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On topologizing measure spaces via differentiation bases. (English) Zbl 0175.34003
Let \((X,\mathfrak M,\mu)\) be a measure space furnished with a differentiation basis \((\mathcal D, \Rightarrow)\). Let \(x\in X\) and let \(E_\alpha\) be a net of sets of positive measure. Write \(E_\alpha\to x\) provided for each \(\alpha\), there exists \(I_\alpha\in \mathcal D$ such that \(x\in E_\alpha\subset I_\alpha\) and \(I_\alpha\Rightarrow x\). Let \(f$ be defined and summable on \(X\). Call \(f\) ``continuous'' provided for every \(x\in X\) \centerline{\( \lim_{E_\alpha\to x} \left(\int_{E_\alpha} f\, d\mu/ \mu(E_\alpha)\right) = f(x)\quad\text{ whenever }E_\alpha\to x.\)} This definition of continuity agrees with the usual notion of continuity of a function in the Euclidean spaces and on a variety of other topological measure spaces, provided the differentiation basis is appropriately compatible with the topology. Let \(\mathcal T\) be the smallest topology for which every ``continuous'' function is, in fact, continuous. The authors study this topology under a variety of hypotheses. In particular, if \((X,\mathfrak M,\mu)\) is a separable measure space, a basis \((\mathcal D, \Rightarrow)\) exists for which the topology is pseudo-metrizable and for which the class of continuous functions is sufficiently large to admit a form of Lusin's theorem.
Reviewer: A. M. Bruckner

MSC:
28-XX Measure and integration
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