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Additional resources for 4-Manifold topology II: Dwyers filtration and surgery kernels
Therefore we have 4o (A) = O (A-), and the is This condition is essential, as the following example shows. Let A be the set of the elements ak(k = 1, 2, ... ) and let W be the or-field of all subsets of A. ; if X is an infinite of A, then let -,o, (X) be the number of elements ak in X with k subset of A, then let w,(X) _ +-. )) is monotone increasing and V(X) = lim cp,(X) - 0 if X is finite, or - + - if X is infinite; and we,have so. (A) _ -1- cc ,o(A)=0. -Then since a ,-(A) is finite, y,,(A;) is also finite.
Bm . ) _ ,p(SBm). ). Am). 22. 22. In order that the set A e 932° be a zero-set for p, it is necessary and suffi- cient that A(=,) A. Necessrrr: Let A =,. A (==,)B. Then we have A - B(_,) A; hence there is a C e 9J2, such that A - B C; C and C =, A, and thus also C(--,) A. Setting B + C = A', we have A' e 91 and A C; A'; thus it is enough to prove that A' =, A. 31, from B(=,)A and C(=,)A it follows that B + C = A'(=,)A. 31 from A'(=,)A and X(=,)X it follows that A'X = X (=,)AX ; thus according to (4) : cp(X) = sa°(AX) ; since AX a D2°, it follows from A =, A that p°(AX) = 0; thus p(X) = 0 for X A A'19, that is, A' =, A.
_, Lim Bm, Lim Am m m m m m m m Lim B,, and, if Lim Am, Lim B. exist, also Lim A, =, Lim B.. 44. If w is additive in the field M, if A e 9)l, B e 9)t, and A =, B, then p(A) ,o(B), p+(A) = o+(B), (A, = V(B), O(:1) = O(B) From A = AB + (A - B), B = AB + (B - A) it follows that (p(A) _ ,p(AB) + p(A - B), ,p(B) = to(AB) + tip(B - A) ; thus since fp(A - B) = 0 and p(B - A) = 0, we have: rp(A) = p(B) _ p(AB). 31. 45. If v is totally additive in the a -field 91, if A e 9)2, and if A = A+ + Aand A = A* + A** are two deeonmpositions of A into disjoint sets of 912, such that +(_4) _ p+(A**) = 0, 0-(A) =' o7(A*) = 0, then A* A+ and A** =, A-.