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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
Full Text:
##### References:
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