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asked
How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?
awarded
 Custodian
reviewed
Approve
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
comment
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
@terceira Ah my $\mu, \mu_n$ are all probability measures.
comment
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
@terceira It seems in your example $\mu_n := \delta_n$. Could you explain what is your $\mu$?
accepted
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
comment
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Thank you so much for your help! I have just got the same conclusion from this answer in which Stone–Weierstrass theorem is appealled.
asked
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
comment
Optimal Transport: how is this transport map Borel measurable?
Thank you so much for your dedicated help! Happy Christmas and new year to you.
comment
Optimal Transport: how is this transport map Borel measurable?
May I ask if the converse, i.e., $x_k \to x \Longrightarrow d_H(\partial h(x_k),\partial h(x)) \to 0$, holds? I tried but to no avail in my attempt.
awarded
 Organizer
revised
Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Added tags
suggested
Approve
Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
revised
A function $f_r$ where $f_r (x)$ is defined as the ratio between $f(x)$ and the average value of $f$ over $B(x, r)$
edited body
asked
A function $f_r$ where $f_r (x)$ is defined as the ratio between $f(x)$ and the average value of $f$ over $B(x, r)$
accepted
Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
comment
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
@MichaelGreinecker Thank you so much for your references! Could you post it as an answer? It seems Theorem 6.1 in Parthasarathy's Probability measures on metric spaces also provides such a version.
asked
A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
asked
Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
accepted
A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space
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