# Tagged Questions

**0**

votes

**1**answer

66 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 ...

**3**

votes

**1**answer

171 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 $X$ ...

**12**

votes

**4**answers

456 views

### Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...

**5**

votes

**0**answers

169 views

### Existence of an universally measurable pullback

Edited: previous version of the question was less general but also less readable.
Let $X,Y$ and $Z$ be standard Borel spaces, that is topological spaces homeomorphic to Borel subsets of complete ...

**6**

votes

**2**answers

232 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 ...

**3**

votes

**1**answer

123 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 ...

**2**

votes

**1**answer

131 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 ...

**4**

votes

**2**answers

175 views

### Obtaining conditional probabilities as pushforwards of [0,1]

It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...

**9**

votes

**1**answer

399 views

### A question on infinite dimensional Gaussian measure and affine tranformations.

Let $\gamma_\infty$ denote the product Gaussian measure on $\mathbb{R}^\mathbb{N}$. Which $a,b \geq 0$ satisfy that for every Borel set $K\subseteq \mathbb{R}^\mathbb{N}$ of positive measure, $a K + ...

**4**

votes

**0**answers

168 views

### Borel selections of partitions of the space of probability measures

The environment:
Suppose that $X$ is a Polish space and $\Delta(X)$ the set of Borel probability measures over $X$ (given the topology of weak convergence, with the Prohorov metric).
Let ...