proposition: Lemma subset_integral (A B : set T) (mA : measurable A) (mB : measurable B) (f : T -> \bar R) : measurable_fun B f -> (forall x, B x -> 0 <= f x) -> A `<=` B -> \int[mu]_(x in A) f x <= \int[mu]_(x in B) f x.
proof: Proof. move=> mf f0 AB; rewrite -(setDUK AB) integral_setU//; last 4 first. - exact: measurableD. - by rewrite setDUK. - by move=> x; rewrite setDUK//; exact: f0. - by rewrite disj_set2E setDIK. by apply: leeDl; apply: integral_ge0 => x [Bx _]; exact: f0. Qed.
imports: HB(structures) mathcomp(all_ssreflect ssralg ssrnum ssrint interval finmap) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality fsbigop ) signed reals ereal topology normedtype sequences real_interval esum measure lebesgue_measure numfun