|
In mathematics, the regularity theorem for Lebesgue measure is a result in measure theory that states that Lebesgue measure on the real line is a regular measure. Informally speaking, this means that every Lebesgue-measurable subset of the real line is "approximately open" and "approximately closed". ==Statement of the theorem== Lebesgue measure on the real line, R, is a regular measure. That is, for all Lebesgue-measurable subsets ''A'' of R, and ''ε'' > 0, there exist subsets ''C'' and ''U'' of R such that * ''C'' is closed; and * ''U'' is open; and * ''C'' ⊆ ''A'' ⊆ ''U''; and * the Lebesgue measure of ''U'' \ ''C'' is strictly less than ''ε''. Moreover, if ''A'' has finite Lebesgue measure, then ''C'' can be chosen to be compact (i.e. – by the Heine–Borel theorem – closed and bounded). 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「Regularity theorem for Lebesgue measure」の詳細全文を読む スポンサード リンク
|