Commit 18b26ad2 authored by Sven Wille's avatar Sven Wille

more probability and measure theory

parent c8b9cf7e
......@@ -52,7 +52,7 @@ theory ProbabilityTheoryDefinitions : base:?Logic =
theory RealValuedRandomVariables : base:?Logic > PS : probabilitySpace ❘ =
include ?ProbabilityTheoryDefinitions/RandomVariables (PS) ❙
include measures?RealBorelMeasurableSpace
include measures?RealBorelMeasurableSpace ❙
isRealValuedRandomVariable : ((PS.universe) ⟶ ℝ) ⟶ prop ❘ = [f] isRandomVariable realBorelMeasurableSpace f❙
realValuedRandomVariable = randomVariable realBorelMeasurableSpace ❙
......
namespace http://mathhub.info/MitM/smglom/Probability ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import meas http://mathhub.info/MitM/smglom/measures ❚
......@@ -121,6 +121,6 @@ theory RealVectorspace : base:?Logic =
realVec1 : realVec ❘ = asVectorspace realField ❘ # ℝ1 ❙
realVec2 : realVec ❘ // = productspace realField ℝ1 ℝ1 ❘ # ℝ2 ❙
realVec3 : realVec ❘ // = productspace realField ℝ1 ℝ2 ❘ # ℝ3 ❙ // takes forever ❙
// finite_real_vectorspace : pos_lit ⟶ RealVec
finite_real_vectorspace : pos_lit ⟶ realVec ❙ // uncommented and changed by sw : why was this one commented out
namespace http://mathhub.info/MitM/smglom/measures ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import top http://mathhub.info/MitM/smglom/topology ❚
import arith http://mathhub.info/MitM/smglom/arithmetics ❚
theory SigmaOperator : base:?Logic =
include ts:?AllSets❙
include ?SigmaAlgebra❙
bigF : {A : type} {Ω : set A} {M : set (set A)} ⊦ M ⊑ ( ℘ Ω) ⟶ set (set (set A))
❘ = [A , O , M , h ] ⟪ ℘ (℘ O) | ([S : set (set A)] M ⊑ S ∧ isSigmaAlgebra S) ⟫ ❘# bigF 2 3 4 ❙
sigmaOperator : {A : type} {Ω : set A} {M : set (set A)} ⊦ M ⊑ (℘ Ω) ⟶ set (set A) ❘ = [A , O, M , h] INTERSECT (bigF O M h) ❘ # σ 2 3 4 ❙
// alt : generatorSigmaAlgebra ❙
theory SigmaOperatorTheorems : base:?Logic =
include ?SigmaOperator ❙
theorem_sigmaOperator_1 : {A : type} {O : set A} {M : set (set A)} {h : ⊦ M ⊑ ℘ O} ⊦ M ⊑ σ O M h ❙
theorem_sigmaOperator_2 : {A : type} {O : set A} {O0 : set A} {M : set (set A)} {N : set (set A)} {h : ⊦ M ⊑ ℘ O} {h0 : ⊦ N ⊑ ℘ O0} ⊦ M ⊑ N ⟶ ⊦ σ O M h ⊑ σ O0 N h0❙
theorem_sigmaOperator_3 : {A : type} {O : set A} {M : set (set A)} {h : ⊦ M ⊑ ℘ O} ⊦ σ O M h ⊑ ℘ O ❙
theorem_sigmaOperator_4 : {A : type} {O : set A} {M : set (set A)} {h : ⊦ M ⊑ ℘ O} ⊦ σ O (σ O M h) (theorem_sigmaOperator_3 A O M h) ≐ σ O M h ❙
theorem_sigmaOperator_5 ❙
theory BorelExample : base:?Logic =
theory bla : base:?Logic =
......@@ -8,7 +8,7 @@ theory Countable : base:?InductiveTypes =
include ?AllSets ❙
include natt:?InductiveNaturalNumbers ❙
prop_countableSet : {A : type} set A ⟶ prop ❘= [A,S] ∃ [f : setastype S ⟶ Nat] is_injective f ❘# is_countable 2 ❙
prop_countableSet : {A : type} set A ⟶ prop ❘= [A,S] ∃ [f : setastype S ⟶ INat] is_injective f ❘# is_countable 2 ❙
lemma_finiteSetIsCountable : {A : type} {s : set A} ⊦ s finite ⟶ ⊦ is_countable s ❙
......@@ -17,7 +17,7 @@ theory CountablyInfinite : base:?InductiveTypes =
include ?Countable ❙
include ?Bijective ❙
prop_countablyInfiniteSet : {A : type} set A ⟶ prop ❘= [A,S] ∃ [f : setastype S ⟶ Nat] is_bijective f ❘# is_countablyInfinite 2 ❙
prop_countablyInfiniteSet : {A : type} set A ⟶ prop ❘= [A,S] ∃ [f : setastype S ⟶ INat] is_bijective f ❘# is_countablyInfinite 2 ❙
lemma_countablyInfiniteSetsAreCountable : {A : type} {s : set A} ⊦ is_countablyInfinite s ⟶ ⊦ is_countable s ❙
......
......@@ -17,5 +17,5 @@ theory FiniteSets : base:?Logic =
theorem_setConsFinite : {A : type, s : set A , a : A} ⊦ s finite ⟶ ⊦ (s <- a) finite❙
theorem_setConsFiniteSplit : {A : type, s : set A , a : A } ⊦ (s <- a) finite ⟶ ⊦ s finite ❘ # setSpFin 2 3 4❙ // error , wenn keine notation dann fehle rin setsum -> bla ❙
theorem_setConsFiniteSplit : {A : type, s : set A , a : A } ⊦ (s <- a) finite ⟶ ⊦ s finite ❘ # setSpFin 2 3 4❙ // error , wenn keine notation dann fehler in setsum -> bla ❙
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