Commit e0a24e7f by SuperDuckDuck

### added vitali theorem (coin toss version)

parent ec870dd0
 namespace http://mathhub.info/MitM/smglom/probability ❚ import base http://mathhub.info/MitM/Foundation ❚ import sets http://mathhub.info/MitM/smglom/typedsets ❚ import seqs http://mathhub.info/MitM/smglom/calculus ❚ import calc http://mathhub.info/MitM/smglom/calculus ❚ // for seqeuences and intervals ❚ import mes http://mathhub.info/MitM/smglom/measures ❚ // theory Vitali : base:?Logic = include sets:?AllSets ❙ ... ... @@ -10,14 +11,3 @@ import seqs http://mathhub.info/MitM/smglom/calculus ❚ ❚ theory VitaliCoinToss : base:?Logic = include sets:?TypedSets ❙ include seqs:?Sequences ❙ include base:?DescriptionOperator ❙ vitaliSet : set (sequence ℕ) ❘= ⟪ (fullset (ℕ ⟶ ℕ)) | ([f] ∀[i] f i ≐ 0 ∨ f i ≐ 1) ⟫ ❙ flipOneZero : ℕ ⟶ ℕ ❘= [n] (if (n ≐ 1) then 0 else (if (n ≐ 0) then (1 : ℕ) else n))❙ flipAt : (sequence ℕ) ⟶ ℕ ⟶ (sequence ℕ) ❘= [oldSeq, n] [i : ℕ] (if (i ≐ n) then (flipOneZero (oldSeq i)) else (oldSeq i)) ❙ // vitaliTheorem : ⊦ (¬ (∃[P] )) ❙ ❚ \ No newline at end of file
 namespace http://mathhub.info/MitM/smglom/probability ❚ import base http://mathhub.info/MitM/Foundation ❚ import sets http://mathhub.info/MitM/smglom/typedsets ❚ import calc http://mathhub.info/MitM/smglom/calculus ❚ // for seqeuences and intervals ❚ import mes http://mathhub.info/MitM/smglom/measures ❚ theory VitaliCoinToss : base:?Logic = // a version of the vitali theorem from Hans-Otto Georgii's "Stochastik" ❙ include sets:?AllSets ❙ include calc:?Sequences ❙ include base:?DescriptionOperator ❙ include calc:?Intervals ❙ include mes:?SigmaAdditive ❙ vitaliSet : set (sequence ℕ) ❘= ⟪ (fullset (ℕ ⟶ ℕ)) | ([f] ∀[i] f i ≐ 0 ∨ f i ≐ 1) ⟫ ❙ vitaliPowerSet : collection (sequence ℕ) ❘= ℘ vitaliSet ❙ flipOneZero : ℕ ⟶ ℕ ❘= [n] (if (n ≐ 1) then 0 else (if (n ≐ 0) then (1 : ℕ) else n))❙ flipAt : ℕ ⟶ (sequence ℕ) ⟶ (sequence ℕ) ❘= [n , oldSeq] [i : ℕ] (if (i ≐ n) then (flipOneZero (oldSeq i)) else (oldSeq i)) ❙ vitaliPowerSetType : type ❘= setastype (vitaliPowerSet)❙ invariancy : (vitaliPowerSetType ⟶ ℝ) ⟶ bool ❘ = [P] ∀[A ] ∀[n : ℕ] A ⊑ vitaliSet ⇒ P (elem vitaliPowerSet A) ≐ P (elem vitaliPowerSet (im (flipAt n) A))❙ vitaliTheorem : ⊦ (¬ (∃[P : vitaliPowerSetType ⟶ ℝ] (∀[x : vitaliPowerSetType] P x ≤ 1 ∧ 0 ≤ P x) ∧ P (vitaliSet) ≐ 1 ∧ prop_sigmaAdditive vitaliPowerSet P) ∧ invariancy P) ❙ ❚ \ No newline at end of file
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