Lebesgue-Stieljes Measures: Difference between revisions

Revision as of 02:06, 19 December 2020

Given $F:R\rightarrow R$ nondecreasing and right contiuous, define an outer measure by

\mu _{g}(E)=\inf \left\{\sum _{i}\mu _{g}(I_{i})\ :\ E\subset \bigcup _{i}I_{i}\right\}

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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 \F(~~b_i~~) \ : \ A\subset \bigcup_i ~~\left[a_i, b_i\right]~~ \right\}</math>		:<math>\mu_g(E) = \inf\left\{\sum_i \mu_g(I_i) \ : \ E\subset \bigcup_i I_i \right\}</math>