III.1 and III.2 The Axioms of Choice and Determinacy, Banach Spaces and III.3 Bayesian Analysis: Pages 158 – 159.

The Axiom of Choice states that given a family of non-empty sets Xi where i ranges over an index set I then there is a function, f, called the choice function, defined on I such that f(i) Î Xi for all i.  For a finite number of sets the existence of the choice function follows by induction.  It turns out that for an infinite number of sets one cannot deduce the existence of the choice function from the usual rules for building sets.

When the choice function is used in a proof then that part of the proof is nonconstructive.

The proof of the Banach-Tarski Paradox requires the axiom of choice.  It shows that there is a way of dividing up the unit sphere into a finite number of subsets and then recombining these subsets, using rotations, reflections and translations, to form two solid unit spheres.  The proof is nonconstructive.  It can be shown that some of these sets are not measurable and this explains the apparent disconnect with the volume aspect of the paradox.

Two aspects of the axiom of choice are: the well-ordering principle and Zorn’s lemma.  The well ordering principle states that every set can be well ordered.  Zorn’s lemma states that under certain circumstances maximal elements exist.  Zorn’s lemma can be used to show that every vector space has a basis.  These two forms are equivalent to the axiom of choice.

In formal set theory how the axiom of choice is related to the other set theory axioms is studied.

In a two player game the game is called determined of one of the two players has a winning strategy.  Using the axiom of choice on can construct nondetermined games.

The axiom of determinacy states that all games are determined.  It contradicts the axiom of choice.  The axiom of determinacy can be added to the Zermelo-fraenkel axioms without choice.  It implies that many sets of reals have good properties such as being Lebesgue measurable.  Variants of the axiom of determinacy are connected with the theory of large cardinals.

Banach spaces will be considered under the normed spaces and Banach spaces category.

Conditional probability is conditioned on some event.  P(A|B) = P(A ,B)/P(B).  A simple argument gives Bayes Theorem: P(A|B) = P(B|A)P(A)/P(B).  This gives the conditional probability of A given B in terms of the conditional probability of B given A.

In statistics random data may be given by an unknown probability distribution.

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