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Results tagged with metric-spaces
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user 477203
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
1
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1
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106
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Conditions that ensure the metric topology of $E$ coincides with the initial topology induce...
Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$,
$$
x_n \to x \iff [f(x_n) \to f(x) \quad \forall …
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4
votes
0
answers
471
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Disintegration of measures: a confusion about an existence proof from a lecture note
I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e.,
Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be Pol …
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1
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0
answers
396
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Portmanteau theorem for finite signed Borel measures
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$,
$\mathcal M_1(X)$ the spa …
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0
votes
1
answer
206
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Complex Borel measures: relation between the total variation norm of a measure and those of ...
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\l …
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0
votes
Accepted
Complex Borel measures: relation between the total variation norm of a measure and those of ...
I represent below @NikWeaver's idea of improving $\frac{1}{2} [\cdot]' \le [\cdot]$ to get $\frac{1}{\sqrt 2} [\cdot]' \le [\cdot]$.
For complex number $z = x + iy$ with $x,y \in \mathbb R$, we have
…
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0
votes
1
answer
217
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A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ whe...
Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ b …
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votes
Accepted
A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ whe...
Below is a counter-example taken from this thread. It works even when $\mathcal C_c$ is replaced by $\mathcal C$, the space of all continuous functions from $X$ from $E$.
Let $X:=[0, 1]$, $E:=\mathbb …
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1
vote
0
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85
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Is this metric on the space of $\mu$-measurable functions complete?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions fr …
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1
vote
0
answers
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Can we modify this extended pseudometric such that its convergence is equivalent to that in ...
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 fun …
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3
votes
1
answer
150
views
Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable …
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4
votes
2
answers
359
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Vague convergence: confusion about the regularity of a signed Radon measure and that of its ...
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ass …
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2
votes
0
answers
138
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A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu...
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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2
votes
1
answer
320
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Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_...
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega …
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2
votes
0
answers
91
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Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of al …
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