Prologue

The length function

We denote the set of real numbers by R. Let R* denote the set of extended

real numbers. (See appendix A for details.)

Let X denote the collection of all intervals of R. If an interval I e l has

end points a and b we write it as /(a, b). By convention, the open interval

(a, a) = 0 V a G R. Let [0,+oo] := {x G R*|x 0} = [0,+00) U {+00}.

Define the function A : X — [0, 00] by

X(I(ab))'=i | k -

a

l i f a , 6 e R ,

' ' +00 if either a = —00 or b = +00 or both.

The function A, as defined above, is called the length function and has

the following properties:

Property (1): A(0) = 0.

Property (2): A(J) A(J) if I C J.

This is called the monotonicity property of A (or one says that A is

monotone) and is easy to verify.

Property (3): Let I G X be such that I = (JILi ^ where Ji fl Jj = 0 for

i 7^ j . Then

A(7) = ^ A ( J

i

) .

2 = 1

This property of A is called the finite additivity of A, or one says that A

is finitely additive.

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http://dx.doi.org/10.1090/gsm/045/01