Tagged Questions

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 ...
2
votes
1answer
34 views

Abstract integral - Borel measures - $L^p$ spaces

Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
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 ...
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 ...