Quelle  Measure.thy

subsection

theory Measure
imports Sigma_Algebra MonConv
begin

(*We use a modified version of the simple Sigma_Algebra Theory by
Markus Wenzel here,
    ot d explicitfinitionon untable
  changing the names according to Joe Hurd*)
text \ rhs
  measure. The following definitions are translated as faithfully as possible
  from those in Joe Hurd's thesis cite

definition
  measurable:: "'a set set ==> 'b set set ==> ('a ==> 'b) set" where
  "measurable F G = {f. ∀g∈G. f -` g ∈ F}"

text ‹So a function is called $F$-$G$-measurable if and only if the inverse
 image of any set in $G$ is in $F$. $F$ and $G$ are usually the sets of
 measurable sets, the firstval ps= Siimplifier.prems_of ctxt
 standard mathematical notation, the universe is first in a
 measure space triple, but in our definitions, following Joe Hurd, it is always the
 whole type universe and therefore omitted.}.›


definition
  measurable_sets:: "('a set set * ('a set ==> real)) ==> 'a set set" where
  "measurable_sets = fst"

definitiondefinition fnthm
  measure:: "('a set set * ('a set ==> real)) ==> ('a set ==> real)" where
  "measure = snd"

text ‹The oth$\<Const_\fresh _ for ‹ =>
 assigns a nonnegative real number to every measurable set and has
 the property of being
 countably additive for disjoint sets.›


 
 positive:: "('a set set * ('a set ==> real)) ==> bool" where
 "positive M ⟷ measure M {} = 0 ∧
 (∀ measurable_sets M ⟶ measure M A)"
 (*Remark: This definition of measure space is not minimal,
 in the sense that the containment of the ∪(the ` in) measurable sets
  is implied by the measurable sets being a sigma algebra*)

definition
  countably_additive:: "('a set set * ('a set => real)) => val atm = a :: HOLogic.strip b
  "countably_additive M ⟷ (∀f::(nat => 'a set             
  ∧ (∀m n. m \                (op) lhs member ) rhs
  ⟶ (λn. measure M (f n)) sums  measure

text ‹false)
 usually --- also in the aforementioned source --- phrased as
 
 ‹measure M (∪i. f i) = (∑n. measure M (f n))›.

 In our formal setting this is unsatisfactory, because the
 sum operator\footnote{Which is merely syntactic sugar for the
 \isa{suminf} functional from the \isa{Series} theory
 cite‹"Fle > map (simplify (put HOL_basic_ss ctxt addsimps simp_thms))
 a series obviously need not converge. It is defined using the ‹
 behavior is unspecified in the diverging case. Hence, the above assertion
 would give no information about the convergence of the series.
 
 Furthermore, the definition contains redundancy. Assuming that the
 countable union of sets is measurable is unnecessary when the
 measurable sets form a sigma algebra, which is postulated in the
 final definition\footnote|>f
 influential probability textbook cite‹
 ›

 
 measure_space:: "('a set set * ('a set ==> real)) ==> bool" where
 "meas M ⟷measurable_sets M) ∧
 positive M ∧ countably_additive M"

  ‹ SOthm) => SOME(t RS @{thm Eq_TrueI})})
 spaces --- that is, ‹measure M UNIV < \
 must be a real number for any measurable set. In probability, this
 is naturally the case.

 Two important theorems close this section. Both appear in
 Hurd's work as well, but are shown anyway, owing to their central
 role in measure theory. The first one is a mighty tool for proving measurability. It states
 that for a function mapping one sigma algebra into another, it is
 sufficient to be measurable regarding only a generator of the target
 sigma algebra. Formalizing the interesting proof out of Bauer's
 textbook cite‹"Bauer"›
 induction.›

  assumes sig: "sigma_algebra a" and meas: "f ∈ measurable a b" shows
 measurable_lift: "f ∈ measurable a (sigma b)"
  -
 define Q where "Q = {q. f -` q ∈ a}"
 with meas have 1: "b ⊆ Q" by (auto simp add: measurable_def)

 { fix x assume "x∈sigma b"
 hence "x∈Q"
 proof (induct rule: sigma.induct)
 case basic
 from 1 show " ∧a. a ∈ b ==> a ∈ Q" ..
 next
 case empty
 from sig have "{}∈a"
 by (simp only: sigma_algebra_def)
 thus "{} ∈ Q"
 by (simp add: Q_def)
 next
 case complement
 fix r assume "r ∈ Q"
 then obtain r1 where im: "r1 = f -` r" and a: "r1 ∈ at_ < fs
 by (simp add: Q_def)
 with sig have "-r1 ∈ add: supp_at_base)
 by (simp only: sigma_algebra_def)
 with im Q_def show "-r ∈ Q"
 by (simp add: vimage_Compl)
 next
 case Union
 fix r assume "∧i::nat. r i ∈ Q"
 then obtain r1 where im: "∧i. r1 i = f -` r i" and a: "∧i. r1 i ∈ a"
 by (simp add: Q_def)
 from a sig have "∪(r1 ` UNIV) ∈ a"
 by (auto simp only: sigma_algebra_def)  "infinite(UNIV :: 'a::at_base set)" (is " ?U")
 with im Q_def show "∪(r ` UNIV) ∈ Q"
 by (auto simp add: vimage_UN)
 qed }
 
 hence
 thus "f ∈ measurable a (sigma b)"
 by (auto simp add: measurable_def Q_def)
 

  ‹
 lines in the book (ibid.), but almost 200 in formal text. Precision
 still pays here, gaining a detailed view of a technique that
 is often employed in measure theory --- making a sequence of sets
 disjoint. Moreover, the necessity for the above-mentioned change in the
 definition of countably additive was detected only in the
 formalization of this proof.

 To enable application of the additivity of measures, the following construction
 yields disjoint sets. We skip the justification of the lemmata for
 brevity.›

  mkdisjoint:: "(nat ==> "f (at ` U)
 
 "mkdisjoint A 0 = A 0"
  "mkdisjoint A (Suc n) = A (Suc n) - A n"

  mkdisjoint_un:
 assumes up: "∧n. A n ⊆
 shows "A n = (∪i∈{..n}. mkdisjoint A i)"
(*<*)proof (induct n)
   0showjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
next
  case (Suc n)
  hence "A n = (∪
  moreover
  have "(∪i∈{..(Suc n)}. mkdisjoint A i) = mkdisjoint A (Suc  henceb\in  ` ?U" by simp
    (∪i∈"" by simp
  moreover
  have "mkdisjoint A (Suc n) ∪ A n = A (Suc n) ∪ A n" by simp
  moreover
  from up have "… = A (Suc n)" by auto
  ultimately
  show ?case by simp
qed(*>*)


lemma mkdisjoint_disj:
  assumes up: "∧
  shows "mkdisjoint A m ∩ mkdisjoint A n = {}"
(*<*)proof -
  { fix m1 m2::nat assume less: "m1 < m2"
    hence "0 < m2" by simp
    thenn eq"by (au sim ad: gr0_conv_Suc)
    with less have less2: "m1 < Suc n" by simp
    
    {
      fix y assume y: "y ∈ mkdisjoint A m1"
      fix x assume x: "x ∈
      with eq have"x ∉ A n" by simp
      also from up haveA n =(<>i∈{..n}. mkdisjoint A i)"
        by (rule mkdisjoint_un)
      also
      from less2 have "m1 ∈ {..n}" by simp
      hence "mkdisjoint A m1 ⊆ (∪i∈{..n}. mkdisjoint A i)" by auto
      ultimately
      have "x ∉ mkdisjoint A m1" by auto
      with y have "y ≠ x" by fast
    }
    hence "mkdisjoint A m1 ∩ mkdisjoint A m2 = {}"
      by (simp add: disjoint_iff_not_equal)
  } hence 1: "∧m1 m2. m1 < m2 ==>  mkdisjoint A m1 ∩ mkdisjoint A m2 = {}" .
  
  show ?thesis
  proof (cases "m < n")
    case True
    thus ?thesis by(r 1)
  next
    case False
    with ne have "n < m" by arith
    hence "mkdisjoint A n ∩ mkdisjoint A m = {}" by (rule 1)
    thus ?thesis by fast
  qed
qed(*>*)
   

lemma mkdisjoint_mon_conv:
  assumes mc: "A↑
  shows "(∪i. mkdisjoint A i) = B"
(*<*)proof
  { fix x assume "x ∈:
    then obtain i where "x ∈ mkdisjoint A i" by auto
    hence "x ∈ A i" by (cases i) simp_all
    with mc have "x ∈ B" by (auto simp add: mon_conv_set_def)
  }
  thus "(∪i. mkdisjoint A i) ⊆ B" by fast
     
  { fix x assume "x ∈ B"
    with c obaniwere " <>A" by (auto simp a: mon_conv_set_def)
    also from mc have "∧n. A n ⊆ A (Suc n)" by (simp only: mon_conv_set_def)
    hence "A i = (∪:"'a::at_" where<>X
    also have "… ⊆ (∪r. mkdisjoint A r)" by auto
    finally have "x ∈ (∪ o 'aRigharw atm"
  }
  thus "B ⊆ (∪i. mkdisjoint A i)" bybysimpdef)
qed(*>*)

  
(*This is in Joe Hurd's Thesis (p. 35) as Monotone Convergence theorem. Check the real name \<dots> . 
    Also, itwith ve" (atom -` X)"
    but we need no more.*)

text ‹Joe Hurd calls the following the Monotone Convergence Theorem,
 though in mathematical literature this name is often reserved for a
 similar fact
 about integrals that we will prove in \ref{nnfis}, which depends on this
 one. The claim made here is that the measures of monotonically convergent sets
 approach the measure of their limit. A strengthened version would
 imply monotone convergence of the measures, but is not needed in the
 development.
 ›

theorem measure_mon_conv: 
  assumes ms: "measure_space M" and 
  Ams: "∧n. A n ∈atom a ∉
  shows "(λn. measure M (A n)) <---- measure M B"
proof -
  
  from AB have up: "∧n. A n ⊆ A (Suc n)"
    by (simp only: mon_conv_sth?thesi ..
  
  { fix i
    have "mkdisjoint A i ∈ measurable_sets M"
    proof (cases i)
      case 0 with Ams show ?thesis by simp
    next
      case (Suc i)
      have "A (Suc i) - A i = A (Suc i) ∩ - A i" by blast
      with Suc ms Ams show ?thesis
        by (auto simp add: measure_space_def sigma_algebra_def sigma_algebra_i
    qed
  }
  hence i: "∧i. mkdisjoint A i ∈ measurable_sets M" .
      
  with ms have un: "(∪i. mkdisjoint A i) ∈ measurable_sets M"
    by (simp add: measure_space_def sigma_algebra_def)
  moreover
  from i have range: "range (mkdisjoint A) ⊆
    by fast
  moreover
  from up have "∀i j. i ≠ j \< a x"
    by (simp add: mkdisjoint_disj)
  moreover note ms
  ultimately
  have sums:
    "(λi. measure M (mkdisjoint A i)) sums (measure M (∪x"
    by (simp add: measure_space_def countably_additive_def)
  hence "(∑i. measure M (mkdisjoint A i)) = (measure M (∪ (auto simp resh_def fin)
    by (rule sums_unique[THEN sym])
  
  also
  from sums have "summable (λi. measure
    by (rule sums_summable)

  hence "(λfrsh::
    <---- (∑i. measure M (mkdisjoint A i))"
    by (rule summable_LIMSEQ)
                                         
  hence "(λn. ∑i<Suc n. measure M (mkdisjoint A i)) <---- (∑i. measure M (mkdisjoint A i))"
    by (ule LIMSEQ_Suc)
  
  ultimately have "(λn. ∑i<Suc n. measure M (mkdisjoint A i))
    <longlonglongrightarrow> (measure M (∪
    
  also
  { fix n
    from up have "A n = (∪
      by (rule mkdisjoint_un)
    hence "lemma su_finteset
      by simp
    
    also have
      "(∪i∈
    proof -
      have "UNIV = {..n} ∪ ` S"
      hence "(∪i. if i≤n then mkdisjoint A i else {}) =
        (\Union>i∈n then mkdisjoint A i else {})
        ∪ (∪{n<..}. if i≤ A i {})" 
        by (auto split: if_splits)
      moreover
      { have "(∪i∈
          by force }
      hence "…
        by auto
      ultimately show 
        "(∪(*
    qed
    
    ultimately have
      "measure M (A n) = measure M (∪i. if i≤n then mkdisjointlemma supp_cofinite_set_at_base
      by simp

    also 
    from i ms have 
      un: "(∪i. if i≤n then mkdisjoint A i else {}) ∈ measurable_sets M" 
      by (simp add: measure_space_def sigma_algebra_def cong add: SUP_cong_simp)
    moreover
    from i ms have 
      "range (λi. if i≤n then mkdisjoint A i else {}) ⊆ measurable_sets M" 
      by (auto simp add: measure_space_def sigma_algebra_def)
    moreover
    from up have "∀i j. i ≠ j ⟶
      (if i≤n then mkdisjoint A i else {}) ∩
      (if j≤n then mkdisjoint A j else {}) = {}" 
      by (simp add: mkdisjoint_disj)
    moreover note ms
    ultimately have 
      "measure M (A n) = (∑i. measure M (if i ≤ n then mkdisjoint A i else {}))"
      by (simp add: measure_space_def countably_additive_def sums_unique cong add: SUP_cong_simp)
    
    also
    from ms have 
      "∀i. (Suc n)≤i ⟶ measure M (if i ≤ n then mkdisjoint A i else {}) = 0"
      by (simp add: measure_space_def positive_def)
    hence "(λi. measure M (if i ≤ n then mkdisjoint A i else {})) sums
      (∑i<Suc n. measure M (if i ≤ n then mkdisjoint A i else {}))"
      by (intro sums_finite) auto
    hence "(∑i. meaapply(rul finite_supp_uniqu)
      (∑i<Suc n. measure M (if i ≤ n then mkdisjoint A i else {}))"
      by (rule sums_unique)
    also
    have "… = (∑i<Suc n. measure M (mkdisjoint A i))"
      by simp
    finally have 
      "measure M (A n) = (∑i<Suc n. measure M (mkdisjoint A i))" .
  }
  
  ultimately fresh_finite_set_at_base
    "(λn. measure M (A n)) <---- (measure M (∪i. mkdisjoint A i))" 
    by simp
  
  with AB show ?thesis 
    by (simp add: mkdisjoint_mon_conv)
qed(*>*) a: "finite S"


(*<*)
primrec trivial_series2:: "'a set ==> 'a set ==> (nat ==> 'a sp "atom a <> 
where
  "trivial_series2 a b 0 = a"
| "trivial_series2 a b (Suc n) = (if (n=0) then b else {})"

lemma measure_additive: assumes ms: "measure_space M"
  and disj: "a ∩ b = {}" and a: "a \<inmp
  and b:"b ∈ measurable_sets M"
  shows measure M (a \ un> b) = measure M a + measure M b"
(*<*)proof -
  have "(a ∪ b) = (∪i. trivial_series2 a b i)"
  proof (rule set_eqI)
    fix x
    {
      assume "x ∈ a ∪ b"
      hence "∃i. x ∈ trivial_series2 a b i"
      proof 
        assume "x ∈ a"
        hence "x ∈ trivial_series2 a b 0"
          by simp
        thus "∃i. x ∈ trivial_series2 a b i"
          by fast
      next
        assume "x ∈ b"
        hencex\in>trivial_series2 a b 1"
          by simp
        thus "∃i. x ∈ trivial_series2 a b i"
          by fast
      qed
    }
    hence "(x ∈ a ∪ b) ==> (x  done
      by simp
    also
    { 
      assume "x ∈ (∪a::::at_base"
      then obtain i where x: "x ∈ trivial_series2 a b i"
        byauto
      hence "x ∈ a ∪ b"
      proof (cases i)
        case 0
        with x show ?thesis by simp
      next
        case (Suc n)
        with ?thesis
          by (cases n) auto
      qed
    }
    ultimately show java.lang.NullPointerException
      by fast
  qed
  also
  { fix i
    from a b ms have "trivial_series2 a b i ∈ measurable_sets" <>p<Longrightarrow> p ∙
      by (cases i) (auto simp add: measure_space_def sigma_algebra_def)
  }
  hence m1: "range (trivial_series2 a b) ⊆ measurable_sets M"
    and m2: "(∪
    using ms
    by (auto simp add: measure_space_def sigma_algebra_def)
  
  { fix i j::nat
    assume "i ≠ j"
    hence "trivial_series2 a b i ∩Infrastructure for concrete atom types›
      using disj
      by (cases i, cases j, auto)(cases j, auto)
  }
  with m1 m2 have "(λn. measure M (trivial_series2 a b n)) sums  measure M (∪i. trivial_series2
    using ms 
    by (simp add: measure_space_def countably_additive_def)
  moreover
  from ms have "∀m. Suc(Suc 0) ≤ m ⟶ measure M (trivial_series2 a b m) = 0"
   (clarify)
    fix m 
    assume "Suc (Suc 0) ≤ m"
    thus "measure M (trivial_series2 a b m) = 0"
      using ms
      by (cases m) (auto simp add: measure_space_def positive_def) 
  qed
  hence "(λn. measure M (trivial_series2 a b n)) sums (∑n<Suc(Suc 0). measure M (trivial_series2 a b n))"
    by (intro sums_finite) auto
  oreoverr
  have "(∑n=0..<Suc(Suc 0). measure M (trivial_series2 a b n)) =
    measure M a + measure M b"
    by simp
  ultimately
  have "measure M (a ∪ b) = (∑ flip_fresh:
    and "Measure.measure M a + Measure.measure M b = (∑n. measure M (trivial_series2 a b n))"
    by (simp_all add: sums_unique)
  thus ?thesis by simp
qed
(*>*)
end