proposition: Lemma sintegralEnnsfun (f : {nnsfun T >-> R}) : sintegral mu f = (\sum_(x \in [set r | r > 0]%R) (x%:E * mu (f @^-1` [set x])))%E.
proof: Proof. rewrite (fsbig_widen _ setT) ?sintegralET//. move=> x [_ /=]; case: ltgtP => //= [xlt0 _|<-]; last by rewrite mul0e. rewrite preimage10 ?measure0 ?mule0//= => -[t _]. by apply/eqP; apply: contra_ltN xlt0 => /eqP<-. 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