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On the regularity for solutions to H-systems by Musina R.

By Musina R.

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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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