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Measure-centering *ultrafilters*

"... Like pure mathematicians in general, measure theorists in the last hundred years have often used ultrafil-ters as a tool. I suppose that the first person to notice that ultrafilters have intrinsic properties expressible in terms of measure theory was Sierpiński (Sierpiński 45), who showed that if ..."

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will be directed to ways in which measure theory can display differences between different classes of

*ultrafilter*. In §538 of my book Fremlin 08, I looked at p-point*filters*, Ramsey*ultrafilters*, rapid*filters*, ‘measure-*converging*’*filters*(an idea due to Matt Foreman), and*filters*with what I call the ‘Fatou### On generalized limits and Toeplitz's algorithm

"... ABSTRACT. We study generalized limits in connection with the Toeplitz linear algorithm of convergence. Keijuiords and phrases. Ultrafilters, generalized limits, Toeplitz sequences and matrices. 1991 Mathematics Subject Classification. Primary 40C05. Secondary 40H05. The problem of assigning a sum, f ..."

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n E No A solution to the problem is given (due to the compactness of i) by the limit limn,u over a non trivial

*ultrafilter*U in N consisting of unbounded*subsets*of N. This limit can be approximated by limits over*filters*F with countable basis in N which are finer than the Frechet*filter*, coarser### H-CLOSED EXTENSIONS OF ABSOLUTES

"... ABSTRACT. Let X be a Hausdorff topological space. Denote by PX (respectively, EX) the "Hausdorff absolute " (respectively Iliadis absolute) of X, i.e., the unique extremally disconnected (respectively, extremally disconnected zero-dimensional) Hausdorff space that can be mapped onto X by a ..."

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] provide the necessary background; also see [21]. Let r(X) denote the lattice of open

*subsets*of the space X (partially ordered by set inclusion). A maximal*filter*o • on r(X) is called an open*ultrafilter*. The maximality of o • ensures that the adherence of o • (defined to be C3 (cœxV: V C o**1 - 3**of

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