- #1

leo.

- 96

- 5

*Abstract Wiener Spaces*by Leonard Gross, which seems to be where these things were first presented.

Now, I'm having a hard time to grasp the idea/motivation behind the very first definition of the author, namely, that of cylinder sets and cylinder set measures. In fact, the author defines it as follows:

Let ##\mathscr{L}## be a locally convex real linear space and ##\mathscr{L}^\ast## its topological dual space. For each finite dimensional subspace ##K## of ##\mathscr{L}^\ast##, we denote by ##\pi_K## the linear map of ##\mathscr{L}## onto the dual space ##K^\ast## of ##K## given by ##\pi_K(x)y=\langle y,x\rangle## for ##x## in ##\mathscr{L}## and ##y## in ##K##. Let ##\mathscr{R}## be the collection of subsets of ##\mathscr{L}## which have the form ##C = \pi_K^{-1}(E)## where ##E## is a Borel set in ##K^\ast##. Such a set ##C## will be called a *tame set* (also known as a cylinder set) and will be said to be based on ##K##. The class ##\mathscr{R}## is a ring and the family ##\mathcal{S}_K## of sets in ##\mathscr{R}## which are based on ##K## is a ##\sigma##-ring.

Definition 1.A real-valued nonnegative finitely additive function##\mu##on##\mathscr{R}##is called a cylinder set measure on##\mathscr{L}##if##\mu##is countably additive on each of the##\sigma##-rings##\mathcal{S}_K##and##\mu(\mathscr{L})=1##.

On the other hand, there seem to be two related definitions on Wikipedia. The first is the definition of a cylinder set,

The second is the definition of a cylinder set measure,Consider the Cartesian product ##X = \prod_\alpha X_\alpha## of some spaces ##X_\alpha##, indexed by some index ##\alpha##. The canonical projection corresponding to some ##\alpha## is the function ##p_\alpha : X\to X_\alpha## that maps every element of the product to its ##\alpha## component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form, $$\bigcap_{i=1}^n p_{\alpha_i}^{-1}(A_{\alpha_i})=\{(x_\alpha)\in X | x_{\alpha_1}\in A_{\alpha_1},\dots, x_{\alpha_n}\in A_{\alpha_n}\}$$

Let ##E## be a separable, real, topological vector space. Let ##\mathcal{A}(E)## denote the collection of all surjective, continuous linear maps ##T : E\to F_T## defined on ##E## whose image is some finite-dimensional real vector space ##F_T##: $$\mathcal{A}(E)=\{T\in \operatorname{Lin}(E;F_T) : \text{$T$ surjective and $\dim_{\mathbb{R}}F_T < +\infty$}\}.$$ A cylinder set measure on ##E## is a collection of probability measures $$\{\mu_T : T\in \mathcal{A}(E)\}$$ wher ##\mu_T## is a probability measure on ##F_T##. These measures are required to satisfy the following consistency condition: if ##\pi_{ST}: F_S\to F_T## is a surjective projection, then the pushforward of the measure is as follows: $$\mu_T = (\pi_{ST})_\ast (\mu_S)$$

Now not only I'm having trouble to understand the motivation behind Gross' definition, I'm also really failing to see how it connects to these other two definitions.

For instance, in Gross' definition I really can't see why one should work with the dual of subspaces of the dual of the original space. This process of taking two duals is looking very weird to me to tell the truth. After all, in general we can map the space to the double dual, so why can't we simply realize all of those subspaces inside of itself?

So, my doubts are:

- How can we really understand the motivation behind Gross' definition? Why define cylinder sets and cylinder set measures as he does? Why first work with finite dimensional subspaces of the dual, and why after with work with the dual of these finite dimensional subspaces?
- How Gross' definition of cylinder sets connects with the definition of cylinder set we find on Wikipedia's page? I can't see how to bridge those at all.
- How Gross' definition of cylinder set measures connect with the definition of cylinder set measure on Wikipedia's page also? I somehow believe that in some sense the finite dimensional vector spaces ##F_T## from Wikipedia's definition are just the ##K^\ast## from Gross definition and that the projections ##T## are the ##\pi_K##. Still I can't see how to make this precise and why on Earth one would not simply work with subspaces of the original space instead of taking two duals.