Difference between Product and Cylindrical $\sigma-$ algebras?

Question

What is the difference between the product and the cylinder $\sigma-$algebra? In wikipedia says that ``the cylindrical $\sigma-$algebra or product σ-algebra is a $\sigma-$algebra often used in the study either product measure or probability measure of random variables on Banach spaces." However, there is no definition or an intuitive explanation of what differs between them or it is me that I can not explain them.

Can anybody give an intuitive example?

I have one in mind that says:

Consider a set of messages $M=\{c,s\}$, where $c$ means continue the conversation and $s$ means stop the conversation Let $H_t=(M\times M)^{t-1}$, $t=1,2,...$ be the set of all pairs of messages in $M$ possibly sent before stage $t$ and let $H_{\infty}=(M\times M)^{\mathbb{N}}$, the standard way to provide these sets with a measurable structure is the following. Let $\mathcal{H}_{t}$ be the algebra over $H_{\infty}$ generated by cylinder sets of the form $h_{t−1}\times H_{\infty}$, where $h_{t−1}$ is a sequence in $H_t$ Ht. Let $\mathcal{H}_{\infty}$ be the $\sigma-$algebra over $H_{\infty}$, generated by the algebras $\mathcal{H}_t$ $t=1,2,...$ and $\mathbb{N}=\{\{1\},\{2\},\{1,2\}\}$ describes the sets of players possibly choosing $s$ at some stage.

If the set of hisoriries in the case of two players is this $M$ which is finite the $\sigma-$ algebras that are generated are cylinder ones according to the paradigm, why? Does this depend on the set $M$? if the set was not finite but it was a set like $[0,1]$ or $[0,+\infty)$ or we had $I$-players, namely $H_t=(\underbrace{M\times M\times\dots\times M}_{\text{# of $M$'s is $I$}})^{t-1}$ would the assumption of cylinder $\sigma-$ algebras still remain?

I have seen this answer is quite good, but are the Borel $\sigma-$ algebras cylindrical ones? can we explain this intuitiveley with the above example?

Answer 1

It seems like the cylindrical $\sigma$-algebra is another name for the product $\sigma$-algebra. The product $\sigma$-algebra on $\prod_{a \in A}X_a$ is the smallest $\sigma$-algebra that makes all the projections $\pi_{a_0} \colon \prod_{a \in A}X_a \to X_{a_0}$ measurable. So it is generated by sets of the form $\pi_a^{-1}(E_a)$, where $a \in A$ and $E_a$ is a measurable subset of $X_a$.

This is in analogy with the product topology, which is the smallest topology on $\prod_{a \in A}X_a$ that makes the projections continuous. We have a result that if $A$ is countable and each $X_a$ is a second countable topological space, then the product sigma algebra equals the Borel sigma algebra on $\prod_{a \in A}X_a$. So for example, $B_{\mathbb{R}^2} = B_{\mathbb{R}} \otimes B_{\mathbb{R}}$.