3

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!

Community
  • 1
Dontknowanything
  • 2,810

1 Answers1

2

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.

user359811
  • 46
  • 1
    I dont think i understand it completely.What is the scalar multiplication here? and Could you give example of countably infinite Boolean Algebra? – Dontknowanything Aug 07 '16 at 09:06
  • 1
    The scalar multiplication is $0 \cdot x = 0$ and $1 \cdot x = x$. For a countably infinite Boolean algebra, you can take the finite subsets of $\mathbb{N}$ and their complements. – user359811 Aug 07 '16 at 16:08