Sigma algebras and measurable maps defined.

diff --git a/helm/software/matita/dama/sigma_algebra.ma b/helm/software/matita/dama/sigma_algebra.ma
new file mode 100644 (file)
index 0000000..94ceb92
--- /dev/null
+++ b/helm/software/matita/dama/sigma_algebra.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/sigma_algebra/".
+
+include "topology.ma".
+
+record is_sigma_algebra (X:Type) (A: set X) (M: set (set X)) : Prop ≝
+ { siga_subset: ∀B.B ∈ M → B ⊆ A;
+   siga_full: A ∈ M;
+   siga_compl: ∀B.B ∈ M → B \sup c ∈ M;
+   siga_enumerable_union:
+    ∀B:seq (set X).(∀i.(B \sub i) ∈ M) → (∪ \sub i B \sub i) ∈ M
+ }.
+
+record sigma_algebra : Type ≝
+ { siga_carrier:> Type;
+   siga_domain:> set siga_carrier;
+   M: set (set siga_carrier);
+   siga_is_sigma_algebra:> is_sigma_algebra ? siga_domain M
+ }.
+
+(*definition is_measurable_map ≝
+ λX:sigma_algebra.λY:topological_space.λf:X → Y.
+  ∀V. V ∈ O Y → f \sup -1 V ∈ M X.*)
+definition is_measurable_map ≝
+ λX:sigma_algebra.λY:topological_space.λf:X → Y.
+  ∀V. V ∈ O Y → inverse_image ? ? f V ∈ M X.
+

diff --git a/helm/software/matita/dama/topology.ma b/helm/software/matita/dama/topology.ma
index 3a7ec0da792c0cafd86e8ae51ab68b8e221791be..e0a926d7673ebbbcdaf9ce8fd5582e3857a5c401 100644 (file)
--- a/helm/software/matita/dama/topology.ma
+++ b/helm/software/matita/dama/topology.ma
@@ -28,16 +28,16 @@ record is_topology X (A: set X) (O: set (set X)) : Prop  ≝
 record topological_space : Type ≝
  { top_carrier:> Type;
    top_domain:> set top_carrier;
-   top_O: set (set top_carrier);
-   top_is_topological_space:> is_topology ? top_domain top_O
+   O: set (set top_carrier);
+   top_is_topological_space:> is_topology ? top_domain O
  }.
 
 (*definition is_continuous_map ≝
  λX,Y: topological_space.λf: X → Y.
-  ∀V. V ∈ top_O Y → (f \sup -1) V ∈ top_O X.*)
+  ∀V. V ∈ O Y → (f \sup -1) V ∈ O X.*)
 definition is_continuous_map ≝
  λX,Y: topological_space.λf: X → Y.
-  ∀V. V ∈ top_O Y → inverse_image ? ? f V ∈ top_O X.
+  ∀V. V ∈ O Y → inverse_image ? ? f V ∈ O X.
 
 record continuous_map (X,Y: topological_space) : Type ≝
  { cm_f:> X → Y;