Abstract
The following result is due to H. Steinhaus [20]: “If A,B⊂R are sets of positive inner Lebesgue measure and if the function f: R x R→R is defined by f(x,y):=x+y (x,yɛR), then the interior of f(A x B) is non void”. In this note there is proved, that the theorem of H. Steinhaus remains valid, if
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(1)
R is replaced by certain topological measure spaces X, Y and a Hausdorff space Z,
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(2)
f is a continuous function from an open set T⊂X x Y into Z and satisfies a special local (respectively global) solvability condition in T,
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(3)
A⊂X is a set of positive outer measure, B⊂Y contains a set of positive measure and A x B⊂T.
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Sander, W. Verallgemeinerungen eines Satzes von H. Steinhaus. Manuscripta Math 18, 25–42 (1976). https://doi.org/10.1007/BF01170533
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DOI: https://doi.org/10.1007/BF01170533