All Questions
7 questions
1
vote
0
answers
30
views
Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?
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
2
votes
0
answers
150
views
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 \...
- 657
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
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 $\Omega$...
- 657
0
votes
0
answers
65
views
Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
- 835
1
vote
0
answers
450
views
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 ...
- 657
1
vote
1
answer
61
views
Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$, and
$\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$.
Then $\mathcal ...
- 835
4
votes
0
answers
2k
views
Does rate of convergence in probability come from a metric?
In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
- 63