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Results tagged with measure-theory
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user 99469
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
2
votes
1
answer
223
views
Is Boltzmann entropy well-defined for arbitrary probability density function?
$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \ln …
- 128k
2
votes
1
answer
794
views
Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb...
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We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that
$$
\ …
- 114k
0
votes
1
answer
99
views
Sequential compactness of a sequence of curves of Borel probability measures
$
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- 1,724
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
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- 825
1
vote
1
answer
116
views
Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < ...
$
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- 23.2k
2
votes
1
answer
61
views
Approximate a non-negative function which is measurable in product $\sigma$-algebra
$
\DeclareMathOperator*{\supp}{supp}
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- 60.6k
1
vote
2
answers
126
views
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integr...
$
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- 825
3
votes
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integr...
$
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- 825
1
vote
1
answer
104
views
Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
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Let $(\Omega, \cA, \mu)$ be a probability space and $\cA_1, \cA_2$ sub $\sigma$-algebras of $\cA$. Let $\c …
- 41.1k
1
vote
1
answer
272
views
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
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- 6,323
2
votes
0
answers
64
views
Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô int...
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- 825
1
vote
1
answer
75
views
Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with ...
For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the Wasser …
- 5,380
0
votes
1
answer
76
views
Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$
We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a meas …
- 825
4
votes
2
answers
242
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\m …
- 5,380
5
votes
1
answer
437
views
Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \...
Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as
$$
\partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle …
- 28k