3
votes
3answers
74 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
2
votes
0answers
17 views

pseudo inverse of a finite-to-one continuous map and measurability

Given that $\pi: X \to Y$ is a continuous onto map between compact metric spaces such that the fiber $\pi^{-1}(y)$ is a finite subset of $X$ for all $y$, is the map $y \mapsto \pi^{-1}(y)$ guaranteed ...
3
votes
1answer
38 views

Is the set of continuous function with Lebesgue zero set a Borel set in continuous space?

Let $D$ be a domain in $\mathbb{R^d}$ and denote the continuous function space on $D$ as $X := C(\overline{D})$ where we can define the $\sigma$-algebra $\mathscr{B}(X)$ of $X$, that is sets in $X$ ...
3
votes
1answer
74 views

Conditional probability independent of one variable

Let $X,Y$ and $Z$ be Borel spaces (that is, Borel subsets of Polish spaces) and let $\mathcal P(X)$ denote the Borel space of all Borel probability measures on $X$. For a product measure $P\in ...
4
votes
2answers
99 views

Cardinality of an algebra

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets. I am asking (1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak ...
2
votes
1answer
33 views

Measurability of level sets of measures

Let $X$ be a standard Borel space, and $\mathcal P(X)$ be the set of Borel probability measures on $X$ with a topology of weak convergence. It is known that $\{p:p(B) = 1\}$ is a Borel subset of ...
3
votes
0answers
69 views

Measurability of one set of measures

Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ ...
1
vote
1answer
55 views

A Lebesgue measurable universal Borel function

In 1918 Sierpiński constructed a Lebesgue measurable real-valued function on $[0,1]$ which isn't bounded above by any Borel function (I couldn't find the original reference, but here is a pdf of a ...
0
votes
1answer
110 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
1
vote
1answer
92 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
2
votes
1answer
63 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
0
votes
1answer
74 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
3
votes
1answer
37 views

Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
0
votes
1answer
54 views

Oxtoby - Thm 6.2 - There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category

I am reading the book Measure and Category of Oxtoby. In Theorem 6.2 it is stated that There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category As in the picture below. My ...
6
votes
2answers
127 views

Who are the measurable sets in $\mathbb{R}$

Is there any Characterization for all measurable sets in $\mathbb{R}$? Can I say that a set is measurable if an only if it has the property of Baire? (differs from an open set by a first category ...
2
votes
2answers
112 views

Measurability of a total variation metric

Let $X$ be a standard Borel space and let us denote by $\mathcal P(X)$ the space of Borel probability measures on $X$ endowed with the topology of weak convergence. Define $d:\mathcal P(X)\times ...
1
vote
0answers
35 views

Relationship between conjugacy class and centralizer for measure preserving transformations

Let $(X, \mathcal{B}, \mu)$ be a Lebesgue probability space. Let $\Phi$ be the space of all invertible measure preserving transformations on $(X,\mathcal{B}, \mu )$, endowed with the weak topology. ...
5
votes
1answer
61 views

Composition of Borel relations

Let $X,Y,Z$ be Polish spaces, or standard Borel spaces, and let us consider two relations $A \subseteq X \times Y$ and $B \subseteq Y \times Z$ that are Borel sets. Define their composition as $$ C ...
1
vote
0answers
42 views

Well - order on reals and not Lebesgue measurable sets

Let $(x_t )_{t<\omega_1} $ be a well ordering of the set of real numbers of the type $\omega_1 $ ( the first uncountable ordinal, we assume the continuum hypothesis ) and let $\leq $ be a natural ...
6
votes
1answer
79 views

Measurability of the pushforward operator on measures

Let $X$, $Y$ and $Y'$ be (standard) Borel spaces. We let $\mathcal B(X)$ be the Borel $\sigma$-algebra of $X$ and $\mathcal P(X)$ to be the space of all Borel probability distributions on $X$ endowed ...
2
votes
0answers
45 views

Example of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$ but $|X|,|Y| \leq |\mathbb{R}|$

I am interested in knowing examples of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$. By allowing $|X|,|Y|$ to be large we can provide a trivial counterexample, as in the one ...
10
votes
3answers
344 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
1
vote
1answer
71 views

A particular measure in the Cantor space $2^\infty$ / How to prove it also defines a $\sigma$-algebra?

Consider the following measure $\mu$ for the Cantor set (seen as the space of infinite sequences of 0's and 1's): $$ \mu\left(E\right) = \lambda \left(g\left(E\right) \right) \tag{1}$$ where ...
2
votes
1answer
43 views

Baire functions are constant?

Is it true that every Baire function $f:\mathbb{R}\to \mathbb{N}$ must be constant? $f:X \to Y$ is a Baire function for $X,Y$ metrizable spaces if $f$ is a member of $F(X,Y)$, where $F$ is the ...
7
votes
3answers
716 views

Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
1
vote
1answer
86 views

Why universally and not just Borel policies

In a famous book Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve they use universally measurable policies that come up with some handy features: e.g. they show that every ...
1
vote
0answers
154 views

Is Ulam's theorem equivalent to Sierpiński's theorem?

Ulam's theorem is stated on page 25, Measure and Category(2ed), John Oxtoby, as: A finite measure $\mu$ defined for all subsets of $X$ of power less than the least weakly inaccessible cardinal ...
6
votes
1answer
427 views

Uncountable union of multiples of measurable sets.

For any $n\in\mathbb{N}$, consider the measurable space $(\mathbb{R}^n,\mathscr{B}^n)$, where $\mathscr{B}^n$ is the Borel $\sigma$-algebra generated by the Euclidean topology on $\mathbb{R}^n$. ...
3
votes
1answer
90 views

Measures on all subsets of $\aleph_0$

A theorem of Ulam says: A finite measure $\mu$ defined on all subsets of a set of cardinality $\aleph_1$ must be $0$ for all subsets if it sends every $1$-element subset to $0$. Will this ...
9
votes
1answer
262 views

From universal measurability to measurability

Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact metrizable space endowed with its Borel $\sigma$-algebra $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally ...
2
votes
2answers
136 views

Any example for a function having domain and range as subset of real line that is NOT Borel function?

Suppose there is a function $f:A\to B$ where $A,\,B\subseteq\mathbb{R}$, is there any example for this function being NOT Borel function? Well the question came up to be when I was reading the ...
5
votes
3answers
406 views

What measure does Lebesgue measure induce on the fat Cantor set?

I know that the fat Cantor set under the subspace topology is homeomorphic to Cantor space $\{0,1\}^{\mathbb N}$ under the product topology induced by the discrete topology on $\{0,1\}$. Call the ...
9
votes
0answers
160 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
1
vote
1answer
100 views

non-Lebesgue-measurable subsets of Smith-Volterra-Cantor sets

For the Smith-Volterra-Cantor set (or simply SVC) we define an equivalence relation R by making each connected component in SVC an equivalence class. It is easy to see that the collection of all ...
2
votes
2answers
213 views

How to show the diagonal of product of Hausdorff spaces is not in the product of its Borel-$\sigma$ algebras?

Let $X$ be a Hausdorff space, with $|X| > \mathfrak{c}$. $\mathcal{B}(X)$, $\mathcal{B}(X \times X)$ are Borel-$\sigma$ Algebras on $X$ and $X\times X$ respectively. $\mathcal{B}(X)⊗\mathcal{B}(X)$ ...
5
votes
1answer
73 views

Universal measurability of a kernel

Let $X$ and $A$ be Borel topological spaces, that is it is they are homeomorphic to Borel subsets of a complete separable metric space. Let further $\pi$ be a universally measurable stochastic kernel ...
9
votes
1answer
145 views

Can an uncountable family of positive-measure sets be such that no point belongs to uncountably many of them?

I would be happy to know whether the following is true: For every uncountable family $\Gamma$ of positive-measure sets in a $\sigma$-finite measure space, there is at least one point that belongs to ...
3
votes
2answers
149 views

Every uncountable Polish Space has a copy of $\{0, 1\}^\mathbb{N}$

I am having trouble verifying corollary 7.8 on p. 6 in this document http://www.math.ucla.edu/~biskup/275b.1.13w/PDFs/Standard-Borel-Spaces.pdf My troubles are with the definition of the "tree" ...
1
vote
0answers
35 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
2
votes
1answer
91 views

Notation related to the Borel hierarchy

I'm reading an article where the author uses many classes in the Borel hierarchy: namely, $\mathcal F_\sigma$, $G_\delta$, $\mathcal F_{\sigma\delta}$, etc. In this context, he mentions the class ...
3
votes
2answers
131 views

Uniform versus product topologies on $[0,1]^\mathbb{N}$, and their Borel $\sigma$-algebras

Let $\tau_U$ and $\tau_P$ be the uniform (i.e. sup-metric) and product topologies on $[0,1]^\mathbb{N}$, respectively. Clearly, these topologies are not the same ($\tau_P$ is separable and $\tau_U$ ...
5
votes
0answers
161 views

Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
3
votes
2answers
135 views

Why for a compact metric probability space, any Borel subset can be approximated by compact set?

Let $X$ be a compact metric space with a Probability Borel measure $\mu$. Let $C$ be any Borel subset of $X$. Then for any small positive number $a$, we can find compact set $K$ such that $K$ is ...
6
votes
2answers
172 views

A Borel algebra containing $\infty$

If I adjoint $\infty$ to the real numbers ($\overline{\mathbb{R}}=\mathbb{R}\cup\infty$) is there a reasonable way to define a $\sigma$-algebra "$\mathcal{B}_{\overline{\mathbb{R}}}$" such that ...
5
votes
1answer
378 views

Do the Baire Sets of $\mathbb{R}$ contain the Borel Sets of $\mathbb{R}$?

I'm trying to gain some intuitive characterization of the Baire subsets of $\mathbb{R}$. I've seen different definitions of Baire subsets (which I presume are equivalent) but here is a ...
19
votes
1answer
588 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
0
votes
4answers
284 views

Is $A$ a Borel set?

Let be $X$ a metric compact space and $(G,+)$ a topological compact abelian group. Let be $\mathcal{A}$ the Borel $\sigma$-algebra of $X$ and $\mathcal{B}$ the Borel $\sigma$-algebra of $G$. ...
3
votes
1answer
218 views

Lebesgue theory and axiom of choice

I have been told that the existence of non-Lebesgue-measurable sets on $\mathbb R$ is impossible without axiom of choice. Do any other well-known results in Lebesgue theory depend on the axiom of ...
22
votes
1answer
651 views

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product ...
1
vote
2answers
765 views

set in $\mathbb{R}$ which is not a Borel-set [duplicate]

Possible Duplicate: Lebesgue measurable but not Borel measurable Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly if i start from the topology of $\mathbb{R}$, i.e. all ...