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revised Origin of the term dual space?
edited tags
May
18
comment Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers
$S_n = X_1 + \dots + X_n$
May
18
comment Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers
You definitely have to use $S_n$, because the statement to be proved is that $S_n/n \to 0$ converges in probability to a constant, not $X_n/n$.
May
18
comment Proving that T(t)x is in the domain
@user3482534: That isn't what I would do. If $x \in D(A^2)$ then $Ax \in D(A)$, so applying your sentence "I know that" with $x$ replaced by $Ax$, you get $T(t) Ax \in D(A)$. But $T(t) A x = A T(t) x$ so you have $A T(t) x \in D(A)$, which means $T(t) x \in D(A^2)$.
May
18
comment Image of collection of probability measures in $C_b(S)'$
Let us continue this discussion in chat.
May
18
revised Image of collection of probability measures in $C_b(S)'$
fix pointwise convergence
May
18
comment Image of collection of probability measures in $C_b(S)'$
@Holymonk: No, in the proof I do not claim a priori that the cluster point $\ell$ is not in the sequence. But it follows from the rest of the proof that it is not, since all the elements of the sequence are in the image of $\iota$ and we prove that $\ell$ is not.
May
18
comment Image of collection of probability measures in $C_b(S)'$
@Holymonk: I am still not sure that I am following your notation (what is $\psi$?) but maybe the issue is this: I am using the definition of "cluster point" found on Wikipedia. In a Hausdorff topological space $X$, I say $p$ is a cluster point of a sequence $\{p_n\}$ if every neighborhood of $p$ contains $p_n$ for infinitely many $n$. I don't require that these $p_n$ be distinct from $p$. So, for instance, given the constant sequence $1,1,1,1,\dots$, I would say that 1 is a cluster point.
May
18
comment Image of collection of probability measures in $C_b(S)'$
@Holymonk: What is $\mathcal{V}$ and what is $\psi$?
May
17
answered Sigma field generated by a Boole algebra and the monotone class theorem
May
17
comment Sigma field generated by a Boole algebra and the monotone class theorem
The examples I know are based on the non-collapsing of the Borel hierarchy. Basically, there is not going to be any simple procedure like this for constructing a $\sigma$-algebra from its generating sets, not in any finite or countable number of steps. Are you reasonably comfortable with product topologies and the Baire category theorem? If so I can write up something that works for this case.
May
17
comment Infinite not the limit of finite processes
What exactly do you mean by the "limit" of a sequence of sets? There isn't an obvious definition that will behave the way you describe.
May
16
comment Sigma field generated by a Boole algebra and the monotone class theorem
In general, it is not true that $\mathcal{G}$ is stable under countable intersection (nor complementation), so this idea does not work.
May
16
answered Image of collection of probability measures in $C_b(S)'$
May
15
answered Proving that T(t)x is in the domain
May
15
answered It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces?
May
15
comment It is true that the relatively compact open subsets cannot exist in infinite-dimensional normed spaces?
en.wikipedia.org/wiki/Riesz%27s_lemma
May
15
revised Space of bounded functions is reflexive if the domain is finite
hausdorff
May
15
awarded  Disciplined
May
14
comment Law of large numbers - almost sure convergence
I think to understand this precisely, you will have to learn measure theory.