Abstract Measure Page: 21
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SI
Using (A C) as a *test* set of m#-ae&surabllityt
io«(A - C) - m*((A - G) n B) + m*((A - C) /i B*),
Since A n c* = <A - C), it follows by substitution that
m*(A) = #(A ^ C) (A C) B) ■#* i^( (A • C) /") B1)
= a«(A C) -mt (A n (8 - C)) •+ M«<(A - C) n B )
^ *(A n (B - C)) + m(A n (2 ~ G)#).
By Z,B&P (B ~ C) is a®-®easurable*
2.26. Theorera. |f ffl*is an oufrey measure defined oa
H, then the union & finite number o£ immeasurable SSM
is
Proof. Let and Bg be m*-measurable sets in H. fhen
m*(A) r m(& n B1) + a# (A n B|) and
IB#(A - B^) = ®«{{A - B^) n Bg) + b*((A - B^) n Bp,
where A is any set ©f 1# It is necessary to prove that
a (A) = ■ <* n <BX c Bg)) + b (A n (Bx c Bg)*).
From the initial statements,
a U) = m*(A n Bx) + s ((A - B^) /i ig)
+ «<<A • Bj) n B|).
How
((A - Bx) n Bg) C (A n Bx) = A /I (B1 C* Bg).
Since m® is subadditive,
m*(A) m«(A n U Bg)) + m ((A - Bx) n 1|)
= «*(A n {31 u Bg)) + **<A n <BX u Bg)*)*
It follows that (Bx U Bg) is ra*-measurable* The proof
follows by induction.
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Bridges, Robert Miller. Abstract Measure, thesis, 1957; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc107950/m1/24/: accessed May 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .