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In mathematics, two positive (or signed or complex) measures ''μ'' and ''ν'' defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets ''A'' and ''B'' in Σ whose union is Ω such that ''μ'' is zero on all measurable subsets of ''B'' while ''ν'' is zero on all measurable subsets of ''A''. This is denoted by A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. ==Examples on R''n''== As a particular case, a measure defined on the Euclidean space R''n'' is called ''singular'', if it is singular in respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure. The Heaviside step function on the real line, : has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set not containing 0, then but . Example. A singular continuous measure. The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous. 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「singular measure」の詳細全文を読む スポンサード リンク
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