Capacitylike set functions and upper envelopes of measures

Upper envelopes of inner premeasures

“…= (6T©)_L. In this case there is also the opposite relation © C (IT'DJL, which in [1] occurs in the overall assumptions. It can thus be said that the added assumption © C (TTI)!.…”

“…However, it is an essential point that we removed the overall assumptions in [1] that © be stable under countable intersections and that f3 be a continuous at 0. On the one hand this produces no loss, because after the fundamentals in MI 6.31 an inner -A-premeasure which for some • == ar is • continuous at 0 proves to be an inner • premeasure.…”

Upper envelopes of inner premeasures

“…< oo then there exists a Radon premeasure (p : Comp(X) -> [0,oo[ such that ^p ^ f3 and supy? = sup (3. Then Adamski [1] transferred the theorem to the frame of abstract measures. He assumed certain pairs of lattices © and 1 of subsets in an abstract set X to take the place of Comp(X) and Op(X).…”

“…Now the present paper wants to show that the frame of MI leads to even more fortified and simpler forms of the results. One of the simplifications is that Adamski [1] assumed G to be stable under countable intersections and the initial set function f3 : © ->• [0, oo [ to be a continuous at 0, which at once leads to the level of measures, whereas we shall see that the adequate level is the so-called finitely additive one, that is the level of contents. Also we shall compare our basic result with the main theorem of MI Section 18, at that place called the extended Henry-Lembcke-Bachman-Sultan-Lipecki-Adamski theorem, this time after Adamski [2].…”

“…The present section combines the basic consequence 3.7 with the ideas of Adamski [1] Lemma 3.2(b)(c) in order to obtain an extended version. We use the ideas in question as expressed in the next two lemmata.…”