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comment Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?
But Q1 is true for Brownian motion; just fix any $y \in A$ and let $\tau = \inf\{t \ge 0 : X_t = y\}$. I think something is wrong with your "strong Markov" logic.
1h
comment Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?
If I have read correctly, isn't Q2 false for a standard Brownian motion? If $B$ is any set of full Lebesgue measure containing $\xi$, it will satisfy $P_t(x,B) = 1$ for every $x \in B$ and every $t \ge 0$.
Mar
23
awarded  Yearling
Mar
20
comment How many subsets of $[0,1)$ are there modulo null sets?
You might just edit the first paragraph to clarify that it doesn't extend Eric's (current) result.
Mar
20
comment Relatively compact sets in Ky Fan metric space
Incidentally, the 3-digit reference numbers are not a typo. Bogachev's books feature a staggering number of references. Volume 1 has 1055 references and Volume 2 has 2038 (!!!)
Mar
20
answered Relatively compact sets in Ky Fan metric space
Mar
18
comment Banach space dual to $L^\infty(I,H^1(M))$
@Alan: That is a very different question - you aren't looking for the dual but the predual. The answer is almost certainly going to be $L^1(I, H^1(M))$ and the proof will probably look very similar to the proof that $L^1(I, \mathbb{R})^* = L^\infty(I, \mathbb{R})$. The canonical reference for any question like "what is the dual of Banach space $X$" is Dunford and Schwartz, Linear Operators. Another possible place to look is Dinculeanu's Vector Measures.
Mar
18
awarded  measure-theory
Mar
17
answered Injective inclusion map from RKHS function space to $L_p(\mu)$
Mar
17
comment Injective inclusion map from RKHS function space to $L_p(\mu)$
I don't see that assertion in what you've written; is it elsewhere in the text? The line you quote after "At a certain point" is an if and only if statement. It doesn't assert that the map is injective. If there is an assertion, somewhere, that the inclusion map is always injective, then it must follow from other conditions you have not mentioned.
Mar
17
comment Injective inclusion map from RKHS function space to $L_p(\mu)$
So I guess I am confused as to what you are really asking. Certainly the inclusion map can be injective. There doesn't seem to be any assertion that it is always injective.
Mar
17
comment Injective inclusion map from RKHS function space to $L_p(\mu)$
Haven't you answered your own question? If, for instance, all the functions in $H$ are continuous, the inclusion map will be injective. For a concrete example, take $X = [0,1]$ and let $H = H^1([0,1])$ be the Hilbert space of all absolutely continuous $f : [0,1] \to \mathbb{R}$ having $f' \in L^2([0,1],m)$, equipped with the inner product $\langle f,g\rangle_H = \int_0^1 (fg + f'g')\,dm$.
Mar
16
comment Parodies of abstruse mathematical writing
Along similar lines: "Nothing is better than eternal happiness. A cheese sandwich is better than nothing. By transitivity, a cheese sandwich is better than eternal happiness."
Mar
16
comment W*-completion of a C*-algebra?
@YemonChoi: That was a dumb comment, I'm way out of my depth here. But let me ask a dumb question, then: say $X = [0,1]$. If we're trying to "complete" $C([0,1])$, and the claim is we should get $L^\infty([0,1], m)$, why should $m$ be Lebesgue measure? Why not some other measure? Or does it somehow not matter?
Mar
16
comment W*-completion of a C*-algebra?
As this ancient question has just been bumped: surely in the commutative case you simply want the bounded Borel functions? If you say "essentially bounded" and "up to equality almost everywhere" you have to specify a measure, or at least a $\sigma$-ideal of null sets, and I don't see that there's a canonical choice.
Mar
15
comment Approximating by independent Poisson random variables
If $W_1, W_2$ are dependent, it seems unreasonable for the law of $(W_1, W_2)$ to be well approximated by a pair of independent Poissons. For example, what happens if $X_i = Y_i$ so that $W_1 = W_2$?
Mar
15
comment Approximating by independent Poisson random variables
What is $\mathcal{L}$?
Mar
14
comment The most cited paper in Mathematics
@ThiKu: Oh, you are right. How silly of me.
Mar
14
comment The most cited paper in Mathematics
For MathSciNet, if you go to ams.org/mathscinet/citations.html, click "Top 10", and select "Journal Articles", you get what appears to be MathSciNet's idea of the most cited papers. The top one listed is Bosma, Cannon and Playoust, "The Magma algebra system", but surprisingly it says it has only 165 citations through 2013. Maybe this reflects that not all papers in MathSciNet have their references indexed?
Mar
14
comment Parodies of abstruse mathematical writing
Ok, hope you'll pardon me for committing self-promotion by adding some details to Deane's answer. Thanks Deane!