0
votes
0answers
12 views

Extension of Premeasures

Here, a premeasure is a countably additive set function whereas a measure is one acting on a sigma-algebra. Not every positive premeasure admits an extension to a positive measure as the following ...
0
votes
1answer
33 views

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring sets?

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring on sets? By a closed-open interval , I mean an interval of the form $[x,y)$ A ring of sets is a non-empty ...
1
vote
2answers
16 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...
0
votes
1answer
20 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
1
vote
3answers
39 views

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
1
vote
1answer
22 views

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions.

Let $X$ be a non-empty set, $A \subseteq X$. Decide the set $\mathcal M(\mathcal{E})$ of $\mathcal{E}$-$\mathcal B(\mathbb R)$-measureable functions $f: X \rightarrow \mathbb R$ in each of the ...
1
vote
1answer
22 views

Let $(X,\epsilon)$ be a measurable space. Show $\mu: \epsilon \rightarrow [0, \infty]$ is a measure given three conditions.

Let $(X,\epsilon)$ be a measure space and $\mu: \epsilon \rightarrow [0, \infty]$. Suppose $\mu$ satisfies $\mu(\emptyset)=0$, $\mu(A \cup B) = \mu(A) + \mu(B)$ for disjoint $A,B$ and $\mu(\cap_{n ...
-2
votes
1answer
41 views

Show that $\mathcal{A}$ is not a field

Suppose that $\mathcal{A}$ is a a class of subset of $\Omega$ s.t. $\Omega \in \mathcal{A}$ $A\in \mathcal{A}$ implies $A^c \in \mathcal{A}$ $\mathcal{A}$ is closed under finite disjoint union. ...
1
vote
2answers
23 views

Show that $\liminf_{n \to \infty}A_n=\{\omega: \lim_{n \to \infty} 1_{A_n}(\omega)=1 \}$

Show that $\liminf_{n \to \infty}A_n=\{\omega: \lim_{n \to \infty} 1_{A_n}(\omega)=1 \}$. Where $\liminf_{n \to \infty}A_n=\bigcup_{n=1}^\infty\bigcap_{k=n}A_n$ and indicator function is defined as ...
2
votes
3answers
25 views

Show limits of function $f(x) := \lambda(B \cap (-x,x])$ and there exist a Borel-set $A$ such that $A \subseteq B$ and $\lambda(A)=a$.

Consider the measure space $(\mathbb R, \mathcal B(\mathbb R), \lambda)$, where $\lambda$ denote the Lebesgue-measure on $\mathbb R$. Let $B \in \mathcal B(\mathbb R)$ and $f:(0, \infty) ...
0
votes
1answer
31 views

How do I see $\mu\left(\bigcap_{n \in \mathbb N} B_n\right) = \lim_{n\rightarrow \infty} \left(\mu \bigcap_{j=1}^n B_j\right)$.

Suppose $\mu : \epsilon \rightarrow [0,\infty]$ is a measure from the $\sigma$-algebra $\epsilon$ into extended $\mathbb R$. How do I see $$\mu\left(\bigcap_{n \in \mathbb N} B_n\right) = ...
0
votes
1answer
27 views

Let $\epsilon = \{A \subseteq \mathbb R : A \ \text {or} A^C \ \text {is countably}\}$. I want to show $\epsilon \subseteq \mathcal B(\mathbb R).$

Let $\epsilon = \{A \subseteq \mathbb R : A \ \text {or} A^C \ \text {is countably}\}$. I want to show $\epsilon \subseteq \mathcal B(\mathbb R).$ As you probably already know $\epsilon$ is a ...
0
votes
1answer
70 views

Is the union of an infinite sequence of finite sets an infinite set?

Given the following question: "Let $X$ be an infinite set and define $\mu^*$ on $\mathcal P \left({X}\right)$ by $\mu^*(A) = 0$ if A is finite and $\mu^*(A)=1$ if A is infinite. Is $\mu^*$ an outer ...
2
votes
1answer
11 views

Let $X \neq \emptyset$ and $B \subseteq X$. Show $\epsilon_B := \{A \subseteq X : B \subseteq A \lor B \subseteq A^C \}$ is a $\sigma$-algebra.

Let $X \neq \emptyset$ and $B \subseteq X$. Show the system $\epsilon_B := \{A \subseteq X : B \subseteq A \lor B \subseteq A^C \}$ is a $\sigma$-algebra in $X$. I've already proved the two ...
1
vote
0answers
19 views

showing that the sets (Banach-Tarski-ish) which comprise $S^1$ are disjoint

Let $S^1$ be the unit circle and consider $S^1 = \cup_{q \in \mathbb{Q}} A_q$ where the sets $A_q$ are constructed as follows: Define the equivalence relation $z \sim w$ if for $z = e^{i\alpha}, w = ...
1
vote
1answer
39 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
1
vote
2answers
31 views

Measures: Sigma-Additivity vs. Continuity

Let $R$ be a ring of sets that contains the empty set and $\mu$ be a positive and finite set function on $R$. If $\mu$ is countable additive, then it is continuous from below and above: $$A_n\uparrow ...
1
vote
2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
4
votes
1answer
43 views

How to prove the following defined collection is a sigma algebra?

Let $\mu$ and $\lambda$ be two measures on a $\sigma$-algbra $\mathfrak{F}$ on $\Omega$, such that $\mu (A)=\lambda(A)$ for any $A\in \mathfrak C$, where $\mathfrak C\subset\mathfrak{F}$ is a ...
0
votes
3answers
35 views

How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
1
vote
0answers
20 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
3
votes
2answers
67 views

finite additivity&countable additivity

Let $\tau$ be a semialgebra of subsets of $\Omega$ and let P: $\tau\rightarrow [0,1]$, with $P(\Omega)=1$, and it satisfies finite additivity: $P\big(\bigcup_{i=1}^{n}D_i\big)=\sum_{i=1}^{n}P(D_i)$ ...
2
votes
2answers
39 views

The set of $x$ where a sequence convergences in terms of set operations

I'm befuddled by this. Suppose $f:\mathbb{R}\to\mathbb{R}$, $f_n:\mathbb{R}\to\mathbb{R}$, $n=1,2,\dots$, and consider the set $$\bigcap_{k\geq 1}\bigcup_{p\geq 1}\bigcap_{m\geq p}\{x\in\mathbb{R} \ ...
1
vote
0answers
40 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...
3
votes
1answer
64 views

Trying to prove lim inf $(A_n) \subseteq$ lim sup $(A_n)$

I am trying to show that for a sequence of sets $(A_n)$ $$ lim \ inf \ (A_n) \subseteq lim \ sup \ (A_n)$$ I can see why this is true intuitively as follows - $x \in lim \ inf \ (A_n)$ $\implies ...
0
votes
0answers
78 views

Proving a result on measures

I have been told in university that: The three following conditions are equivalent for an additive function defined over a $\sigma$-algebra (in fact, over a ring, but let's make life easier ...
0
votes
1answer
21 views

Notation for a collection of sets under a certain condition

I am looking for the notation to describe "A collection of sets that are the union of a finite number of intervals". Is this correct - $A = \{A_i\}_{i \in I}$ where each $A_i = \bigcup_{n \in N} ...
2
votes
1answer
19 views

Measurable set indicator functions - need clarification on a book's statement

A book I'm reading says the following about indicator functions $\chi_A$ : But unless I'm missing something, how can that that be? If $B$ is for example the set $(-2,2)$, $1 \in B$, but since $B$ ...
1
vote
1answer
29 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
1
vote
2answers
87 views

What does $\vee$ mean in set theory?

The following proof is from Probability by Davar Khoshnevisan. There is a symbol $\vee$ in the third sentence of the proof. What does this symbol mean, please? There seems no definition about it in ...
0
votes
0answers
32 views

Equivalence relation in measure theory

Every set of positive measure has non measurable subsets and, moreover: $\mathcal{P}(A) \subseteq \mathcal{L} \iff \lambda(A)=0$ How would you go about proving this? Cant get my head around it ...
0
votes
0answers
15 views

Show that the completion of a ring $\xi$ is a $\sigma$ - algebra

$\mathcal{N}$ be the class of $\mu$ - null sets. Let $\mathcal{S} \bigtriangleup \mathcal{N}:= \{ E \bigtriangleup N : E \in \mathcal{S}, N \in \mathcal{N} \}$. Similiarly $\mathcal{S} \cup ...
0
votes
0answers
34 views

Lebesgue measure and symmetric difference inequality

Suppose the conclusion holds. Therefore $\begin{align} \lambda(A \bigtriangleup B) < \epsilon &= \lambda(A \B \cup B\A) < \epsilon \\ &= \lambda(A \cup B \ - A \cap B) < \epsilon ...
1
vote
2answers
65 views

Why is this not a $\sigma$ - algebra

Why is $\displaystyle \{\cup_{i=1}^{\infty} (a_i, b_i] : a_i < b_i, i=1,...,n, n \in \mathbb{N_0} \}$ not a $\sigma$-algebra? It looks ok to me, the only way I can see that this would fail is ...
2
votes
1answer
32 views

Semialgebra logical error

I was following a proof on Rosenthal's, A first look at rigorous probability theory book, but I believe that one step flawed. Let me set up: $\mathcal{J}$ is a semialgebra of subsets of $\Omega$. ...
1
vote
1answer
29 views

if $\mathcal{A}$ is a $\sigma$-algebra, for $y \in \mathcal{A}$ show that $\mathcal{A}_y=\{y \cap A|A\in \mathcal{A}\}$ is also a $\sigma$-algebra.

If $\mathcal{A}$ is a $\sigma$-algebra, for $y \in \mathcal{A}$ show that $$\mathcal{A}_y=\{y \cap A|A\in \mathcal{A}\}$$ is also a $\sigma$-algebra. So far I have shown that $\emptyset \in ...
-1
votes
1answer
63 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
2
votes
1answer
47 views

Describe measurable functions

I have the following exersice: Describe the Borel-measurable functions $f:X\to\mathbb{R}$, where $X$ has the $\sigma$-algebra of subsets $A\subseteq X$ such that $A$ is countable or $X\setminus A$ is ...
2
votes
2answers
69 views

Correspondence as a graph of a multifunction

Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do ...
1
vote
0answers
32 views

How to show $\mathcal{B}(\mathbb{R}^2)=\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})$?

I would like to show that $$ \mathcal{B}(\mathbb{R}^2) = \mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}). $$ Can you say me what is a good way to show that? My idea is to show it via generators ...
2
votes
3answers
103 views

Is the cardinality of the rationals the same on the unit interval?

Let $\mathscr{G}$ be a compact subset of $\mathbb{R}$. Denote by $\mathbb{Q} ( \mathscr{G})$ the set of all rational numbers in $\mathscr{G}$. Does $| \mathbb{Q} ( \mathscr{G}_1)| = |\mathbb{Q} ...
0
votes
1answer
37 views

Is there any good text introducing a part of Borel-hierarchy which is in need in measure theory

Is there any good text introducing a subpart of Borel-hierarchy which is in need in measure theory, which can be done in short time? Say, 1~3 days if possible. (Assuming i'm studying about 14hours a ...
1
vote
0answers
45 views

Pi-system generating a tail sigma-field

I have the following problem. Let $(X_n)_{n\in Z_+}$ be a sequence of random variables with values in $\{-1,1\}$. Let $\mathcal{F}:=\sigma(X_0,X_1,...)$ be the product sigma-field on $\{-1,1\}^{Z_+}$ ...
0
votes
2answers
43 views

Wikipedia Definition: $\sigma$-algebra

Wikipedia defines (see wiki): If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra... Is this really equivalent to the usual ...
1
vote
2answers
55 views

Show that $\left\{A\subset\Omega: A~\text{ is countable or }A^C\text{ is countable}\right\}$ is a $\sigma$-algebra

Let $\Omega$ a set and $$ C=\left\{A\subset\Omega: A~\text{ is countable or }A^C\text{ is countable}\right\}. $$ Show that $C$ is a $\sigma$-algebra and that $$ ...
1
vote
1answer
127 views

A basic property of the Lebesgue outer measure

If $G$ is a measurable set and satisfies $m^*(G)<\infty$, then for all $\varepsilon>0$ there exists a closed set $F\subset G$ such that $m^*(F)>m^*(G)-\varepsilon$ Edit: I know that: ...
1
vote
2answers
424 views

Example of $\sigma$-algebra

I understood the definition of a $\sigma$-algebra, that its elements are closed under complementation and countable union, but as I am not very good at maths, I could not visualize or understand the ...
0
votes
1answer
56 views

Why isn't $ P( \cup_n A_n) = \text{lim}_n P(A_n)$ obvious?

Let ( $ \Omega, \cal A$, P ) be a probability space. Let $(A_n)_n $ be an increasing sequence of events. I am reading a proof using $\sigma$-additivity to prove that $$ P( \cup_n A_n) = \text{lim}_n ...
4
votes
1answer
84 views

Show identity between product-$\sigma$-algebra and a set

Let $T$ be any index set and $(\Omega_i,\mathcal{A}_i)_{i\in T}$ a family of measurable spaces and $\mathcal{A}:=\bigotimes_{i\in T}\mathcal{A}_i$. Show that $$ \mathcal{A}=\left\{A\subset ...
5
votes
1answer
49 views

Show that $\mathfrak{Z}$ is a semi ring

Consider measurable spaces $(\Omega_t,\mathcal{A}_t), t\in T$ ($T$ is any index set). With $\mathcal{E}(T)$ we the set of all finite, not-empty subsets of $T$. Show that $$ ...