# Tagged Questions

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 ...
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 ...
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?
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? $(\Omega^T,\mathcal{A}^T)$
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 ...
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 ...
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 ...
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 ...
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
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 ...
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 ...
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 ...
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\{ ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\sup_n \{ A_n \}$ ...
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 ...
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 ...
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 ...
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].
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 ...
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} ...
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 ...
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: $a_n>0$, $b_n >1$ ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
2answers
748 views

### Finite additivity follows from countable additivity!

How to prove that finite additivity follows from countable additivity!!?
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?
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 $Y$. ...
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 ...
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 ...
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