proposition: Lemma Lnorm_eq0_eq0 r f : (0 < r)%R -> measurable_fun setT f -> 'N_r%:E[f] = 0 -> ae_eq mu [set: T] (fun t => (`|f t| `^ r)%:E) (cst 0).
proof: Proof. move=> r0 mf; rewrite unlock (gt_eqF r0) => /poweR_eq0_eq0 fp. apply/ae_eq_integral_abs => //=. apply: measurableT_comp => //. apply: (@measurableT_comp _ _ _ _ _ _ (@powR R ^~ r)) => //. exact: measurableT_comp. under eq_integral => x _ do rewrite ger0_norm ?powR_ge0//. by rewrite fp//; apply: integral_ge0 => t _; rewrite lee_fin powR_ge0. 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 lebesgue_integral numfun exp convex itv