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

Question

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.

Answer 1

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.

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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.