2
votes
2answers
75 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
1answer
62 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
0
votes
0answers
48 views

elegant proof of Radon–Nikodym theorem

Do you know about an elegant proof of Radon–Nikodym theorem, which is not as cumbersome as the usual ones?
3
votes
1answer
66 views

A neat proof that the Lebesgue measure is rigid motion invariant.

I'm busy doing a small undergraduate maths project on the Banach Tarski paradox and I was hoping I could prove that a lebesgue measure is rigid motion invariant but I can't find an eloquent proof ...
0
votes
1answer
643 views

Converges in measure then a subsequence converges almost everywhere

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $\{f_n\}_{n\in\mathbb{N}}$ a sequence of measurable functions which converge in measure to a function $f$. Prove that exists a subsequence ...
2
votes
1answer
133 views

Is my proof showing $Q$ is non-measurable complete?

Is my proof valid or complete? if not, what is missing?? Define $R$ on $[0, 1)$. When $x,y\in [0, 1)$, $xRy \iff y-x$ is a rational number. Then $R$ is an equivalence relation. Let $Q$ be the set ...
3
votes
2answers
791 views

$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable

Problem. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a monotone increasing function. Show that $f$ is measurable. Solution. We know that the set of discontinuites of any monotone increasing ...
4
votes
1answer
350 views

Extension of Fatou's lemma

let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$. I want to show if (1) holds, then (2) holds too. $$\lim_{n \rightarrow ...
3
votes
2answers
847 views

Question about $L^1$-$L^2$ integrable functions

Can somebody tell me what's wrong with the following argument? If $f$ is $L^1$ Lebesgue-integrable, say $f$ positive, then it is bounded almost everywhere by some bound $M$. Then $f^2 < M\cdot f$ ...
2
votes
2answers
208 views

Prove that this function is measurable

I cannot write a neat proof of this result, so I would like to see how to be precise in these kinds of arguments.. Here is the problem Let $I=[0,1]$ and let $f\colon I\times\mathbb R\to \mathbb R$ ...
1
vote
3answers
106 views

lebesgue measure of $\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, 0 \leq y \leq 10, z \in \mathbb Z \} $

Find the lebesgue measure of the set: $$ \Bigl\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, \quad 0 \leq y \leq 10, \quad z \in \mathbb Z \Bigr\} $$ I think is a null set but for some reason I ...
2
votes
1answer
121 views

Help with proof writing.

How do I use this the following result if $f$ is a non-negative measurable function on $X$, then $\int_X f~d\mu =0$ if and only if $f=0$ a.e. on $X.$ to prove that if $f$ be an ...
9
votes
0answers
577 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...