I noticed that there is some similarity between the Baire property and the property of measurable:
- We say that a set $X$ has the Baire property if there is an open set $U$ such that $U\Delta X$ is meager. The collection of all sets with the Baire property turns out to be the smallest $\sigma$-algebra containing all of the open sets and the meager sets.
- In fact, we can define the collection of Lebesgue measurable sets to be the smallest $\sigma$-algebra containing the open sets and the nullsets.
My questions:
I wonder if there are some interesting results from this observation.
Is there a general rule to form a useful notion by forming the smallest $\sigma$-algebra containing all of the open (or closed) sets and the $\sigma$-ideal (or other small sets)?
The notion of measurable is induced by some value function, i.e., the outer measure. Is there a value function that can induce the notion of the Baire property?