proposition: Lemma ecvg_approx (f0 : forall x, D x -> (0 <= f x)%E) x : D x -> EFin \o approx^~x @ \oo --> f x.
proof: Proof. move=> Dx; have := leey (f x); rewrite le_eqVlt => /predU1P[|] fxoo. have dvg_approx := dvg_approx Dx fxoo. have : {homo approx ^~ x : n m / (n <= m)%N >-> n <= m}. by move=> m n mn; have := nd_approx mn => /lefP; exact. move/nondecreasing_dvgn_lt => /(_ dvg_approx). by rewrite fxoo => ?; apply/cvgeryP. rewrite -(@fineK _ (f x)); first exact: (cvg_comp (cvg_approx f0 Dx fxoo)). by rewrite ge0_fin_numE// 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