Theory of Statistics - George Mason University

638 0 Statistical Mathematics
Ordering the Reals
The field of real numbers IR is complete and Archimedean ordered; in fact,
the reals can be defined as a (any) complete Archimedean ordered field. The
ordering properties correspond to our intuitive understanding of ordering.
The more useful ordering of the reals is the linear ordering that results
from the usual inequality relation. On the other hand, IR d cannot be linearly
ordered in an intuitive or universally useful way. An ordering of x, y ∈ IR d
based on notions of x < y or x ≤ y are rarely useful in statistical applications
(see Section 0.0.1.4 on page 614). Most useful orderings are based on relations
between x and y, but their usefulness depends on the application. (See
Gentle (2009), pages 538 to 549, for some discussion of this.)
In order to simplify the following discussion, we will focus on subsets of
IR. Because the reals are not a well-ordered set using the usual inequality
relation, we find it convenient to define limiting maxima and minima based
on that relation.
For X ⊆ IR, the supremum of X or least upper bound of X, is the number
defined by
and
x ∗ = sup(X), (0.0.36)
∀x ∈ X, x ≤ x ∗
∀ɛ > 0, ∃x ∈ X ∋ x > x ∗ − ɛ.
The infimum of X, that is, the greatest lower bound of X, is written as
and is defined similarly, with the inequalities reversed.
Examples:
x∗ = inf(X) (0.0.37)
• Let A = {x}. Then sup(A) = inf(A) = x.
• Let A = {x | x = 1/i, i ∈ ZZ+}. Then sup(A) = 1 and inf(A) = 0. Notice
that inf(A) /∈ A.
• Let A = {x | i, i ∈ ZZ+}. Then sup(A) = ∞ and inf(A) = 1. Alternatively,
we may say that sup(A) does not exist. In any event, sup(A) /∈ A.
An important fundamental property of the reals is that every bounded
set of reals has a supremum that is a real number. This property is often
called Dedekind completeness. In the usual axiomatic development of the reals,
Dedekind completeness is an axiom.
The maximum of a well-ordered set is the largest element of the set, if it
exists; likewise, the minimum of a well-ordered set is the smallest element of
the set, if it exists. The maximum and/or the minimum may not exist if the
set has an infinite number of elements. This can happen in two ways: one, the
set may have no bound; and another, the bound may not be in the set.
Theory of Statistics c○2000–2013 James E. Gentle