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](https://www.worldcat.org/title/decomposition-of-multivariate-probabilities/oclc/1094828) 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.



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


