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
Famous quotes containing the word proofs:
“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”
—David Hume (1711–1776)
“A man’s women folk, whatever their outward show of respect for his merit and authority, always regard him secretly as an ass, and with something akin to pity. His most gaudy sayings and doings seldom deceive them; they see the actual man within, and know him for a shallow and pathetic fellow. In this fact, perhaps, lies one of the best proofs of feminine intelligence, or, as the common phrase makes it, feminine intuition.”
—H.L. (Henry Lewis)
“Trifles light as air
Are to the jealous confirmation strong
As proofs of holy writ.”
—William Shakespeare (1564–1616)