Proofs
The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom, and its interaction with the remaining two axioms. When studying axiomatic probability theory, many deep consequences follow from merely these three axioms.
In order to verify the monotonicity property, we set and, where for . It is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that
Since the left-hand side of this equation is a series of non-negative numbers, and that it converges to which is finite, we obtain both and . The second part of the statement is seen by contradiction: if then the left hand side is not less than
If then we obtain a contradiction, because the sum does not exceed which is finite. Thus, . We have shown as a byproduct of the proof of monotonicity that .
Read more about this topic: Probability Axioms
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