All Questions
Tagged with borel-sets real-analysis
9 questions
0
votes
1
answer
361
views
$f_\epsilon=\inf\{f(y):|y-x|<\epsilon\}$ is measurable Borel [closed]
I found this problem I have tried but it has been a bit complicated for me,
Let $f:\mathbb{R}\to\mathbb{R}$ a bounded function. For each $\epsilon>0$, let $f_\epsilon (x)=\inf\{f(y):|y-x|<\...
- 143
15
votes
2
answers
530
views
Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets
Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
- 807
0
votes
1
answer
136
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A link between continuity and 0-borelian? [closed]
Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ ...
- 4,074
3
votes
1
answer
568
views
Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?
Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$.
Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows:
$$\mathcal{...
- 1,997
1
vote
1
answer
869
views
Borel $\sigma$-algebra on the space of Hölder continuous functions
Let
$(M,d)$ be a separable metric space
$E$ be a $\mathbb R$-Banach space
$\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|_{C^{0+\alpha}(K,\:E)}:=\sup_{x\in K}\left\|f(x)\right\|_E+\sup_{\substack{...
- 167
1
vote
1
answer
245
views
Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Let $X$ be a metric space.
In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
- 623
2
votes
2
answers
233
views
Is the domain of symmetric derivative borel set?
Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
- 337
1
vote
1
answer
321
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A question about Borel sets on the unit interval
It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
- 5,902
4
votes
2
answers
767
views
Possible subsets of reals that equal the set of continuity of a function
This should be an easy question, but I don't quite know how to approach it. It may be somewhat related to the concepts mentioned in the context of this past question, though it was motivated mainly by ...
- 7,320