,1]} f^n = C$, then show $f(x) = \chi_{\{ f = 1 ...
4
votes
1answer
106 views

Convolution of an $L^1$ function and a function that tends to $0$ results in a function that tends to $0$

I'm trying to solve the following problem in review for a test, but have only partly succeeded: Let $K \in L^1(\mathbb{R})$ and $f$ be a bounded, measurable function on $\mathbb{R}$, with ...
2
votes
5answers
161 views

Continuity on $\Bbb R$

$f:\Bbb R\to\Bbb R$ is continuous on $\Bbb R$. $\lim\limits_{x\to\infty}f(x)=0$, and $\lim\limits_{x\to-\infty}f(x)=0$. Prove that $f$ is bounded on $\Bbb R$ and attains either an absolute maximum ...
3
votes
0answers
66 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
0
votes
2answers
122 views

continuous mapping is determined by its values on a dense subset of its domain

Question: If f and g are continuous mappings of a metric space X into a metric space Y, let E be a dense subset of X. if g(p) = f(p) for all p $\in$ E, prove that g(p)= f(p) for all p$\in$ X. Answer: ...
3
votes
1answer
107 views

Prove that $\displaystyle{\int{(F(x+c)-F(x))}\,\mathrm dx=c\mu (\mathbb{R})}$

Let $(X,\mathcal{F},\mu)$ be a finite measure space ($\mu$ is finite Borel measure on $\mathbb{R}$) If $F(x)=\mu ((-\infty,x])$ and $c>0$ prove that $$\displaystyle{\int{(F(x+c)-F(x))}\,\mathrm ...
2
votes
1answer
113 views

Extension of a Bounded Operator on $L^p$ to $L^r$

Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
7
votes
1answer
189 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
1
vote
1answer
63 views

Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$

If anyone has a solution to the following exercise, I would be grateful. Thanks. Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
1
vote
1answer
46 views

Finding pointwise limit

My question is where did "$1+r\cos$","$1+r\cot$" come from?
2
votes
2answers
123 views

Show $\int_X f d\nu = \int_X fgd\mu$ if $\nu(E)=\int_E g d\mu$ .

$f$ and $g$ are both non-negative functions where the integral of non-negative function is defined as the supremum over all simple functions dominated by the non-negative function. Would going ...
3
votes
1answer
121 views

Math Analysis - Problem dealing with bounded variation

Let $f\colon[0,1] \rightarrow \mathbb R$ (all real numbers) be defined by $\displaystyle f(x) = x \sin \left(\frac{\pi}{2x}\right)$ if $0 \lt x \le 1$ and $f(x) = 0$ if $x=0$. Determine ...
1
vote
2answers
423 views

Question 2.1 of Bartle's Elements of Integration

The problem 2.1 of Bartle's Elements of Integration says: Give an exemple of a function $f$ on $X$ to $\mathbb{R}$ which is not $\boldsymbol{X}$-mensurable, but is such that the function $|f|$ ...
4
votes
2answers
139 views

Question from Folland on modes of convergence

I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated. Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow ...
2
votes
1answer
135 views

Real Analysis Qual Problem 2

This shouldn't be a hard problem, but I am stuck on it. I just need to prove the statement or come up with a counterexample. Any help will be appreciated. Let $f: [0, 1] \rightarrow [0, \infty)$ be ...
3
votes
1answer
316 views

Real Analysis Qualifying Exam Problem

I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you. Suppose that $f_j$ is a ...
4
votes
0answers
121 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
5
votes
2answers
241 views

Checking of a solution to How to show that $\lim \sup a_nb_n=ab$

In course of solving the problem How to show that $\lim \sup a_nb_n=ab$ I feel that I've probably made some mistake in my solution for I didn't use the fact that $a_n>0$ $\forall$ $n\geq1.$ ...
3
votes
2answers
196 views

Question from Folland, criteron for a function to belong to $L^p$

This question is from Folland 6.38, Show that $f \in L^p $ iff $\sum_{k=-\infty}^ {\infty} 2^{pk} \mu \{{x: |f(x)|>2^{k}}\} \lt \infty$ If $f \in L^p $, I applied the Chebyshev's inequality But ...