By Flemming Topsøe (auth.)
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Additional info for Topology and Measure
If this were not so, we would be able to find a > 0 such that to any K there would exist G ~ K with ~(G) < ~(X)-a. In other terms, we could find a family (GK)KE ~ ( X ) such that GK ~ K for each K and such that lim ~ ( ~ Gb~~) > e for each K. Then ~a(CGK) ~ ¢, eventual~y holds for each K. ,Kn, we have min ~ ( C G K ) > s, eventually. ,n v -This contradiction proves the assertion. 3). If we ask the corresponding question for subsets of ~+(X;t), much less is known.
2. e. ~ (X)). ths_____n_n ~ exists, and ~ is given by the formula ~(A) = sup Inf ~G; KcA G~K A C ~ (X). e. 6) ~(A) = Inf sup ~K; A E ~ (X). ::~. Kc_g In (1) it is the measures of the compact sets we compute by approximation from outside. 5) for all subsets of X then we obtain the formula for ~e" "Dual" remarks applies to (ii). Proo f • (i) is a special case of Theorem 2, . (il) : With the given set-functlon ~ we associate ~. a set-function ~ (x)~+ defined by v ~s=sup~; Kc_S s¢ (x). It is not difficult to see that ~ is finite (as indicated), monotone, additive and subadditlve.
Thus there exists a continuous h with O