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Results tagged with descriptive-set-theory
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user 11768
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
1
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Measurable $\epsilon$-optimal selection with an analytically measurable stochastic kernel
Some thoughts. It does not seem likely that you can achieve existence for non-analytic $\eta$, hence I'd suggest trying to show that it is - or finding a counterexample. Let's say $c = 0$ and $u = 1_B …
- 1,655
11
votes
1
answer
944
views
Uniformization/measurable selection theorems
Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. …
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3
votes
1
answer
596
views
Inverse of a Borel surjection
Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$.
I …
- 1,655
4
votes
1
answer
1k
views
Quotients of standard Borel spaces
Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\su …
- 1,655
4
votes
1
answer
152
views
Analytic enlargement of an analytic set
Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus …
- 1,655
0
votes
1
answer
137
views
Existence of a map connecting two marginals of a product measure
Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by …
- 1,655
1
vote
Accepted
Particular neighborhoods of analytical sets
If I am correct, one can proceed as follows. Consider a set $A\subset X\times \mathcal P(X)^2$ given by
$$
A = \{(x,p,q):(x,p)\in \Gamma,\rho(p,q)\leq\varepsilon\}
$$
then we obtain $\Gamma^\varepsilo …
- 1,655
2
votes
1
answer
192
views
Particular neighborhoods of analytical sets
Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ e …
- 1,655
5
votes
0
answers
358
views
Existence of an universally measurable pullback
Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not emp …
- 1,655
6
votes
2
answers
317
views
Borel kernel over an analytic set implies existence of a Borel map
Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel $ …
- 1,655
3
votes
1
answer
143
views
Maps that are a.e. equal have almost the same graphs
Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two …
- 1,655
5
votes
1
answer
463
views
Universally measurable map coincides a.e. with a Borel map
Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\ …
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