Suppose we have a sample space Ω = {1, . . . , M } with collection C := 2Ω, the
power set of Ω. To determine P, the probability measure, we employ the following
empirical procedure:
i. Collect N data points taking values in Ω (e.g., N rolls of an M -sided die).
Call these observations x1, . . . , xN .
ii. For each S ⊆ Ω, define
P(S) := number of i-values such that xi ∈ S
N .
As an example, suppose M = 2 and we flip a coin N = 10 times getting 6 heads
and 4 tails, where 1 denotes head and 2 denotes tail. Then
P(∅) = 0, P({1}) = 0.6, P({2}) = 0.4, and P({1, 2}) = 1.
If P is defined using the above procedure, is it a valid probability measure? Either
prove that it is, or give a counterexampl