Math probability problem, help please.?

Question

Let C denote a finite sample space. i.e., the cardinality of C is equal to n, which is an element of the set of all positive integers. I need to obtain an expression for the cardinality of the set of all sigma-fields on C.

I'm trying to do that using the application of Stirling numbers of the second kind, if possible.

thanks in advance

Answer 1

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C is a sigma algebra with n elements.

Let B = { all subsets of C, including the null set and C itself }

because C is a sigma algebra both the null set and itself are in the list of possible subsets.

the cardinality of B will be the sum of all the combinations of n choose r where 0 ≤ r ≤ n. When r = 0 you have the empty or null set and when r = n you have C itself.

the sum, Σ n! / (r! (n -r)! ) for 0 ≤ r ≤ n is a known sum that equals 2^n.

in terms of Stirling numbers, just write the sum as:

Σ S(n,r) for 0 ≤ r ≤ n

http://mathworld.wolfram.com/StirlingNumberoftheSe...