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15 (1930), p. 176. 5. Theorems about limits. 1. , D. +, A m ; then D. J. , ; thus because of D. ,) p(Am). ). 11. ) ? lim v(AA). A We set A = Lim A.. , S. = A. +I + in ; then S. 21, we have 9(A) = lim tp(S A therefore, since (v(SA) k p(A A), we get v(A) urn cp(AA). 8. Example: Let SDI be the system of all subsets of a non-countable set E and let a(A) be the number of elements of A. ) are finite), is shown by the example of footnote 5, p. 1; for we then have , (Lim la. l) - m(A) -° 0,sr(lam1) - 1.

A; for otherwise we would have Lun X. = (a I. 4, {a} e 912, contradictory to the assumption. But since Lim X. ' a 9)2; and, since a is a continuity-point,,p({ a}) = 0 and we get p(X4,) to I V(Xm) I == p. ,p(Xm), so that I y (X;) I z p. 2 lim cp(X,) = p({a}) = 0. 61. 6, we can substitute O(X) < p for the condition I ,c(X) I < p. BIBLIOGRAPHY. The notion of the continuous set functions is due to J. RADON, Sitzungs- bericbte Akad. Wise. Wien 122 (1913), p. 6, for which another proof was given by W.

BIBLIOGRAPHY. The notion of the continuous set functions is due to J. RADON, Sitzungs- bericbte Akad. Wise. Wien 122 (1913), p. 6, for which another proof was given by W. BIBaPIfvsal, Ann. Soc. Polon. Math. 7 (1928), p. 76; cf. also E. SzPILRAJN, Fund. math. 22 (1934), p. 309. Besides, as to No. 3, we mention especially H. HAHN (1], p. 408; also R. S. PHILLIPS, Bull. Amer. Math. Soc. 46 (1940), p. 274; A. SOBCZYK and P. C. HAi[xsR, Duke Math. Journ. 11 (1944), p. 839. 4. Atoms. Again let 9)2 be a o-field and let p be totally additive in W't.