All Questions
Tagged with lebesgue-measure gn.general-topology
8 questions
7
votes
2
answers
448
views
Uncountable collections of distinct subsets of an interval (existence)
Throughout, $\mu$ is just the Lebesgue measure.
Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
- 101
1
vote
1
answer
183
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
- 307
4
votes
1
answer
203
views
Generalized limits in Boolean algebras
Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
- 1,151
5
votes
0
answers
364
views
Computing the infinite dimensional Lebesgue measure of "cubes"
There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
- 9,330
0
votes
0
answers
66
views
Coincidence Topologies for $L^p$ spaces
If $X$ and $Y$ are compact metric spaces then it is well-known that the compact-open topology on $C(X,Y)$ coincides with the topology of uniform convergence on compacts. Therefore, the latter is ...
- 5,405
3
votes
2
answers
236
views
Can a continuous surjection from a Hilbert cube to a segment behave bad wrt Lebesgue measures?
Suppose $\hat{I}=[0,1]^\mathbb{N}$ is a Hilbert cube and $I=[0,1]$.
Consider Lebesgue measures $m_1$ and $m_2$ on $\hat{I}$ and $I$ correspondingly. By Lebesgue measure on the Hilbert cube I mean the ...
- 173
1
vote
0
answers
331
views
Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203
4
votes
1
answer
2k
views
Lebesgue measure of boundary of Caccioppoli set
Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
- 320