MathDB

suppose that

S_{\mathbb N}

is the set of all permutations of natural numbers. finite permutations are a subset of

S_{\mathbb N}

that behave like the identity permutation from somewhere. in other words bijective functions like

\pi: \mathbb N \longrightarrow \mathbb N

that only for finite natural numbers

i

\pi(i)\neq i

. prove that we cannot put probability measure that is countably additive on

\wp(S_{\mathbb N})

(family of all the subsets of

S_{\mathbb N}

) that is invarient under finite permutations.

Problem Statement