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In mathematics, specifically in measure theory, the trivial measure on any measurable space (''X'', Σ) is the measure ''μ'' which assigns zero measure to every measurable set: ''μ''(''A'') = 0 for all ''A'' in Σ. ==Properties of the trivial measure== Let ''μ'' denote the trivial measure on some measurable space (''X'', Σ). * A measure ''ν'' is the trivial measure ''μ'' if and only if ''ν''(''X'') = 0. * ''μ'' is an invariant measure (and hence a quasi-invariant measure) for any measurable function ''f'' : ''X'' → ''X''. Suppose that ''X'' is a topological space and that Σ is the Borel ''σ''-algebra on ''X''. * ''μ'' trivially satisfies the condition to be a regular measure. * ''μ'' is never a strictly positive measure, regardless of (''X'', Σ), since every measurable set has zero measure. * Since ''μ''(''X'') = 0, ''μ'' is always a finite measure, and hence a locally finite measure. * If ''X'' is a Hausdorff topological space with its Borel ''σ''-algebra, then ''μ'' trivially satisfies the condition to be an tight measure. Hence, ''μ'' is also a Radon measure. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on ''X''. * If ''X'' is an infinite-dimensional Banach space with its Borel ''σ''-algebra, then ''μ'' is the only measure on (''X'', Σ) that is locally finite and invariant under all translations of ''X''. See the article There is no infinite-dimensional Lebesgue measure. * If ''X'' is ''n''-dimensional Euclidean space R''n'' with its usual ''σ''-algebra and ''n''-dimensional Lebesgue measure ''λ''''n'', ''μ'' is a singular measure with respect to ''λ''''n'': simply decompose R''n'' as ''A'' = R''n'' \ and ''B'' = and observe that ''μ''(''A'') = ''λ''''n''(''B'') = 0. 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「Trivial measure」の詳細全文を読む スポンサード リンク
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