All Questions
19 questions
1
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176
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If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$.
Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
- 641
3
votes
1
answer
177
views
Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
- 131
0
votes
0
answers
80
views
Measurable Extension
Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
- 136
2
votes
2
answers
307
views
Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
- 113
1
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0
answers
87
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$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?
Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
- 309
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
0
votes
0
answers
47
views
Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
0
votes
0
answers
113
views
The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
2
votes
0
answers
202
views
Prove or disprove that $u=0$ a.e. on $\Bbb R^d$
Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
- 1,101
3
votes
0
answers
252
views
Characterization of a Bochner/strongly measurable function solely as a random element
Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively.
This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
- 31
3
votes
1
answer
242
views
Sobolev embedding into measurable functions
Consider the fractional Sobolev space
$$
W^{k,2}(\mathbb R^n):=\big\{f \in \mathcal S'\,\big|\,(1+\|\xi\|)^k\hat f(\xi)\in L^2(\mathbb R^n)\big\}
$$
for some $k\in\mathbb R$, and let $\mathcal M$ ...
- 43.2k
3
votes
1
answer
148
views
Measurability of superposition operator with non-separable Banach space
Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have
$$t \mapsto f(t,x) \text{ is measurable}$$
$$x \...
- 107
6
votes
1
answer
1k
views
Are Bochner measurablity and Borel measurability compatible?
Let $(X,\mathfrak{M})$ be a measurable space and $E$ be a Banach space and $f:(X,\mathfrak{M})\rightarrow E$ be a function.
Question
Are Borel-measurable condition on $f$ and Bochner-measurable ...
- 337
2
votes
1
answer
756
views
Functional representation of adapted jointly measurable stochastic processes
It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...
- 1,773
6
votes
0
answers
365
views
Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
- 1,396
4
votes
1
answer
819
views
The Notion of Strong Measurability for Separable Banach Spaces
Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
- 1,396
12
votes
4
answers
11k
views
The image of a measurable set under a measurable function.
Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are ...
- 5,407
3
votes
2
answers
946
views
Opinions on the Multiplication of Measures
A few questions, hopefully to spark some discussion.
How can one define a product of measures?
We could use Colombeau products by embedding the measures into the distributions? I'm not sure why this ...
- 213