Abstract Measure Page: 35
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35
M M SSL H, tfegu Sfifir, s§t A in
H(R) has ft MIS§1-
Proof, Let C be a measurable cover of A £ H(H). By
definition* A cr C 6 S(R). Also let E be a measurable cover
of (C - A) and let B - (C ~ I).
K Z3> <C - A) and B = (C - E) c= (C - <C - A)) = A.
Consider 8 if 8 <^-<A - B), G £ S(R).
G <^r (A • B) = (A - (C - E)) - A n E = (B - A*)
c E - <C - A).
Since Eiia measurable cover of (C • A> ®ad 0 e S<H)#
5<&) - G by Cb> of 3.S, Thus B is a measurable kernel of A.
Corollary, If si Is M R then £or §£L 2*1
A €■ H<B) there exists 1 Igt B c- S(B) such ttot
2£t In S(R) with A C => B ii follows that I(C - B) = 0.
Proof. Let A £ H(R) and let B be a measurable kernel
of A in S(R). If C * S(R) with A Q -=> B„ then
(C - B) & S(R) and (C - B) <cr (A - B). fh® theorem asserts
St(C — B) - 0.
3.7. SteJM* I£ m is ,g-rfMlt M H* thei* a necessary
sufttclant coition for a set A H(R) to |&
l^^as^rft^t M BSZS. SUlk sets B, C C- S(R),
B ^ A d C with 1{B - C) = 0.
Proof, Sufficiency* Assume there exist sets Bt G e S(R)
with B dAoc and S(B - C) =0. Recall that B end C are
-measurable. Let F be an arbitrary set in H<R).
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Bridges, Robert Miller. Abstract Measure, thesis, 1957; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc107950/m1/38/: accessed May 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .