1
vote
0answers
88 views

Whats the purpose : Hilbert's problems in measure space

This may sound a very newbie question, anyway I would like to ask here and to make it more clear for me. I've got an assignment to consider boundary problems in space of finite measures W, where the ...
2
votes
1answer
128 views

A sequence of functions $f_n \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$

Consider a sequence of functions $\{f_n \}\in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ , convergent to $f$ in $L^1(\mathbb{R})$ and to $g$ in $L^2(\mathbb{R})$. Prove that $f=g$ a.e. What I understood ...
2
votes
0answers
410 views

Is the measurability of the set E required for this problem to be right or have a solution? [closed]

This is one problem from my text book and since this book is new edition, I have been finding many typos or errors in this book. So I am not sure if this problem has an error that it should have ...
4
votes
1answer
183 views

Problem about absolute continuity of a function

$f:\mathbf{R} \to \mathbf{R}$ is an increasing function with $\lim_{x\to -\infty}f=0$ ,$\lim_{x\to \infty}f=1$, and $\int_{R}f'=1$. Prove that $f$ is absolutely continuous on every interval ...
1
vote
1answer
148 views

Uniformly continuous $f$ in $L^p([0,\infty))$

Assume that $1\leq p < \infty $, $f \in L^p([0,\infty))$, and $f$ is uniformly continuous. Prove that $\lim_{x \to \infty} f(x) = 0$ .