All Questions
Tagged with fa.functional-analysis measure-theory
738 questions
4
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1
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412
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Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?
I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
- 617
4
votes
1
answer
2k
views
Operator topologies on $L^{\infty}(X,\mu )$
Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ ...
- 1,270
4
votes
1
answer
393
views
How can I show that "almost all function" have property P?
The following is cross-posted from
https://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept
since I didn't (yet) get an answer there.
(I hope that's okay?)
...
- 151
4
votes
2
answers
479
views
When does a function space allow for point evaluations? [closed]
Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.
What are necessary and sufficient conditions for point evaluations at arbitrary $x \...
- 5,327
4
votes
2
answers
378
views
Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
is the space of all borel measures on $\...
- 3,738
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
4
votes
1
answer
579
views
Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $...
- 8,512
4
votes
1
answer
860
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k
4
votes
2
answers
378
views
A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, ...
- 41
4
votes
2
answers
198
views
Is the conditional expectation of a Caratheodory function a Caratheodory function?
Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
- 165
4
votes
1
answer
597
views
Meaning of Alberti rank-one theorem
Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
user123457
4
votes
1
answer
151
views
Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
4
votes
1
answer
364
views
$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$
I asked this question on MSE here some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction ...
- 261
4
votes
1
answer
78
views
Bound for the generalised Rényi dimension of a measure
If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for $j_1,\ldots,...
- 6,206
4
votes
2
answers
558
views
Is a specific sequentially closed subset of $M([0,1])$ closed?
Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
- 95
4
votes
1
answer
523
views
On Radon measures with values in Banach space
It is known that continuous linear functionals on the space $C_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C_0({\mathbb{R}^n})$ is uniform closure of the space of ...
- 43
4
votes
1
answer
315
views
Doubt on Morrey spaces of measures according to T. Giga and Y. Miyakawa
I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the ...
- 41
4
votes
2
answers
427
views
Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
- 125
4
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1
answer
184
views
Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
- 391
4
votes
1
answer
253
views
Regarding outer functions
Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
- 111
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
4
votes
1
answer
209
views
Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative
Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
- 14.4k
4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
- 1,649
4
votes
1
answer
370
views
Norms for complex measures
I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
- 123
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
4
votes
0
answers
495
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
4
votes
0
answers
117
views
If $f:U_1\to\mathcal L^p(\mu;E_2)$ is Fréchet differentiable, can we say anything about the Fréchet differentiability of $u\mapsto f(u)(\omega)$?
Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $p\ge1$, $E_i$ be a $\mathbb R$-Banach space, $U_1\subseteq E_1$ be open and $f:U_1\to L$ be Fréchet differentiable at $x\in U_1$, ...
- 167
4
votes
0
answers
213
views
Classification of Euclidean-invariant measures?
Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely,
By ...
- 63.9k
4
votes
0
answers
115
views
Box counting dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it?
The analogous question for the ...
- 839
4
votes
0
answers
115
views
point-wise approximation of the identity in hereditary Lindelof spaces
Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...
- 4,058
4
votes
0
answers
143
views
A point concerning Fremlin's example on Borel sets in non-separable Banach spaces
Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...
- 4,058
4
votes
0
answers
585
views
Dual of the space of all bounded functions, $B(X, \mathbb{R}).$
Let $X$ be a non compact separable metric space. Denote
by $B(X, \mathbb{R})$ the set of all bounded real functions
endowed with the sup norm, this is a Banach space. Denote by
$C_b(X,\mathbb{R})\...
- 757
4
votes
0
answers
123
views
Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,616
4
votes
0
answers
185
views
A strongly open set which is not measurable in the weak operator topology
Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
- 4,058
4
votes
0
answers
451
views
Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
4
votes
0
answers
309
views
Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,287
3
votes
3
answers
776
views
Radon-Nikodym property for space of signed measures
Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with ...
- 2,247
3
votes
2
answers
808
views
Growth of $L^p$ norms as $p \to \infty$
Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
- 1,211
3
votes
2
answers
766
views
Borel vs measure for all Borel measures
Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
- 18.7k
3
votes
2
answers
589
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
3
votes
2
answers
410
views
Is a bounded sequence of $H^1(\Omega)$ tight?
Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$.
Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
- 1,101
3
votes
2
answers
322
views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user139844
3
votes
1
answer
376
views
Does the space of Lipschitz functions have the Radon-Nikodym property?
Context.
Space of Lipschitz functions. Denote by $Lip_0(D)$ the space of all Lipschitz functions on a metric space $D$ vanishing at some base point $e \in D$. The norm in $Lip_0$ is defined as ...
- 437
3
votes
2
answers
370
views
Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations
In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is ...
- 597
3
votes
1
answer
199
views
A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$
I asked this at math.stackexchange, but nobody answered.
Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\...
- 7,426