Definitions of Cylinder Sets and Cylinder Set Measure

[leo.]

I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper 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,

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, $$\bigcap_{i=1}^n p_{\alpha_i}^{-1}(A_{\alpha_i})=\{(x_\alpha)\in X | x_{\alpha_1}\in A_{\alpha_1},\dots, x_{\alpha_n}\in A_{\alpha_n}\}$$

The second is the definition of a cylinder set measure,

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:

1. 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?
2. 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.
3. 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.

Any help in understanding better these definitions is highly appreciated!

 

[andrewkirk]

I don't think I could understand Gross's definition without putting a great deal of work into it, either.

But I notice that you say the paper is the one that introduced the field. If so then it is very likely a bad place to learn from. Seminal papers like that are usually hard to understand, because the author has only just worked out the concepts themself, and has not had an opportunity to work out the best way to clearly explain them.

Consider Newton's writings on calculus, which are almost completely opaque, and the early writings on relativity and quantum mechanics, which did not have the benefit of the modern highly structured and logical concepts and notation (spacetime as a 4D manifold, and Hilbert spaces supported by Dirac's bra-ket notation respectively). I am told that Godel's original presentation of his incompleteness theorems is much harder to understand than the modern version I read.

The paper you linked appears - based on the references - to be dated in the late 1950s. Notation and concepts have moved on a lot in measure and probability theory since then. I think a modern text on the topic would be much easier to understand. It may even be that some of the concepts he uses, such as using duals of spaces, have turned out to be unnecessary and are no longer used in modern presentations.

 

[soloenergy]

Hi there,

I understand your confusion with the different definitions of cylinder sets and cylinder set measures. Let me try to explain the motivation behind Gross' definition and how it connects with the definitions on Wikipedia's page.

First, let's consider the definition of a cylinder set. It is essentially a set that is formed by taking a finite intersection of preimages of canonical projections in a Cartesian product space. This definition is useful in many contexts, but it may not be the most general or abstract definition. Gross' definition aims to provide a more general framework for cylinder sets, which can be applied to various types of spaces, not just Cartesian product spaces.

Now, let's look at Gross' definition of cylinder sets and cylinder set measures. The motivation behind this definition lies in the fact that these objects are closely related to Gaussian measures. In fact, Gaussian measures on abstract Wiener spaces can be defined as cylinder set measures. This is why Gross starts by defining cylinder sets as preimages of Borel sets in finite-dimensional subspaces of the dual space.

You may be wondering why Gross works with the dual space instead of the original space. The reason for this is that Gaussian measures are typically defined on the dual space, and by working with the dual space, we can easily extend our results to Gaussian measures. Additionally, by working with finite-dimensional subspaces of the dual space, we can capture the idea of "approximating" a Gaussian measure on the original space with simpler measures on these subspaces.

Now, how does Gross' definition of cylinder sets connect with the definitions on Wikipedia's page? As you mentioned, the finite-dimensional vector spaces $F_T$ in Wikipedia's definition are indeed the same as the $K^\ast$ in Gross' definition. The projections $T$ in Wikipedia's definition are also related to the linear maps $\pi_K$ in Gross' definition. However, there is a key difference between the two definitions: Wikipedia's definition is specific to separable, real, topological vector spaces, while Gross' definition is more general and can be applied to any locally convex real linear space.

Finally, let's consider the connection between Gross' definition of cylinder set measures and Wikipedia's definition. As you mentioned, the finite-dimensional vector spaces $F_T$ in Wikipedia's definition are the same as the $K^\ast$ in Gross' definition. However, the consistency condition in Wikipedia's definition is what sets it apart from Gross' definition. This condition ensures