Lebesgue-Stieljes Measures: Difference between revisions
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Given <math> F : R \rightarrow R </math> nondecreasing and right contiuous, define an outer measure by | Given <math> F : R \rightarrow R </math> nondecreasing and right contiuous, define an outer measure by | ||
:<math>\mu_F^{*}(A) = \inf\left\{\sum_i \mu_F^{*}(\left(a, b\right]) \ : \ A\subset \bigcup_i \left(a, b\right] \right\}</math> | :<math>\mu_F^{*}(A) = \inf\left\{\sum_i \mu_F^{*}(\left(a, b\right]) \ : \ A\subset \bigcup_i \left(a, b\right] \right\}</math> | ||
where <math> \mu_F^{*}(\left(a, b\right]) = F(b) - F(a) </math> and the infimum taken over all coverings of A by countably many semiopen intervals. By Carathéodory's Theorem, we know that <math>\mu_F := \left.\mu_F^{*}\right|_{M_{\mu_F^{*}}}</math> is a measure. This measure is sometimes called the Lebesgue–Stieltjes measure associated with F. | where <math> \mu_F^{*}(\left(a, b\right]) = F(b) - F(a) </math> and the infimum taken over all coverings of A by countably many semiopen intervals. By Carathéodory's Theorem, we know that <math>\mu_F := \left.\mu_F^{*}\right|_{M_{\mu_F^{*}}}</math> is a measure. This measure is sometimes called the Lebesgue–Stieltjes measure associated with F.<ref name="Folland">Folland, Gerald B. (1999). ''Real Analysis: Modern Techniques and Their Applications'', John Wiley and Sons, ISBN 0471317160, Second edition.</ref> | ||
==References== | ==References== | ||
Revision as of 06:26, 19 December 2020
Given nondecreasing and right contiuous, define an outer measure by
![{\displaystyle \mu _{F}^{*}(A)=\inf \left\{\sum _{i}\mu _{F}^{*}(\left(a,b\right])\ :\ A\subset \bigcup _{i}\left(a,b\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426f2df40dd3f9b86da35165f7a52ba887196cfc)
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]
![{\displaystyle \mu _{F}^{*}(\left(a,b\right])=F(b)-F(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf603aad0fe2f4fd3be8821bbbd477146bb759c)
References
- ↑ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.