2
votes
2answers
59 views

What does this set look like? [on hold]

Consider the set of points $A \subset \mathbb{Q}^2$ in the unit square with both coordinates rational: $$A=\{(x,y) \in \mathbb{Q^2} \mid \, 0 \leq x \leq 1,\,\, 0 \leq y \leq 1 \} $$ If we color the ...
-1
votes
1answer
53 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
40 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 ...
0
votes
0answers
5 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
31 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
87 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
31 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
37 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
40 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
42 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 $$ ...
0
votes
1answer
96 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
69 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
53 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
80 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
43 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 $$ ...
1
vote
1answer
62 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
1
vote
1answer
56 views

Oxtoby, measure and category, Vitali sets

In page 22 of that book (see picture below), Vitali set is described. And, the author remarks at the end of the page that Given a Vitali set $V$, then, $V=A \cup B$, where $B$ lacks the property of ...
0
votes
1answer
36 views

What is the difference between property of Baire and Second Category in $\mathbb{R}$

I am reading Oxtoby's book Measure and category. I have a question regarding the assumptions of theoren 4.8 in there: Here are the definitions: Linear set: Any subset of $\mathbb{R}$ Second ...
1
vote
1answer
31 views

Show that $\mathbb P$ is finitly additive and that $\mathcal A$ is an algebra

Given $\mathcal F=\{A \subset\Omega \mid A \mbox{ or } A^c \mbox{ is finite}\}$, show that $\mathcal F$ is an algebra. Second, set: $\mathbb P(A) = \begin{cases} 0, & \mbox{if } A \mbox{ is ...
0
votes
1answer
21 views

Measure and empty set second version

If A is a nonempty open set, the measure of A is not 0 ...first open set is uncountable, I pick up one interval irrational intersect [ 0,1 ] with measure 1 ...however this is not open...I am wondering ...
0
votes
0answers
16 views

Elementary measure theory exercise about $\liminf_{n \to \infty} \mu (E_n)$ [duplicate]

I'm going through the Fremlin's book (vol 1) on my own, and frankly, I don't have any good idea where to begin with the following exercise: Assuming that ($X, \Sigma, \mu$) is an measure space and ...
1
vote
1answer
210 views

Sigma field generated by Borel sets is the same as sigma field generated by intervals

Let $\mathcal{R} = \{ B_1 \times B_2 : B_1,B_2 \in \mathcal{B} \} $ where $\mathcal{B}$ is the sigma field of Borel sets. Let $\mathcal{I} = \{ I_1 \times I_2 : I_1,I_2 \; \; \text{are intervals} \} ...
1
vote
1answer
40 views

What is this set?

I have the following set, ${F}' = \{ A \subset \Omega :^\exists B,N \in {F}$ s.t. $\mu(N) = 0$ and $A \Delta B \subset N\}$ Here, $F$ is a collection of the subsets of $\Omega$, precisely ...
1
vote
1answer
26 views

Question about some basic concepts

A $\pi-$system a closed under the formation of finite intersections, $$A,B\in\mathcal{P}\Rightarrow A\cap B\in\mathcal{P}$$ so is this implies that any field is a $\pi-$system, therefore a ...
3
votes
2answers
107 views

$\sigma$- ideal

Let $(\Omega,\mathcal{A})$ be a measurable space. $\mathcal{N}\subset\mathcal{P}(\Omega)$ is called a $\sigma$ ideal, if $$ (1)~\emptyset\in\mathcal{N},~~~~~(2) N\in\mathcal{N}, M\subset ...
1
vote
1answer
101 views

Number of sigma algebras for set with 4 elements

I am supposed to watch out for sigma algebras that belong to the set $X=\{1,2,3,4\}$. I found 15(now with the new set even more) of them. I was wondering whether there is some nice proof how to see ...
1
vote
2answers
78 views

cantor middle thirds set

Let $f: [0,1] \to \mathbb{R}$ be defined by letting $f = 0 $ on $\mathcal{C}$, the Cantor set and $f(x) = k $ for every $x$ in each interval of lenght $\frac{1}{3^k}$ which has been removed from ...
1
vote
2answers
64 views

Borel sets defined for closed sets

This is adapted from 1.7.7 in Friedman's "Foundations of Modern Analysis": Let $\mathscr{B}$ be the $\sigma$-ring generated by the class of open subsets of $X$ [a fixed set], and $\mathscr{D}$ the ...
0
votes
1answer
53 views

Does $\bigcup _{j\in\mathbb{N}}A_j=A$

I'm reading R. Schilling's Measure, Integrals and Martingales and in a proof he makes the following statement. "Since $A=\bigcup _{j\in\mathbb{N}}A_j$...", is this allways true? The context: The ...
0
votes
2answers
46 views

$A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$.

Construct a sequence of measurable sets $A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$. My thoughts: Since $\lim\sup ...
0
votes
2answers
94 views

sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$.

Give an example in $\mathbb{R}$ of a sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$. My thoughts: By definition, ...
0
votes
1answer
41 views

Proving $m\lambda(E_m)\le\sum_{k=1}^{\infty}\lambda(A_k)$

Assume $A_1,A_2,...$ are measurable sets. Let $m\in\mathbb{N}$, and let $E_m$ be the set defined as follows: $x\in E_m\iff$ $x$ is a member of at least $m$ of the sets $A_k$. Prove that $E_m$ is ...
0
votes
2answers
34 views

$\cup A_n$ is infinite and has infinite complement

This is exercise 1.1.9 in Friedman's "Foundations of Modern Analysis": Let $\mathscr{D}$ consist of those sets [in $2^X$?] which are either finite or have a finite complement... If $X$ is not ...
1
vote
1answer
47 views

Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then

Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then $h(x) = F( f(x), g(x) ) $ is also measurable. Proof: For all $a$, $ \{ x : ...
0
votes
2answers
109 views

$m ( \{ x : f(x) > 0 \} ) = 0 \implies f = 0 $ almost everywhere

Suppose $f$ is non-negative measurable function. Put $E = \{ x : f(x) > 0 \} $. Say $m(E) = 0$. In other words, $E$ is null set. Then does it follow that $f = 0 $ almost everywhere ?
0
votes
2answers
207 views

Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions

Consider $\mathbb R^{[0,1]}$ the space of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. The question is: how to prove that $C[0,1]\notin \mathcal ...
0
votes
0answers
82 views

For $A\subset X$, prove that the characteristic function $\chi_A$ (which maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$.

For $A\subset X$, prove that the characteristic function $\chi_A$ (which maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$. I got the forward direction (i.e, proving $A\subset M$). For the ...
1
vote
1answer
175 views

About a proof that functions differing from a measurable function on a null set are measurable

Consider the following problem: Say $f: M \to \mathbb{R} $ is measurable function, $M$ a measurable set and $g : M \to \mathbb{R}$ such that $ Y = \{ x : f(x) \neq g(x) \} $ is a null set. We ...
0
votes
1answer
34 views

Proving set equality

Let $\{ f_n \}$ be a sequence of functions, then I am having hard time trying to see why $$ \{ x : ( \max_{n \leq k} f_n) (x) > a \} = \bigcup_{n=1}^{k} \{ x : f_n(x) > a \} $$ these two sets ...
3
votes
1answer
86 views

Simple proof about induced $\sigma$-algebra

Let $\cal F$ be a $\sigma$-algebra of subsets of $\Omega$ and $A$ a subset of $\Omega$. Show that $\mathcal C=\{A\cap B:B\in\mathcal F\}$ is a $\sigma$-algebra of subsets of $A$. Show that it is not a ...
1
vote
1answer
66 views

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras:

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras: (a) $M=\{\emptyset,X\}$ (b) $M=2^{X}$ (c) For certain disjoint sets $E_1,...,E_N$, ...
0
votes
0answers
60 views

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$.

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without ...
2
votes
1answer
132 views

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$.

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$. In fact, prove that $N'$ may be chosen to be a $G_{\delta}$, a countable ...
2
votes
2answers
121 views

Lim sup of sequence of sets and theirs unions [closed]

I have to prove the following equality: Can somebody help me to prove this?
1
vote
1answer
63 views

Let $M$ be the $\sigma$ algebra generated by $N$ and let $M'$ be the algebra generated by $N$.

Let $N\subset 2^X$. Let $M$ be the $\sigma$ algebra generated by $N$ and let $M'$ be the algebra generated by $N$. What relation must hold between $M$ and $M'$? My answer: $M$ is the intersection of ...
3
votes
2answers
607 views

Give an example of two $\sigma$ algebras whose union is not an algebra

Give an example of two $\sigma$ algebras in a set $X$ whose union is not an algebra. I've considered the sets $\{A|\text{A is countable or $A^c$ is countable}\}\subset2^\mathbb{R}$, which is a ...
0
votes
1answer
68 views

how to prove: if $\mathscr{F}_i, i\in I$ are $\sigma$-algebras, then $\bigcap\limits_{i\in I}\mathscr{F}_i$ is

I ran into this conclusion on $\sigma$-algebra: If $\mathscr{F}_i, i\in I$ are $\sigma$-algebras, then $\bigcap\limits_{i\in I}\mathscr{F}_i$ is also a $\sigma$-algebra. Note here $I$ is an ...
0
votes
1answer
137 views

Understanding the supremum limit of a set

Given a sequence ${A_n}$, we define the set lim sup $A_n = \{x : x$ belongs to infinitely many $A_n$'s$\}$ That is - lim sup $A_n = \bigcap_{m=1}^\infty (\bigcup_{n=m}^\infty A_n)$ I can't see how ...
0
votes
0answers
38 views

What does an algebra induced by $\mathcal{A}$ on a subset $E$ mean?

I am trying to understand what an algebra induced by $\mathcal{A}$ on a subset $E$ means. Let $X = \{1, 2, 3, 4, 5, 6, 7, 8\}$ Let $\mathcal{A}$ be an algebra of subsets of $X$ Let $E \subset X$ ...
0
votes
1answer
24 views

What is $E \cap A$ where A is an algebra of subsets of $X$

I am having some confusion around the intersection of sets when the elements of one of the sets are sets themselves. Let $X = \{1, 2, 3, 4, 5, 6\}$ Let $\mathcal{A} = \{\varnothing, \{1, 2, 3\}, ...