INTEGRAL CHOQUET; CHOQUET INTEGRAL

PUTU KARTIKA DEWI, Rini Indrati

2014 | Disertasi | PROGRAM STUDI S2 MATEMATIKA

Let ? be a nonempty set and ? ? 2? be a ?-algebra. A non additive measure ? on a measurable space ??, ? is a monotone function ?: ? ? 0,?? with ??? ? 0. In this thesis we discuss an integral with respect to a non additive measure, that is called the Choquet integral. In general, the Choquet integral is not additive. By using interpreters, we show the relationship of comonotonicity of functions and additivity of its integral. By using that relationship and the Urysohn Lemma we prove that, in case ? is a locally compact Hausdorff space and ? is a regular non additive measure, the Choquet integral of a non negative function can be approximated by the Choquet integral of a continuous function with compact support. Further, the monotone convergence theorem holds in the Choquet integral if the non additive measure ? is semicontinuous from below. Meanwhile, the dominated convergence theorem holds in the Choquet integral if the non additive measure ? is semicontinuous from above and subadditive.

Kata Kunci : non additive measure, Choquet integral