Tagged Questions
0
votes
1answer
39 views
Showing that a union of the subsets of two $\sigma$-algebras is a $\sigma$-algebra.
I got back an assignment for a first course in analysis and I have made a very basic error, and I'm having a lot of trouble pinpointing exactly what piece of information I'm missing.
You have two ...
1
vote
1answer
18 views
why S' is closed with respect to the formation of finite unions?
$Proposition$ : Let S be a semiring of subsets of a set X. Define S' to be the collection of
unions of finite disjoint collections of sets in S. Then S' is closed with respect to the formation
of ...
5
votes
2answers
57 views
The cardinality of Lebesgue sets
Suppose $A=\{S\;|\;S \subset \mathbb R^n, S\text{ is Lebesgue measurable}\}$. What is the cardinality of $A$? Is it the same as the cardinality of all of the real numbers?
1
vote
1answer
47 views
what does $(\Omega^T,\mathcal{A}^T)$ mean?
Let $(\Omega_t,\mathcal{A}_t), t\in T$ be a collection of measurable spaces. What does the notation mean? $(\Omega^T,\mathcal{A}^T)$
1
vote
1answer
52 views
Additive set function properties
I am reading an introduction to measure theory, which starts by defining $\sigma$-rings then additive set functions and their properties which are given without proof. I was able to prove two of them ...
2
votes
0answers
28 views
What does the notation mean? $A_t\times \boldsymbol\times_{s\neq t}\Omega_s $
Let $(\Omega_t,\mathcal{A}_t), \;t\in T$ be a collection of measurable spaces. What does the following notation mean? $$A_t\times \boldsymbol\times_{s\neq t}\Omega_s $$ where $t\in T$ and $A_t\in ...
2
votes
1answer
37 views
Why is $\{x:f(x)\ge a\}=\bigcap_{n=1}^\infty\{x: f(x)\gt a-\frac{1}{n}\}$ and $\{x: f(x)\gt a\}=\bigcup_{n=1}^\infty\{x: f(x)\ge a+\frac{1}{n}\}$?
It is known from measure theory that for all $a \in \mathbb{R}$, the following are equivalent:
$f$ is measurable.
The set $\{x : f(x) \ge a\}$ is measurable.
The set $\{x : f(x) \gt a\}$ is ...
0
votes
0answers
38 views
Question on integrable set
What is an integrable set?
"If $T$ is a Hausdorff topological space and $\mu$ is a positive
Radon measure on $T$, and if $F$ is a measurable multifunction, then
for each integrable set ...
5
votes
3answers
71 views
How to derive a union of sets as a disjoint union?
$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$
The results is obvious enough, but how to prove this
5
votes
1answer
47 views
simple question from set theory/measure theory
This is a simple question.
On pages 5-6 of Measure Theory,Vol 1, Vladimir Bogachev he writes that:
for $E=(A\cap S)\cup (B\cap (X-S))$
Now, he writes that:
$X-E = ((X-A)\cap S) \cup ((X-B)\cap ...
1
vote
1answer
44 views
Halmos Measure Theory section 5
Th5.B E is a class of sets. $R_o (E)$ is the ring generated by E. Any set in $R_o (E)$ can be covered by a finite union of sets in E.
The proof: The class of all sets that can be covered by a finite ...
1
vote
2answers
73 views
Proof that the pre-image is a sigma algebra
We shall prove that for sets X,Y and a map f: X $\rightarrow$ Y, if B is a $\sigma$-algebra on Y, then
$\{f^{-1}(E) ; E \in B\}$ is a $\sigma$-algebra on X.
I have shown all the properties except ...
3
votes
0answers
79 views
$\left(0,1\right]\neq\biguplus_{k=1}^{n}\bigcap_{j=1}^{k}G_{j},$ Proving elegantly
How can I show (without making many distinctions by cases) that the equality
$$
\left(0,1\right]=\biguplus_{k=1}^{n}\bigcap_{j=1}^{k}G_{j},
$$
can't hold, if $G_{j}\in\mathcal{A}\cup\left\{ ...
4
votes
2answers
92 views
What does the completed graph of a function mean
zab said:
the Levy metric between two distribution functions $F$ and $G$ is simply the Hausdorff distance $d_C$ between the closures of the completed graphs of $F$ and $G$.
I have difficulty in ...
3
votes
1answer
43 views
Finite Partitions of the Unit Interval
Does the unit interval have a finite partition $P$ such that no element of $P$ contains an open interval? I would think that the answer is no, because each element of $P$ would have Lebesgue measure ...
2
votes
0answers
63 views
How are algebras and rings of subsets generated in this paragraph?
From ncatlab
What is missing is a simple description of the σ-algebra generated by ℬ.
For a mere algebra, this is easy; any ℬ can be taken as a subbase of an algebra, the symmetric unions ...
6
votes
1answer
146 views
Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
There are some set systems with algebraic titles, such as "field",
"algebra", "ring" and "semi-ring" (and possibly other titles), in
their names. Examples are
a sigma field (aka sigma algebra, ...
6
votes
1answer
221 views
Bolzano-Weierstrass and measures
Let $\{\mathcal{A}_n\}$ be an infinite sequence of sets with $\mathcal{A}_n \subset \mathcal{M}$, where $\mathcal{M}$ is a bounded subset of $\mathbb{R}$ (for simplicity). Is there a "nice" limit ...
4
votes
1answer
159 views
When does it make sense to define a base of a set system?
In a topology, a base is defined to be a class of subsets such that
every open set is the union of some members of it.
In a convexity
structure, a base is defined to be a class of subsets ...
2
votes
2answers
85 views
When does it make sense to define a generator of a set system?
In a set system, such as a topology, sigma algebra or
convexity structure, a generator is defined to be a class of
subsets such that the given set system is the coarsest such set
system ...
11
votes
2answers
266 views
Hyperreal measure?
If AC be accepted, then there exists a Lebesgue unmeasurable set called Vitali Set. However, I'm curious about measure valued in hyperreal numbers. Argument in disproof of unmeasurability of Vitali ...
3
votes
1answer
65 views
Difference between “measure on” and “measure over”
I want to make sure I understand the difference between the terms "measure on" and "measure over," assuming there is one. Is a measure on the set $X$ the same as a measure over its power set ...
0
votes
1answer
60 views
Generating a Sigma Algebra from an Algebra
Let $\mathscr{A}$ be an algebra of sets over $X$. Let $\mathscr{A}_\sigma$ contain $\mathscr{A}$ as well as any countable union of any sequence of members of $\mathscr{A}$. I'm trying to figure out ...
0
votes
1answer
57 views
Algebra generated by countable family of sets is countable?
If I have a countable family of sets $\mathcal{A}=\{A_1,A_2,...\}$ and construct the Algebra generated by $\mathcal{A}$. Will it also be countable?
My intuition screams YES, but I cannot seem to ...
4
votes
4answers
221 views
Cardinality of Vitali sets: countably or uncountably infinite?
I am a bit confused about the cardinality of the Vitali sets.
Just a quick background on what I gather about their construction so far:
We divide the real interval $[0,1]$ into an uncountable number ...
1
vote
2answers
192 views
Example of strictly subadditive lebesgue outer measure
One of the properties of the Lebesgue outer measure is that it is subadditive and not countably additive. In fact, I have read that even when the sets A_i are disjoint, there is still generally ...
1
vote
2answers
102 views
What does the supremum of a sequence of sets represent?
I'm trying to understand more about the limits of sequences of sets in Measure Theory.
Given a sequence of sets $\{A_n\}_{n\in \mathbb{N}} = \{ A_1,A_2, \ldots \}$, what does $\sup_n \{ A_n \}$ ...
2
votes
2answers
93 views
Is there an efficient Hausdorff Distance algorithm?
Two sided Hausdorff distance is calculated as
$$H(r_1,r_2)=\max\{h(r_1,r_2),h(r_2,r_1)\}$$
where
$$h(r_1,r_2)=\max_{a \in r_1}\min_{b\in r_2}\|r_1-r_2\|$$ and vice-verse
$r_1$ and $r_2$ are two ...
0
votes
2answers
260 views
Closed under countable union
I am reading a tutorial on measure theory and it states: "Given an interval $E = [a, b]$ and a set $S$ of subsets of $E$ which is closed under countable unions, we define the following..."
I was ...
2
votes
1answer
201 views
Limit Inf/Sup of Sequence of Set Example
In "A Probability Path", they have an example that states that the lim inf and lim sup of [0,n/(n+1)) is equal to [0,1). I guess I don't see how [0,1) is in all the sets except a finite number of ties ...
5
votes
1answer
226 views
Why is the Vitali set not necessarily equal to the interval e.g. [0,1]?
I don't seem to get the special properties of Vitali sets which makes them different from the intervals, e.g. [0,1].
4
votes
4answers
582 views
Interpretation of limsup-liminf of sets
What is an intuitive interpretation of the 'events'
$$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$
and
$$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$
when $A_n$ are ...
2
votes
1answer
123 views
Finding an irrational not covered in standard proof that $\mu(\mathbb{Q} \cap [0,1]) = 0$ [duplicate]
Possible Duplicate:
How would one go about proving that the rationals are not the countable intersection of open sets?
Constructing a number not in $\bigcup\limits_{k=1}^{\infty} ...
2
votes
2answers
86 views
Generated $\sigma(X)$ where $\Omega \neq \mathbb{R}$
Simple question here. I am trying to enumerate the sigma field generated by the random variable: $$X(\omega)=2+1_{\left\{a,b\right\}}(\omega)$$ where $\Omega=\left\{a,b,c,d\right\}$.
I think what is ...
0
votes
1answer
617 views
limsup liminf of sequence of sets
Following up from the discussion here:
Liminf and Limsup of a sequence of sets
I wanted to confirm my understanding of these concepts with another example. Suppose we have: $a_n>0$, $b_n >1$ ...
5
votes
1answer
346 views
Number of $\sigma$ -Algebra on the finite set
Let $X$ is a nonempty set with $m$ members . How many $\sigma$ -algebra can we make on this set?
1
vote
3answers
188 views
A problem in Sigma algebra.
How do I conceptualise this expression :
Let {$A_n$}$^{n=\infty}_{n=1}$ belong to sigma algebra $A$. Define, $\limsup\{A_{n}\}=\bigcap_{n=1}^{\infty}\{\bigcup_{m=n}^{\infty}A_{n}\}
$ and similarly ...
5
votes
1answer
248 views
“Converse” to composition of measurable functions is measurable
Here is a restatement of a problem in a textbook I encountered. I'm
well beyond the age of doing homework and this is purely for
self-study.
Exercise: Let $f : (X,\Sigma_1) \to (Y, \Sigma_2)$ and
...
3
votes
1answer
224 views
$A$ uncountable thus $\mu(A)>0$
I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure - but I am ...
0
votes
1answer
310 views
Sigma algebra example
I am having a bit of trouble understanding this definition. "A collection $\Sigma$ of subsets of S is called a $\sigma$-algebra on S if $\Sigma$ is an algebra on S such that whenever $F_n \in\Sigma (n ...
3
votes
1answer
271 views
Outer measure defined on sets of $\mathbb R$ - help with showing if outer measure or not
I have a long list of definitions of outer measures that I am trying to (a) show IF they are outer measures, and (b) if it is, determine its outer measurable subset of $\mathbb{R}$.
My books ...
3
votes
0answers
92 views
Semi-partition or pre-partition
For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$.
Does anybody met the name for a ...
0
votes
0answers
103 views
algebra of sets [closed]
1)prove that an algebra of sets is closed under finite union and intersections.
2)prove that for a semiring S,countable unions and finite intersections of sigma sets are sigma sets.
8
votes
3answers
214 views
Intersection of two 'huge' sets in the plane
Consider two sets on the plane $A=\mathbb{Q}\times \mathbb{R}$ and $B=\mathbb{R}\times \mathbb{Q}$.
We know that $A\cap B=\mathbb{Q}\times \mathbb{Q}\neq\emptyset$. What about the general cases?
That ...
15
votes
2answers
771 views
Infinite product of measurable spaces
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product ...
1
vote
2answers
748 views
Finite additivity follows from countable additivity!
How to prove that finite additivity follows from countable additivity!!?
1
vote
1answer
122 views
What's a closed countably infinite intersection?
When operating with sigma algebras, what does it mean when we talk about a countably infinite set? And, what are then closed countably infinite interesections?
5
votes
3answers
867 views
Preimage of generated $\sigma$-algebra
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
2
votes
1answer
527 views
Field of sets and Sigma algebra of sets
(1). According to Wikipedia, a field of subsets of X is defined to be a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of ...
0
votes
1answer
821 views
limit superior and limit inferior of the given sequence of sets
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
