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Set Functions by Hans Hahn, Arthur Rosenthal

By Hans Hahn, Arthur Rosenthal

1948 HARDCOVER EX-LIBRARY replica OF vintage GRADUATE research textual content through one of many FOUNDERS of recent research AND TRANSLATED by means of ARTHUR ROSENTHAL;LIBRARY name quantity AND CARD ON inside of entrance COVER;OTHERWISE NO obvious DEFECTS OF ANY type! nice COLLECTIBLE!

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Hence inc S g, or m c . 81. If p is totally additive and finite$ in the a-field fit, then in fit there are only eountably many disjoint sets with 9(X) , 0. 8. BTBLioORAPHY: The notions of positive-, negative-, absolute-function are due to H. LnBasoun, Ann. Ec. Norm. (3) 27 (1910), p. 381. Furthermore Of. J. RADON, Sitsungsberichte Akad. Wise. Wien 122 (1913), p. 71 is due to if. HAHN [11, p. 404; other proofs for it were given by R. FRANCK, Fund. math. 5 (1924), p. 262; W. StsRPIhSKI, Fund. math.

So**- _ (,)** _ v*. 1) it follows immediately that a,*P* = am+. 3), (Sc+)** = ao. 44. 45. P-)*; c** _ (,D)** _ (m+)** + (w-)**. 71. 35 we have a*(M) = af(M+) + a'(M-). 1) we have a*o(M+) = a*(M+), ai(M--) _ 8;(M-). S. If M = R + S (R a 92, S e 5) is a singular decomposition of M and if R'eM19t(orS'eMI ), then R' For since R' - R Q S, we have R' - R e TS, and thus R' - R =, A. 51. If M = R + S (R e 9i,SsS)andM=R'+8'(R's92,S'e6) are singular decompositions of M, then R = o R', S =, 8'. 5 we have R'cwR,RcwH';S'C;w8,ScoS'.

In order that A be a 9-atom, it is necessary and sufficient that A be a 0-atom. 4. 2 5. If A is a p-atom, then A is also a jp+-a1om and a jp -atwn. 4. 2 5 is not true. Example: Let AB = A, IN = { A, A, B, A + B 1, and cp(A) = 0, rp(A) = 1, 49(B) = -1, rp(A + B) = 0. Then A + B is a rp+-atom and a j -atom, but not a c-atom.

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