In a answer of this question it's said that "A ($\sigma$-)algebra is a boolean algebra. In particular, it is an $\Bbb F_2$ vector space."
Unfortunately i have never seen this term Boolean Algebra.Could someone please explain me what is a Boolean Algebra and how is this a $\Bbb F_2$ vector space?
Thanks for your help!
A Boolean algebra of subsets of $X$ is a collection of subsets of $X$ that is closed under finite union, finite intersection and taking complements, and contains $\emptyset$ and $X$ itself.
(Of course, there is also an abstract definition of Boolean algebras as well.)
Any Boolean algebra becomes an $\mathbb{F}_2$-vector space: the "addition" is symmetric difference $$A + B = (A \backslash B) \cup (B \backslash A),$$ and the zero element is $\emptyset.$
The answer you linked to does not seem to be valid, though: infinite-dimensional $\mathbb{F}_2$-vector spaces do not need to be uncountable sets, and there do exist countably infinite Boolean algebras.