Unanswered 'measure-theory+borel-sets' Questions

10 votes

0 answers

744 views

Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...

Wolfgang Loehr

888

asked Aug 12, 2012 at 22:48

5 votes

0 answers

95 views

Is there an equivalent condition for Borel projections being Borel?

Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, ...

J.R.

291

asked Jun 8 at 9:12

5 votes

0 answers

364 views

Computing the infinite dimensional Lebesgue measure of "cubes"

There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...

truebaran

9,330

asked Jul 23, 2020 at 0:46

4 votes

0 answers

414 views

Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...

Longyearbyen

541

asked Jun 7, 2016 at 15:59

3 votes

0 answers

143 views

Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators $$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$ where $\mathbb U_n$ is the ...

Exodd

201

asked Sep 21, 2021 at 7:00

2 votes

0 answers

115 views

Borel measurability

Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...

A beginner mathmatician

1,879

asked Apr 21, 2020 at 20:36

2 votes

0 answers

859 views

Regularity of locally finite Borel measure

Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...

Michael

143

asked Aug 4, 2019 at 13:53

1 vote

0 answers

97 views

Borel structure/sets coming from strong operator topology vs norm topology

Let $X, Y$ be Banach spaces. Moreover, let $\mathcal{L}(X,Y)$ be the space of bounded linear operators equipped with the standard operator norm topology, and $\mathcal{L}_{\mathrm{s}}(X,Y)$ the same ...

Marek Kryspin

139

asked Aug 3, 2023 at 15:28

1 vote

0 answers

154 views

Polish spaces and analytic sets

Can we conclude that an analytic subset $A$ of a Polish space $X$ is also Polish? Let $\mathcal{M}(R^d)$ denotes the family of Borel probability measures on $R^d$ equipped with the Lévy-Prokhorov ...

B-S

39

asked Jul 29, 2023 at 10:05

1 vote

0 answers

399 views

Weak topology on spaces of measures and Borel sets

Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...

Damian Sobota

1,151

asked Sep 17, 2020 at 13:02

0 votes

0 answers

113 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...

S-F

63

asked Nov 22 at 16:01

0 votes

1 answer

78 views

Intersection of sigma algebras generated by shifts

EDIT: Iosif's answer showed that my motivation for this question was mislead. To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...

Florian R

257

asked Nov 13 at 13:49

0 votes

0 answers

38 views

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...

S-F

63

asked Oct 25 at 15:41

0 votes

0 answers

78 views

What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...

S-F

63

asked Oct 19 at 14:39

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All Questions

Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Is there an equivalent condition for Borel projections being Borel?

Computing the infinite dimensional Lebesgue measure of "cubes"

Topology on the space of Borel measures

Is the singular value decomposition a measurable function?

Borel measurability

Regularity of locally finite Borel measure

Borel structure/sets coming from strong operator topology vs norm topology

Polish spaces and analytic sets

Weak topology on spaces of measures and Borel sets

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?

Intersection of sigma algebras generated by shifts

Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?

What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?

All Questions

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