Skip to content

Commit

Permalink
Minkowski for p=1
Browse files Browse the repository at this point in the history
  • Loading branch information
hoheinzollern committed Sep 30, 2023
1 parent 048badf commit 5211206
Showing 1 changed file with 12 additions and 3 deletions.
15 changes: 12 additions & 3 deletions theories/hoelder.v
Original file line number Diff line number Diff line change
Expand Up @@ -440,10 +440,19 @@ Qed.

Lemma minkowski (f g : T -> R) (p : R) :
measurable_fun setT f -> measurable_fun setT g ->
(1 < p)%R ->
(1 <= p)%R ->
'N_p%:E[(f \+ g)%R] <= 'N_p%:E[f] + 'N_p%:E[g].
Proof.
move=> mf mg p1.
move=> mf mg.
rewrite le_eqVlt => /orP [/eqP <-|p1].
rewrite !Lnorm1.
apply: (le_trans (y:=\int[mu]_x ((`|f x|)%:E + (`|g x|)%:E))).
apply: ge0_le_integral => //.
- by repeat apply: measurableT_comp => //; apply: measurable_funD.
- apply: measurableT_comp => //.
by apply: measurable_funD; apply: measurableT_comp.
- by move=> x _; rewrite lee_fin ler_normD.
by rewrite le_eqVlt ge0_integralD ?eqxx//; repeat apply: measurableT_comp => //.
have [->|Nfoo] := eqVneq 'N_p%:E[f] +oo.
by rewrite addye ?leey// -ltNye (lt_le_trans _ (Lnorm_ge0 _ _ _)).
have [->|Ngoo] := eqVneq 'N_p%:E[g] +oo.
Expand Down Expand Up @@ -548,4 +557,4 @@ rewrite -EFinM divff ?mule1; last by
by rewrite ?fineK// fin_numElt (@lt_le_trans _ _ 0 -oo _ _ (Lnorm_ge0 _ _ _)).
Qed.

End minkowski.
End minkowski.

0 comments on commit 5211206

Please sign in to comment.