Locally Compact and Euclidean Spaces
Except in the examples, the set S on which the measures have been defined (more precisely, on certain subsets of which it has been defined) has been an abstract set, devoid of any special structure. In the particular case in which S has additionally the structure of a topological space, e.g., when S is euclidean space, it is natural to consider the relations between the topological and measure-theoretic features of S. For example, elementary analysis suggests that continuous functions should be measurable; this depends, however, on the existence of a suitable relationship between the measure and the topology.
KeywordsEuclidean Space Lebesgue Measure Compact Space Compact Hausdorff Space Finite Linear Combination
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