Next: 5.3 Monotonic Functions
Up: 5. Area
Previous: 5.1 Basic Assumptions about
Index
5.2 Further Assumptions About Area
In this section we will introduce some more assumptions about area.
The assumptions in this section can actually be proved on the
basis of the basic assumptions we have already made, and in fact
the proofs are easy (the proofs are given in appendix B).
The reason I have made assumptions out of them is that they are
as intuitively plausible as the assumptions I have already made,
and I do not have time to do everything I want to do. I am omitting
the proofs with regret because I agree with Aristotle
that
It is manifest that it is far better to make the principles finite
in number. Nay, they should be the fewest possible provided
they enable all the same results to be proved. This is what
mathematicians insist upon; for they take as principles things
finite either in kind or in number.[25, page 178]
5.14
Assumption (Addition rule for area.)
For any bounded sets
and
in

(5.15) 
and consequently
5.16
Assumption (Subadditivity of area.)
Let
, and let
,
,
,
be bounded sets in
. Then

(5.17) 
5.18
Assumption (Monotonicity of area.)
Let
be bounded sets such that
. Then
.
5.19
Definition (Zeroarea set.)
We
will call a set with zero area a
zeroarea
set.
From the normalization property it follows that every horizontal or vertical
segment
has area equal to . Thus every horizontal or vertical segment is a zeroarea
set.
5.20
Corollary (to assumption 5.18.) ^{5.1}Every subset of a zeroarea set is a zeroarea set. In particular the empty
set is
a zeroarea set.
5.21
Corollary (to assumption 5.16.) The union of a finite number of zeroarea sets is a zeroarea set.
5.22
Definition (Almost disjoint.)
We will say that two bounded subsets
of
are
almost disjoint if
is a zeroarea set.
5.23
Examples.
If
are real numbers with
, then since
the boxes
and
are almost disjoint.
Any zeroarea set is almost disjoint from every set  including itself.
5.24
Assumption (Additivity for almost disjoint sets.)
Let
be a finite
set of
bounded sets such that
and
are almost disjoint whenever
.
Then

(5.25) 
5.26
Notation (Area functions or area
)
Any real valued function
, whose domain is
the family of bounded subsets of
, and which satisfies all of
the assumptions listed in sections
5.1 and
5.2 will
be called an
area function. In these notes I will use the
names ``
'' and ``area'' to denote area function.
Thus
Next: 5.3 Monotonic Functions
Up: 5. Area
Previous: 5.1 Basic Assumptions about
Index
Ray Mayer
20070907