Lebesgue-Stieljes Measures: Difference between revisions
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:<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> | where <math> \mu_F(\left(a, b\right]) = F(b) - F(a) </math> | ||
{{math|''w''((''s'',''t'']) {{=}} ''g''(''t'') − ''g''(''s'')}} and | {{math|''w''((''s'',''t'']) {{=}} ''g''(''t'') − ''g''(''s'')}} and | ||
{{math|''w''((''s'', ''t'']) {{=}} ''g''(''t'') − ''g''(''s'')}} | {{math|''w''((''s'', ''t'']) {{=}} ''g''(''t'') − ''g''(''s'')}} | ||
Revision as of 05:56, 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/58a4bdda36095e649c75b403c6476824d041ca67)
where Template:Math and Template:Math
![{\displaystyle \mu _{F}(\left(a,b\right])=F(b)-F(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ed1f24bfb5dbe0eeac6368fa2d297e287d741b)