Let S be the set of all functions from N to N Prove that S

Let S be the set of all functions from N to N. Prove that |S| = c.

Solution

let N denote the set of natural numbers and c be the collections of subsets in N where Nbelongs to c. we define a finitely additive probability charge c to be a function where k:c->[0,1]. which takes on the value 1 at N and has the property for finite collection of disjoint subsets G1,G2...

let S:N->N be the shift function which sends n to n+1 and c is shift invariant .Let Sc denote the set of finitely additive probability charges on c. we have that mod of S will contain all elements which belong to c. we have the collection of all subsets of N is shift invariant .we can extend Sc to an element in S. we hace c is the collection of subsets of N with N belonging to c and it can be extended to a probability charge on N there for we have that mod S=c