Continuity sets as generator of the $\sigma$-algebra generated by cylinders

On $(\mathbb{R}, \mathcal{B})$ given any finite measure $\mu$ the sets of the form (continuity sets) $$\{A \in \mathcal{B} : \mu(\partial A) = 0\}$$ generate the Borel $\sigma$-algebra $\mathcal{B}$. The same is true in $\mathbb{R}^n$ with some measure $\mu$ in $(\mathbb{R}^n, \mathcal{B}_{\mathbb{R}^n})$. "Decomposition of Multivariate Probability", Roger Cuppens or https://math.stackexchange.com/questions/1634436/sigma-field-generated-by-the-continuity-sets-of-a-measure/1634839.

I'm wonder if this is just an easy result:

Consider $(\mathbb{R}^\infty, \mathcal{B})$, where $\mathcal{B}$ is the $\sigma$-algebra generated by the product topology, which agree with the one generated by cylindrical sets: Given some finite measure $\mu$ on $(\mathbb{R}^\infty, \mathcal{B})$ is enough to use cylindrical based on continuity sets? i.e: by sets of the form $$\{(x_k): \pi_n((x_k)) \in A, n >0, \mu_n(\partial A)=0, A \in \mathcal{B}_n\}$$ where $\pi_n$ projects $(x_1, x_2, x_3, \cdots )$ to $(x_1, x_2, \cdots, x_n)$ and $\mu_n$ is the push of $\mu$ through $\pi_n$?.

Similar idea would work for $X^{[0,1]}$, when $X$ is a compact metric space with a finite measure defined on the cylindrical $\sigma$-algebra?

I feel the argument given in the answer linked above is enough, since this set will contain a base for the topology of the projections. I haven't seen this anywhere and it seems quite useful, is there any reason? any reference actually using this?

Bye.

• There is some typo above, in your description of cylinder sets, $\mu_n(\partial A)$ is not a condition (yet). Also, I'm confused about what you mean by $X^{[0,1]}$, which is not even metrizable under the obvious interpretation (all functions from $[0,1]$ to $X$, endowed with the product topology) if $X$ is nondegenerate. Commented Oct 31, 2016 at 9:43
• @MichaelGreinecker $\mu_n$ is defined from $\mu$ using the projections to $\mathbb{R}^n$, so given $\mu$ on the cylindrical $\sigma$-algebra, such condition make sense. It is possible to see $\mu_n$ as a measure on $\mathbb{R}^n$ or as a measure on $\mathbb{R}^\infty$ restricted to $\sigma(\pi_n)$. In the second case, the usual cylindrical $\sigma$-algebra is only a sub $\sigma$-algebra of the one generated by the product topology, I'm interested in checking if the former is constructable using the "continuity sets" as base of the cyliders. Commented Oct 31, 2016 at 12:41
• There is still a plain typo the way you write your cylinder sets; I presume you want to have $\mu_n(\partial A)=0$. Since continuity-sets are defined in terms of a topology, you would still have to specify which topology to put on $X^{[0,1]}$. Commented Oct 31, 2016 at 14:16
• @MichaelGreinecker Oh, yes, thank you. Commented Oct 31, 2016 at 14:18
• @MichaelGreinecker Is it not enough to have a topology (hence the Borel $\sigma$-algebra) on $X$ to be able to define the cylindrical $\sigma$-algrebra? Having the product topology on $X^{[0,1]}$ would allow us to construct a bigger $\sigma$-algebra no?, I'm so far only interested in the former case. Commented Oct 31, 2016 at 14:49

Fix a finite measure $\mu$ on $\mathbb{R}^\infty$. For each $n\in\mathbb{N}$, there are at most countably many $x\in\mathbb{R}$ such that $$\mu\big(\mathbb{R}^{n-1}\times\{x\}\times\mathbb{R}\times\mathbb{R}\times\ldots\big)>0.$$ So for each $n$ there is a countable dense set $D_n$ such that $$\mu\big(\mathbb{R}^{n-1}\times\{x\}\times\mathbb{R}\times\mathbb{R}\times\ldots\big)=0$$ for all $x\in D_n$. Now the Borel-$\sigma$-algebra on $\mathbb{R}^\infty$ is generated by cylinder sets of the form $$\bigcap_{n\in F\\ a_n\in D_n\\ b_n\in D_n}\pi_n^{-1}\big([a_n,b_n]\big)$$ for some finite set $F\subseteq\mathbb{N}$. These cylinder set are all $\mu$-continuity sets, are closed under finite intersections, and there are only countably many of them.
It is also possible to metrize $\mathbb{R}^\infty$ by a metric $\rho$ given by $$\rho\big(\langle x_n\rangle,\langle y_n\rangle\big)=\sum_{k=1}^\infty 1/2^k\frac{d(x_k,y_k)}{1+d(x_k,y_k)}$$ with $d(a,b)=|b-a|$. Fix a countable dense set $D\subseteq\mathbb{R}$. For each $x$, there are only countably many $r$ such that the open ball $B_r(x)$ is not a continuity set. Let $G_x$ be a countable dense set of radii. Then the family of finite intersections of Balls $B_r(x)$ with $x\in D$ and $r\in G_x$ forms a countable family of $\mu$-continuity sets that generates the Borel $\sigma$-algebra and is closed under finite intersections. Instead of $\mathbb{R}$, one can use any separable metric space.