Abstract Measure Page: 37
iii, 44 leavesView a full description of this thesis.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
$T
Considering (B • C) as a set in l(R)i
(B - C) - (B - A) U (A - C) and
5(B - C) = i(B - A) +• 1CA . C) = 0.
<2) If (A) is not finite let A =JTkn with isCAjj) * 00.
This is simply a statement of the ^finite property of a«,
Let fla/ and be sequences in S(B) such that
Ba Z> cn and m(&n - Cn> = 0* This is permissible by
CI). Let B = J{, Bn and C - Jt. Cfi. Thus B => A "=> C.
i(B - C) - i<JTt Ba . JJCn)
^ ffi(B - C ) ^ 0.
f> n n
3.8, Definition* The inner measure # Induced on H(R)
by m on & is the set function such that for A * H(R)
m^(A) = max A s B, B f S(K},J.
It is clear that % is an extended real valued, non-negative,
aonotone set function with %<$} = 0. It is also clear that
a#(A) - ® (A}.
Lemma 1. M & 6 HCB) SQt B Al ft measurable kernel of A,
1(B) - ■ <*).
Proof, Since
m#CA) = max $m(C)t 6 cz At G & S(R)j
it is elear that «<B) - m^A). Assume 5(8} << m*(A). Then
5(B) ^ cao. By the definition of m*(A) there must be a
Set C such that A => 0 = B C * S(R), and a(C) > 1(B).
But (C - B) cz (A - B) and (C - B) <^S(B). Thus, by the
Upcoming Pages
Here’s what’s next.
Search Inside
This thesis can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Thesis.
Bridges, Robert Miller. Abstract Measure, thesis, 1957; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc107950/m1/40/: accessed May 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .