# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

2,632
questions

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### Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A_1,A_2)
&=&\sup\...

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### What is compatibility?

This is rather subjective. But when we say "a measure is compatible with the topology" what do we mean exactly?
Disclaimer:
I'm not being sarcastic. I'm not being mathematically hostile. ...

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1
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### Integral and inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...

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1
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158
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### Integral with inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...

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2
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### Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...

2
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1
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129
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### Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]:
$$||\mu||_0:= \...

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### Compatibility between a radon measure and the topology of the space [closed]

According to Wikipedia, the definition of Radon measures guarantee a compatibility with the topological properties of the space. Could someone elaborate on this, possibly giving examples?

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### Measurability of a process defined by an integral

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{P} \subset \mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F}$ the $\sigma$-algebra generated by $\{\{0\}\times F_0\}...

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### Vector space of random variables [closed]

Let $(\Omega, \mathfrak{S}, \mu)$ be a probability space and let $R(\Omega)$ be the space of all real-valued random variables $X: \Omega \to \mathbb{R}$ w.r.t. the above probability space having ...

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### Measurability in a product space of a set defined only along its fibers

Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...

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### Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...

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2
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### Measurability of Brjuno numbers

A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...

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### Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...

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### Remainder-balancedness of primes

Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...

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### Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...

6
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1
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242
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### A characterisation of continuous real functions

Let $f: \mathbb R^n \to \mathbb R$ be a measurable function.
We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...

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1
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### Sums of powers of measures of $p$-adic balls

Let $(a_n,k_n) \in \mathbb{Z}_p \times \mathbb{N}$ for $n \in \mathbb{N}$ and consider the sequence of closed $p$-adic balls $B(a_n,k_n) = a_n + p^{k_n}\mathbb{Z}_p$. I assume that the $(a_n,k_n)$ are ...

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75
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### Differentiation under the integral sign in higher dimensions [migrated]

Let's say I have a function $f(\mathbf{x},t)$, $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n$. Are there conditions under which the following holds:
$$
\frac{\partial}{\partial\mathbf{x}^T}\int_\...

6
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2
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141
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### Bisector of two points in a Riemannian manifold has measure $0$

Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...

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2
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123
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### When can we extend a function on a $\lambda$-system to a probability measure?

Let $\Omega$ be a nonempty set and let $\mathcal{L}$ a $\lambda$-system on $\Omega$. That is,
(i) $\Omega \in \mathcal{L}$,
(ii) if $A, B \in \mathcal{L}$ and $A \subseteq B$, then $B \setminus A \in \...

1
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1
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### References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a good introductory treatment to tilted distributions using the following notion of "tilting" (or equivalent):
...

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3
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183
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### More natural example of measurable but not progressive process

All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean.
Consider ...

2
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0
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109
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### A technical question concerning convolution product

Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$.
Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...

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### Overview of generalizations of outer measures?

One approach to generalizations of the concept of outer measures taking values in topological spaces/commutative semigroups is in
L.Ja. Savel'ev, "Extension of outer measures and measures"
...

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1
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### Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?

Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e.,
...

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1
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### Motivation for Ionescu-Tulcea extension theorem (as opposed to Kolmogorov's)

I recently asked a question on the differences between Ionescu-Tulcea and Kolmogorov extension theorems (ITET and KET for short). A lot of my confusion has been cleared there and what I understood ...

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1
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### Meyer's example of a separable process with no path regularity

This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link.
In the following excerpt from ...

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1
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### Does the set of square numbers adhere to Benford's law in every base?

Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to Benford's law for the first digit in every base $b\geq 2$?
Precise formulation of what it means for a set $T\subseteq \omega$ to "...

4
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2
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### Kolmogorov vs Ionescu-Tulcea extension theorem (again)

Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there.
I've recently ...

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1
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124
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### Progressive measurability intuition from Bichteler's *Stochastic integration with jumps* book

In the Stochastic Integration with Jumps Bichteler gives a very intuitive definition of progressive measurability I've never seen before:
Although I like this intuition very much, I cannot find a ...

4
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1
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### Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...

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1
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### Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable?

Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$.
$f:X \to \overline{\mathbb R}$ is called $\mu$-...

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### If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?
Context: I was ...

4
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1
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### Does Szemerédi's theorem hold for sets with positive upper Banach density?

We say that a set of natural numbers $A\subseteq \omega$ has positive upper density if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$
Szeméredi's theorem states that every $A\subseteq \omega$ ...

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1
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60
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### Upper density versus upper Banach density on $\omega$

For $A\subseteq\omega$ we define the upper density by $$d_u(A) = \lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1}.$$ For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ ...

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### Measure on the places of $\bar{\mathbb Q}$

Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...

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### Is there a measure theory for proper classes?

This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes?
Of course when one tries to define measures on "large sets" ...

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### Finding bound of ratios of complex integral over a 7-sphere

We define the following integral:
$$
I(m, n, k)=\int_{\mathbb{S}^7}\left|z_1^m z_2^n\left(z_1 z_4-z_2 z_3\right)^k\right| d S, \quad \text{where } z=\left(z_1, z_2, z_3, z_4\right) \in \mathbb{C}^4\\ \...

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1
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### Textbook definition for "path measure" or "probability measure over paths"

I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?

3
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1
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112
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### A level set of non-constant real analytic function

Assume that $u: B\to [0,1]$, where $B$ is an open ball in $\mathbb{R}^n$, is a nonconstant real analytic function and let $t\in[0,1]$ and let $\mu(t)=\operatorname{Volume}(\{x\in B: u(x)>t\})$. Why ...

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2
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### Is there something like a "self-avoiding Markov chain" on a continuous space?

If stumbled accross self-avoiding walks. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants.
However, as far as I can see they are ...

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0
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### Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space

Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...

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0
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### Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?

Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...

1
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0
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### Does the constrained Wasserstein barycenter admit a blue noise property?

Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...

2
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### A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$

Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...

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103
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### Approximate range of Radon-Nikodym derivative in a dynamical system

Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...

4
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1
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226
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### Supremum of infimum of measure of members of a free ultrafilter

For a set $A\subseteq \omega$ we let the upper density of $A$ be defined as $d^+(A) := \lim\sup_{n\to\infty}\frac{|A\cap(n+1)|}{n+1}$. Let $\text{FrU}(\omega)$ be the collection of free ultrafilters ...

3
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1
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102
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### Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...

4
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2
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169
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### 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 $x\in X$ ...

1
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0
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112
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### Wiener Integral and its distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...