Recent content by wayneckm

Thanks so much for the explanation! It is the book by Doob on stochastic process, it tried to regard a stochastic process as a function of two variables. So it is true that the dependence of second component on the first one does not affect the condition of measurability in a product space as...

Thanks for the reply. First of all, I want to know if the second component \Omega_t depends on the first component T_0 , how can we show/prove its measurability? Secondly, indeed I read this from a book, and the aurthor simply stated that "as there is T_0 such that for each fixed t\in...

Hello all, I have some difficulty in determining the measurability in product space. Suppose the product space is T \times \Omega equipped with \mathcal{T} \otimes \mathcal{F} where ( T , \mathcal{T} , \mu ), ( \Omega , \mathcal{F} , P) are themselves measurable spaces. Now, if there...

Hello all, I am a bit unclear about the infimum under the system of extended real number. More precisely, I am wondering if it is a sqeuence of infinity, \left\{ -\infty, -\infty, \ldots \right\} or \left\{ +\infty, +\infty, \ldots \right\}, should I say the infimum of each of them is...

Hello all, My question is if f : X \mapsto Y is an isometry which preserves norm, i.e. \left\| f(x) \right\| _{Y} = \left\| x \right\|_{X} , does this imply \left\| f(x_2) - f(x_1) \right\| _{Y} = \left\| x_2 - x_1 \right\|_{X} ? Or, essentially is it sufficient to gurantee...

Oops...sorry, i misunderstood the term codomain. So codomain here should be E as stated.

Domain of f is \mathbb{R}^{+} Codomain of f is \mathbb{R}

Hello all,Here is my question while reading a proof. For a compact set K in a separable metrizable spce (E,\rho) and a continuous function t \mapsto f(t) , if we define D_{K} = \inf \{ t \geq 0 \; : \; f(t) \in K \} then, D_{K} \leq t if and only if \inf\{ \rho(f(q),K) : q \in...

the symbol 1 here means the indicator function. Thanks.

Hello all, I have a few questions in my mind: 1) \lim_{n\rightarrow \infty}[0,n) = \cup_{n\in\mathbb{N}}[0,n) = [0,infty) holds, and for \lim_{n\rightarrow \infty}[0,n] = \cup_{n\in\mathbb{N}}[0,n] = [0,infty) is also true? It should not be [0,infty] , am I correct? 2)...

Argh, you are right. I made a mistake in assuming that the bound can be attained within the finite set in the infimum but indeed it is not necessary true. Thanks. Wayne

I think the argument is, first of all, I assume \{t_{n}\} take values in \mathbb{R} , then, due to the existence of limit, \inf is indeed \min and so it should be > - \infty . Somehow I think it is also an if-and-only-if statement. Wayne

Is it because, due to the dense set, any element h in \mathcal{H} is the limit of a sequence of elements \{h_{n}\}\ in H , so this forms a Cauchy sequence, and then, under the isometric map, we can obtain a Cauchy sequence in \mathcal{G} , however, I think we should assume the space...

Hello all, May someone help me on this question: Suppose the map F is an isometry which maps a dense set H of a semi-normed space \mathcal{H} to a normed space \mathcal{G} , now the theorem said we can extend this isometry in a unique manner to a linear isometry of the semi-normed...

Hello all, Sometimes I come across the situation that a topology of a space is defined indirectly through some convergence mode. I can understand when we are given a topology, we can define the convergence of a sequence w.r.t this topology. However, if we start with saying the space is...