proposition: Lemma exp_le_bd mu F (M : R) : 0 <= M -> (forall x, `|F x| <= M) -> \E_[mu] F <= M.
proof: Proof. move=> ge0M bd; apply/(@le_trans _ _ (\E_[mu] (fun _ => M))). + apply/le_exp. + by apply/bounded_has_exp; exists M. + by apply/has_expC. + by move=> x; apply/(le_trans _ (bd x))/ler_norm. by rewrite exp_cst ler_piMl // le1_pr. Qed.
imports: mathcomp(all_ssreflect all_algebra) mathcomp.classical(boolp classical_sets mathcomp_extra) xfinmap ereal reals discrete topology realseq realsum