proposition: Lemma abse_integralP d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) (D : set T) (f : T -> \bar R) : measurable D -> measurable_fun D f -> (`| \int[mu]_(x in D) f x | < +oo <-> \int[mu]_(x in D) `|f x| < +oo)%E.
proof: Proof. move=> mD mf; split => [|] foo; last first. exact: (le_lt_trans (le_abse_integral mu mD mf) foo). under eq_integral do rewrite -/((abse \o f) _) fune_abse. rewrite ge0_integralD//;[|exact/measurable_funepos|exact/measurable_funeneg]. move: foo; rewrite integralE/= -fin_num_abs fin_numB => /andP[fpoo fnoo]. by rewrite lte_add_pinfty// ltey_eq ?fpoo ?fnoo. 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