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 -
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
) .
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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