|
In mathematics, the support (sometimes topological support or spectrum) of a measure ''μ'' 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. ==Motivation== A (non-negative) measure ''μ'' on a measurable space (''X'', Σ) is really a function ''μ'' : Σ → (). Therefore, in terms of the usual definition of support, the support of ''μ'' is a subset of the σ-algebra Σ: :''p'', but it would definitely not work for ''λ'': since the Lebesgue measure of any point is zero, this definition would give ''λ'' empty support. # By comparison with the notion of strict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure: ::: :(or the closure of this). It is also too simplistic: by taking ''N''''x'' = ''X'' for all points ''x'' ∈ ''X'', this would make the support of every measure except the zero measure the whole of ''X''. However, the idea of "local strict positivity" is not too far from a workable definition: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Support (measure theory)」の詳細全文を読む スポンサード リンク
|