Lebesgue-Stieljes Measures

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Given nondecreasing and right contiuous, define an outer measure by

where and the infimum taken over all coverings of A by countably many semiopen intervals. By Carathéodory's Theorem, we know that is a measure. This measure is sometimes called the Lebesgue–Stieltjes measure associated with F.[1]


  1. Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.