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Generalized Smital’s lemma and a theorem of Steinhaus. (English) Zbl 0587.28008
Smital’s lemma states that for any set $$E\subset R$$ of positive outer Lebesgue measure and any dense subset D of R the inner Lebesgue measure of the complement $$R\setminus (E+D)$$ is equal to zero. This is generalized in two directions simultaneously: we replace the usual addition in R by a fairly general two-place function on a separable metric space and the Lebesgue measure by an abstract one. A topological analogue and a theorem on the images of Lebesgue density points are also presented.

##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28A12 Contents, measures, outer measures, capacities