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19 votes
1 answer
465 views

Large Borel antichains in the Cantor cube?

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
Taras Banakh's user avatar
  • 41.9k
18 votes
3 answers
1k views

Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural ...
Tom LaGatta's user avatar
  • 8,512
18 votes
1 answer
11k views

Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow. In non-Hausdorff topology it is standard to ...
polmath's user avatar
  • 321
6 votes
1 answer
922 views

Correspondence between functions on a set and "states" on its power set

Let $L$ be the poset (ordered by set inclusion) that is the power set of some set $X$. A state is a function $s:L \rightarrow [0,1]$ satisfying i) for {$p_1,p_2,...$}, $p_i \in L$ a pairwise ...
tortortor's user avatar
  • 207
5 votes
2 answers
478 views

Generalization of the concept of a measure

Consider the following generalization of the concept of a measure: Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice. Let $M = (Y, \bullet, e)$ be a commutative monoid. An $(L, M)$-measure is ...
user76284's user avatar
  • 2,203
4 votes
2 answers
274 views

Does strong stochastic ordering exist?

For two probability measure $\mu$ and $\nu$ on $\mathbb{R}$, we call $\mu$ is stochastically smaller than $\nu$ (i.e., $\mu\leq\nu$) , if $\int f \, d\mu\leq\int f \, d\nu$ for any nonnegative bounded ...
Jinxiang Yao's user avatar
4 votes
1 answer
119 views

Antisymmetry of the stochastic order

An ordered topological space is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ ...
Tobias Fritz's user avatar
  • 6,406
4 votes
1 answer
236 views

Measurable total order

Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$? By "measurable total ordering" we ...
Aryeh Kontorovich's user avatar
3 votes
1 answer
251 views

Supremum with respect to the order of measures on $(X,A)$

Suppose that $(X,\leq )$ is an ordered set, we can define the maximum and the infimum of this set,now let $(X,A)$ be a measurable space and let $M(X,A)$ be the set of all measures on $(X,A)$, we now ...
YuerWu's user avatar
  • 415
2 votes
1 answer
215 views

Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
Dominic van der Zypen's user avatar
2 votes
0 answers
78 views

$\sigma$-fields as closure systems

Let $(\Omega,\mathcal A, P)$ be a probability space and let $\Sigma(\mathcal A) \subset 2^{\mathcal A}$ be the collection of all sub-$\sigma$-fields of $\mathcal A$. Then, $\Sigma(\mathcal A)$ is ...
passerby51's user avatar
  • 1,731
1 vote
1 answer
211 views

Can we order random variables in a measurable way in a general setup?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E)$ be a measurable space $n\in\mathbb N$ $X_1,\ldots,X_n$ be $(E,\mathcal E)$-valued random variables on $(\Omega,\...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
36 views

Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?

Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$. A total preorder $\preceq$ on $\...
aduh's user avatar
  • 869
0 votes
2 answers
435 views

Ordered measurable spaces

Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
daon's user avatar
  • 239
0 votes
1 answer
90 views

Countable sup property of extended measurable functions

Let $(S,\Sigma,\mu)$ a $\mu$-finite measure space. Denote by $\bar{L}^0(\Sigma)$ the set of extended-real valued $\Sigma$-measurable functions. Does this set have the countable sup property when ...
Giuliosky's user avatar