Tagged Questions

2
votes
1answer
48 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 ...
4
votes
1answer
36 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
52 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
32 views

Finding pointwise limit

My question is where did "$1+r\cos$","$1+r\cot$" come from?
2
votes
2answers
69 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 ...
2
votes
1answer
75 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
154 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
60 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
85 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 ...
2
votes
1answer
163 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 ...
5
votes
2answers
129 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
139 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 ...