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In mathematics, a regulated function (or ruled function) is a "well-behaved" function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Georg Aumann in 1954; the corresponding regulated integral was promoted by the Bourbaki group, including Jean Dieudonné. ==Definition== Let ''X'' be a Banach space with norm || - ||''X''. A function ''f'' : (''T'' ) → ''X'' is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true : * for every ''t'' in the interval (''T'' ), both the left and right limits ''f''(''t''−) and ''f''(''t''+) exist in ''X'' (apart from, obviously, ''f''(0−) and ''f''(''T''+)); * there exists a sequence of step functions ''φ''''n'' : (''T'' ) → ''X'' converging uniformly to ''f'' (i.e. with respect to the supremum norm || - ||∞). It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways: * for every ''δ'' > 0, there is some step function ''φ''''δ'' : (''T'' ) → ''X'' such that :: * ''f'' lies in the closure of the space Step((''T'' ); ''X'') of all step functions from (''T'' ) into ''X'' (taking closure with respect to the supremum norm in the space B((''T'' ); ''X'') of all bounded functions from (''T'' ) into ''X''). 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「regulated function」の詳細全文を読む スポンサード リンク
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