Talk:Cylinder set

Remove example

I'm removing this example, added by anon some months ago:

The n-dimensional cylinder sets of C[0,1] are defined by

{\mathcal {C}}_{n}=\{x\in C[0,1]:x_{t_{1}}\in A_{1},\ldots x_{t_{n}}\in A_{n}\}

where the $A_{i}$ for i from 1 to n are Borel subsets of R. The cylinder sets of C[0,1] are then defined by the union

{\bar {\mathcal {C}}}=\bigcup _{1}^{\infty }{\mathcal {C}}_{n}

The σ algebra generated by the cylinder sets is defined to be the intersection of all σ algebras over C[0,1] which contains ${\bar {\mathcal {C}}}$ . This σ algebra ${\mathcal {C}}$ is frequently considered as the σ algebra of C[0,1] functions and is important in the development of the theory of continuous stochastic processes.

I'm removing it because I can't figure out what its trying to say. I think C[0,1] is supposted to be the set of all discrete-time stochastic processes, or something like that; not sure. The union also does not make sense. linas 21:18, 27 October 2006 (UTC)[reply]

definition question

On this page I read,

Consider the cartesian product $X=\prod _{\alpha }X_{\alpha }$ of topological spaces $X_{\alpha }$ , indexed by some index $\alpha$ . The canonical projection is the function $p_{\alpha }:X\to X_{\alpha }$ that selects out the $\alpha$ component of the product. Then, given any open set $U\subset X_{\alpha }$ , the preimage $p_{\alpha }^{-1}(U)$ is called an open cylinder.

Where the definition of open set is members of a topology. So is any collection of these open sets (i.e. U) also an open set? is there some property U must have to be an open set? Is it not the same definition as I stated? PDBailey (talk) 02:51, 22 January 2009‎ aka O18 (talk)

Each topological space

X_{\alpha }

defines its own idea of what an open set is. Thus,

U\subset X_{\alpha }

is open with respect to that definition. That's all; nothing more. Well, there's a bit more: the open cylinder

p_{\alpha }^{-1}(U)

is an open set in the cartesian product, and then the usual rules of topology apply: any finite intersection of open sets is open; any countable union of open sets is open. 67.198.37.16 (talk) 19:02, 11 September 2020 (UTC)[reply]

Cylinder sets - topology vs measure theory

The article is focused on cylinder sets in a product topology, while they play an important role also in a product sigma-algebra, at least in the countable case. I think the article should deal with the two concepts equally. Markov Odometer (talk) 06:18, 22 March 2018 (UTC)[reply]

Why? There are already extensive articles on cylindrical σ-algebra and cylinder set measure which treat those topics; I don't think that they should be merged into this. 67.198.37.16 (talk) 19:08, 11 September 2020 (UTC)[reply]

Cylinder sets on Hilbert spaces

For the TODO List: The article abstract Wiener space provides a definition of cylinder sets on Hilbert spaces that is slightly more general than the definitions in this article. Articulation would be beneficial.

Also, this article should start not with the general definition, but with a simple motivating example. 67.198.37.16 (talk) 18:16, 29 January 2024 (UTC)[reply]