Support (Measure Theory)
Topological Space, Neighbourhood (Mathematics), Lebesgue Measure, Dirac Measure, Strictly Positive Measure
Herausgegeben: Aloysius, Dewayne Rocky
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In mathematics, the support (sometimes topological support or spectrum) of a measure on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.A (non-negative) measure on a measurable space (X, ) is really a function : [0, + ]. Therefore, in terms of the usual definition of support, the support of is a subset of the -algebra : \mathrm{supp} (\mu) := \overline{\{ A \in \Sigma \mu (A) 0 \}}.However, this defi...
In mathematics, the support (sometimes topological support or spectrum) of a measure on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.A (non-negative) measure on a measurable space (X, ) is really a function : [0, + ]. Therefore, in terms of the usual definition of support, the support of is a subset of the -algebra : \mathrm{supp} (\mu) := \overline{\{ A \in \Sigma \mu (A) 0 \}}.However, this definition is somewhat unsatisfactory: we do not even have a topology on ! What we really want to know is where in the space X the measure is non-zero.
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