Proof with auto.
intros.
unfold Ravg.
repeat rewrite map_length.
unfold Rdiv.
apply Rle_trans with (g (Rsum (map m u)) * / INR (length u))...
apply Rmult_le_compat_r.
apply Rlt_le.
apply Rinv_0_lt_compat.
apply lt_0_INR...
apply Rle_trans with (Rsum (map g (map m u)))...
rewrite map_map.
apply Rsum_map_le.
intros.
apply H2.
unfold Rdiv in H0.
destruct u.
simpl in H3.
elimtype False. apply (lt_irrefl _ H3).
simpl @length.
apply H0...
rewrite <- (Rmult_0_r (INR (length (map m (u :: u0))))).
apply le_Rsum.
intros.
destruct (fst (conj_prod (in_map_iff _ _ _)) H4).
destruct H5.
subst...
Qed.

Lemma Ravg_concat v x:
(forall l, In l x -> length l = v) ->
Ravg (concat x) = Ravg (map Ravg x).