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,
, To see this, we fix a sequence (? n ) n?N of positive numbers tending to 0. Next, we define an increasing sequence of sets (C n ) n?N satisfying µ(C n ) y for every n ? N as follows. Set C 0 = ? and observe that by the definition of ?, for each i 1 there exists a measurable set C i such that C i?1 ? C i ? B and µ(C i ) ?(C i?1 , B, y) ? ? i, ? J are measurable sets satisfying µ(B 1 ) y µ(B 2 ), we define ?
Since x µ(A) < ?, we thus know that there exists a measurable set C 1 ? A such that µ(C 1 ) = ?(C 1 , A, x) x. Further, as µ(A) ? x µ(A \ C 1 ) < ?, we also know that there exists a measurable set C 2 ? A \ C 1 such that µ(C 2 ) = ?(C 2 , A \ C 1 , µ(A) ? x) µ(A) ? x. As it turns out, the set S = A \ (C 1 ? C 2 ) is an atom unless it has measure 0, This upper bound on the supremum ?(C, B, y) is in particular reached by C, so µ(C) = ? ,
, + µ(C 1 ? T ) while on the other hand A is the disjoint union of