High Quality Content by WIKIPEDIA articles! In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonne. One must check that this limit exists and is independent of the chosen sequence, but this is an immediate consequence of the continuous linear extension theorem of elementary functional analysis: a bounded linear operator T0 defined on a dense linear subspace E0 of a normed linear space E and taking values in a Banach space F extends uniquely to a bounded linear operator T : E ? F with the same (finite) operator norm.

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