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Oct
17
comment (Reference) Asymptotics of hitting probability by Brownian motion
What is a "solid object"? What if $A$ is the set $\{p : 1 \le |p| \le 2\}$ (a spherical shell) and $x=0$?
Oct
16
comment (Reference) Asymptotics of hitting probability by Brownian motion
In dimensions 1 and 2, I think you mean to say $\lim_{t \to \infty} P_x(T_A > t) = 0$. In dimensions 3 and higher, we certainly have $\lim_{t \to \infty} P_x(T_A > t) = P_x(T_A = \infty) < 1$. Depending on exactly what you mean by "exterior", you may even get $P_x(T_A = \infty) = 0$. But it is clearly true that $P_x(T_A > t) > 0$ for all finite $t$: consider a ball around $x$ disjoint from $A$, and there is a positive probability that the Brownian motion remains in this ball up to time $t$.
Oct
15
awarded  Nice Answer
Oct
15
revised Is it possible to formulate the axiom of choice as the existence of a survival strategy?
ultrafilter argument applies when |C|=2
Oct
15
comment Is it possible to formulate the axiom of choice as the existence of a survival strategy?
@AsafKaragila: Thanks. Fixed, I think.
Oct
15
revised Is it possible to formulate the axiom of choice as the existence of a survival strategy?
added 105 characters in body
Oct
14
revised Is it possible to formulate the axiom of choice as the existence of a survival strategy?
lebesgue measurability
Oct
14
revised Is it possible to formulate the axiom of choice as the existence of a survival strategy?
added 671 characters in body
Oct
14
answered Is it possible to formulate the axiom of choice as the existence of a survival strategy?
Oct
14
comment Is it possible to formulate the axiom of choice as the existence of a survival strategy?
I've usually heard this question phrased in terms of prisoners and hats. Certainly in the case $|G| = \kappa = \aleph_0$ (where the question is well defined in ZF), it is known to be consistent with ZF+DC that the giraffes cannot win (infinitely many will be eaten), even when $|C|=2$. I will try to find the reference, but it comes down to the fact that it is consistent with ZF+DC that all maps from $C^G$ to $C$ have some nice measurability property.
Oct
14
revised A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
fix tyop in title
Oct
12
comment Connected graph as connected space
I'm afraid I still don't understand. Maybe I am naive, but the only definition of "connected graph" I know is that for every pair of vertices $a,b$, there is a finite list of vertices $x_0, x_1, \dots, x_n$ such that $x_0 = a$, $x_n = b$, and $x_i \sim x_{i+1}$. Is there a different definition which you are using?
Oct
12
comment Connected graph as connected space
What part of the proof for the finite case goes wrong in the locally finite case? I don't have the book to check the definitions, but since in a connected graph any two vertices are joined by a path of finite length, it seems to me that you should only have to look at finitely many vertices at a time.
Oct
10
comment Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Is it even true for $G = B = \mathbb{R}$?
Oct
10
comment Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space
Interesting question. It is sufficient to consider the case $f = 1_A$ where $A \subset G \times G$ is Borel, since linear combinations of those give you all simple functions on $G \times G$, and every measurable function is a uniform limit of simple functions. I think it is true, by a monotone class argument, that the measurable functions are the smallest set of functions containing all $\sum b_k \chi_{E_k \times F_k}$ and closed under pointwise convergence of sequences. But your condition is more like a Baire class question.
Oct
9
answered Learning roadmap: 'combinatorial' probability
Oct
3
revised About the Dimension of a complete local ring
typo in title
Oct
2
comment Does there exist a supersmooth non-polynomial function?
@AlexandreEremenko: I don't think it's that obvious, because we get to choose $x$ (or a conull/comeager set of $x$) depending on the sequence $a_n$
Oct
2
comment Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?
I'm confused; can you clarify how this question is different from Null sets in PDE? Also, can you elaborate on "the usual assumptions"?
Oct
1
comment Can a mathematical definition be wrong?
@JBL: Presumably because most developments of the foundations of mathematics start by constructing the natural numbers, and then define the integers as natural numbers with signs (treating 0 specially if needed). Using "nonnegative integer" instead of "natural number" introduces a circularity problem.