diff --git a/Lebesgue/Subset_system.v b/Lebesgue/Subset_system.v index 1d8e5901a4f4e575defe983bf3cdc2b9081e6066..b18dd676c7a0afd85cb121ad1367978643324c25 100644 --- a/Lebesgue/Subset_system.v +++ b/Lebesgue/Subset_system.v @@ -77,7 +77,8 @@ Inductive Sigma_ring : (U -> Prop) -> Prop := | Sigma_ring_Union_seq : Union_seq Sigma_ring. (* Sigma_algebra is the sigma-algebra generated by gen, - ie the smallest sigma-algebra containing gen. *) + ie the smallest sigma-algebra containing gen. + From Definitions 474 p. 84 and 482 p. 85 (v2) *) Inductive Sigma_algebra : (U -> Prop) -> Prop := | Sigma_algebra_Gen : Incl gen Sigma_algebra | Sigma_algebra_wEmpty : wEmpty Sigma_algebra @@ -532,6 +533,7 @@ Proof. intros; apply Ext_equiv; split; now apply Lsyst_lub_alt. Qed. +(* Lemma 501 p. 87 (v2) *) Lemma Sigma_algebra_ext : Incl gen0 (Sigma_algebra gen1) -> Incl gen1 (Sigma_algebra gen0) -> @@ -1624,6 +1626,7 @@ apply Sigma_algebra_Compl. apply Sigma_algebra_Union. Qed. +(* From Lemma 475 p. 84 (v2) *) Lemma Sigma_algebra_wFull : wFull (Sigma_algebra gen). Proof. @@ -1696,6 +1699,7 @@ Proof. rewrite <- Sigma_algebra_is_Algebra; apply Algebra_Union_disj_finite. Qed. +(* From Lemma 475 p. 84 (v2) *) Lemma Sigma_algebra_Inter_seq : Inter_seq (Sigma_algebra gen). Proof. @@ -1722,6 +1726,7 @@ Proof. apply Union_seq_disj, Sigma_algebra_Union_seq. Qed. +(* From Lemma 502 p. 87 (v2) *) Lemma Sigma_algebra_gen_remove : forall A, Sigma_algebra gen A -> Incl (Sigma_algebra (add gen A)) (Sigma_algebra gen). diff --git a/Lebesgue/measurable.v b/Lebesgue/measurable.v index 518d2f12254bff321b2102bdae8002f387cd047e..1c7e0b10b9e631a9f1fc2a59ec113f456bf6004c 100644 --- a/Lebesgue/measurable.v +++ b/Lebesgue/measurable.v @@ -39,7 +39,6 @@ Section measurable_Facts. Context {E : Type}. (* Universe. *) Variable genE : (E -> Prop) -> Prop. (* Generator. *) -(* From Definitions 474 p. 84 and 482 p. 85 *) Definition measurable : (E -> Prop) -> Prop := Sigma_algebra genE. Definition measurable_finite : (nat -> E -> Prop) -> nat -> Prop := @@ -73,7 +72,6 @@ Proof. apply Sigma_algebra_wEmpty. Qed. -(* From Lemma 475 p. 84 *) Lemma measurable_full : measurable fullset. (* wFull measurable. *) Proof. apply Sigma_algebra_wFull. @@ -133,7 +131,6 @@ Proof. apply Sigma_algebra_Union_seq. Qed. -(* From Lemma 475 p. 84 *) Lemma measurable_inter_seq : forall A, measurable_seq A -> measurable (inter_seq A). (* Inter_seq measurable. *) @@ -155,7 +152,7 @@ intros A N HA n Hn; apply measurable_union_finite. intros m Hm; apply HA; lia. Qed. -(* From Lemma 480 pp. 84-85 *) +(* From Lemma 480 pp. 84-85 (v2) *) Lemma measurable_DU : forall A, measurable_seq A -> measurable_seq (DU A). Proof. @@ -164,11 +161,6 @@ apply measurable_diff; try easy. now apply measurable_union_finite. Qed. -Lemma measurable_gen : Incl genE measurable. -Proof. -apply Sigma_algebra_Gen. -Qed. - End measurable_Facts. @@ -179,6 +171,11 @@ Section measurable_gen_Facts1. Context {E : Type}. (* Universe. *) Variable genE : (E -> Prop) -> Prop. (* Generator. *) +Lemma measurable_gen : Incl genE (measurable genE). +Proof. +apply Sigma_algebra_Gen. +Qed. + Lemma measurable_gen_idem : is_Sigma_algebra (measurable genE). Proof. apply Sigma_algebra_idem. @@ -203,7 +200,6 @@ Proof. apply Sigma_algebra_lub_alt. Qed. -(* Lemma 501 p. 87 *) Lemma measurable_gen_ext : forall genE', Incl genE (measurable genE') -> Incl genE' (measurable genE) -> @@ -212,7 +208,6 @@ Proof. apply Sigma_algebra_ext. Qed. -(* From Lemma 502 p. 87 *) Lemma measurable_gen_remove : forall A, measurable genE A -> Incl (measurable (add genE A)) (measurable genE). @@ -249,8 +244,10 @@ Variable f : E -> F. Variable PE : (E -> Prop) -> Prop. (* Subset system. *) Variable PF : (F -> Prop) -> Prop. (* Subset system. *) +(* From Lemma 524 p. 93 (v2) *) Definition Image : (F -> Prop) -> Prop := fun B => PE (preimage f B). +(* From Lemma 523 p. 93 (v2) *) Definition Preimage : (E -> Prop) -> Prop := image (preimage f) PF. End measurable_gen_Image_Def. @@ -311,6 +308,7 @@ Variable genF : (F -> Prop) -> Prop. (* Generator. *) Variable f : E -> F. +(* Lemma 524 p. 93 (v2) *) Lemma is_Sigma_algebra_Image : is_Sigma_algebra (Image f (measurable genE)). Proof. apply Sigma_algebra_equiv; repeat split. @@ -319,6 +317,7 @@ intros B HB; apply measurable_compl; easy. intros B HB; apply measurable_union_seq; easy. Qed. +(* Lemma 523 p. 93 (v2) *) Lemma is_Sigma_algebra_Preimage : is_Sigma_algebra (Preimage f (measurable genF)). Proof. @@ -354,6 +353,7 @@ Variable genF : (F -> Prop) -> Prop. (* Generator. *) Variable f : E -> F. +(* Lemma 527 pp. 93-94 (v2) *) Lemma measurable_gen_Preimage : measurable (Preimage f genF) = Preimage f (measurable genF). Proof. @@ -474,8 +474,10 @@ Section Borel_subsets. Context {E : UniformSpace}. (* Uniform universe. *) +(* Definition 517 p. 91 (v2) *) Definition measurable_Borel := measurable (@open E). +(* From Lemma 518 p. 91 *) Lemma measurable_Borel_open : Incl open measurable_Borel. Proof. intros A HA; now apply measurable_gen. @@ -515,6 +517,7 @@ Context {E F : UniformSpace}. (* Uniform universes. *) Let genExF := Gen_Prod (@open E) (@open F). +(* From Lemma 711 p. 137 (v3) (with m := 2 and Y_i := X_i). *) Lemma measurable_Borel_prod_incl : Incl (measurable genExF) measurable_Borel. Proof. apply measurable_gen_monot. @@ -537,6 +540,7 @@ Let genE1xE2 := Gen_Prod (@open E1) (@open E2). Hypothesis HE1 : is_second_countable E1. Hypothesis HE2 : is_second_countable E2. +(* From Lemma 711 p. 137 (v3) (with m := 2 and Y_i := X_i). *) Lemma measurable_Borel_prod_incl_alt : Incl measurable_Borel (measurable genE1xE2). Proof. @@ -554,7 +558,7 @@ apply measurable_Prop. apply measurable_prod; apply measurable_gen; easy. Qed. -(* From Lem 701 p. 135,136 (RR-9386-v3) (with m := 2 and Y_i := X_i). *) +(* From Lemma 711 p. 137 (v3) (with m := 2 and Y_i := X_i). *) Lemma measurable_Borel_prod_eq : measurable_Borel = measurable genE1xE2. Proof. intros; apply Ext_equiv; split. @@ -562,7 +566,7 @@ apply measurable_Borel_prod_incl_alt; easy. apply measurable_Borel_prod_incl. Qed. -(* From Lem 701 p. 135,136 (RR-9386-v3) (with m := 2 and Y_i := X_i). *) +(* From Lemma 711 p. 137 (v3) (with m := 2 and Y_i := X_i). *) Lemma measurable_Borel_prod_eq_alt : measurable_Borel = measurable_Prod (@open E1) (@open E2). Proof.