src/HOL/Probability/Caratheodory.thy
 author haftmann Sat Jun 28 09:16:42 2014 +0200 (2014-06-28) changeset 57418 6ab1c7cb0b8d parent 56994 8d5e5ec1cac3 child 57446 06e195515deb permissions -rw-r--r--
fact consolidation
1 (*  Title:      HOL/Probability/Caratheodory.thy
2     Author:     Lawrence C Paulson
3     Author:     Johannes Hölzl, TU München
4 *)
6 header {*Caratheodory Extension Theorem*}
8 theory Caratheodory
9   imports Measure_Space
10 begin
12 text {*
13   Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
14 *}
16 lemma suminf_ereal_2dimen:
17   fixes f:: "nat \<times> nat \<Rightarrow> ereal"
18   assumes pos: "\<And>p. 0 \<le> f p"
19   assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
20   shows "(\<Sum>i. f (prod_decode i)) = suminf g"
21 proof -
22   have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
23     using assms by (simp add: fun_eq_iff)
24   have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
25     by (simp add: setsum.reindex[OF inj_prod_decode] comp_def)
26   { fix n
27     let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
28     { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
29       then have "a < ?M fst" "b < ?M snd"
30         by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
31     then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
32       by (auto intro!: setsum_mono3 simp: pos)
33     then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
34   moreover
35   { fix a b
36     let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
37     { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
38         by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
39     then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
40       by (auto intro!: setsum_mono3 simp: pos) }
41   ultimately
42   show ?thesis unfolding g_def using pos
43     by (auto intro!: SUP_eq  simp: setsum.cartesian_product reindex SUP_upper2
44                      setsum_nonneg suminf_ereal_eq_SUP SUP_pair
45                      SUP_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
46 qed
48 subsection {* Characterizations of Measures *}
50 definition subadditive where "subadditive M f \<longleftrightarrow>
51   (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
53 definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow>
54   (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
55     (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
57 definition outer_measure_space where "outer_measure_space M f \<longleftrightarrow>
58   positive M f \<and> increasing M f \<and> countably_subadditive M f"
60 definition measure_set where "measure_set M f X = {r.
61   \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
64   "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
67 subsubsection {* Lambda Systems *}
69 definition lambda_system where "lambda_system \<Omega> M f = {l \<in> M.
70   \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
72 lemma (in algebra) lambda_system_eq:
73   shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
74 proof -
75   have [simp]: "!!l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
76     by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
77   show ?thesis
78     by (auto simp add: lambda_system_def) (metis Int_commute)+
79 qed
81 lemma (in algebra) lambda_system_empty:
82   "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
83   by (auto simp add: positive_def lambda_system_eq)
85 lemma lambda_system_sets:
86   "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
87   by (simp add: lambda_system_def)
89 lemma (in algebra) lambda_system_Compl:
90   fixes f:: "'a set \<Rightarrow> ereal"
91   assumes x: "x \<in> lambda_system \<Omega> M f"
92   shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
93 proof -
94   have "x \<subseteq> \<Omega>"
95     by (metis sets_into_space lambda_system_sets x)
96   hence "\<Omega> - (\<Omega> - x) = x"
97     by (metis double_diff equalityE)
98   with x show ?thesis
99     by (force simp add: lambda_system_def ac_simps)
100 qed
102 lemma (in algebra) lambda_system_Int:
103   fixes f:: "'a set \<Rightarrow> ereal"
104   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
105   shows "x \<inter> y \<in> lambda_system \<Omega> M f"
106 proof -
107   from xl yl show ?thesis
108   proof (auto simp add: positive_def lambda_system_eq Int)
109     fix u
110     assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
111        and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
112        and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
113     have "u - x \<inter> y \<in> M"
114       by (metis Diff Diff_Int Un u x y)
115     moreover
116     have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
117     moreover
118     have "u - x \<inter> y - y = u - y" by blast
119     ultimately
120     have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
121       by force
122     have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
123           = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
124       by (simp add: ey ac_simps)
125     also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
126       by (simp add: Int_ac)
127     also have "... = f (u \<inter> y) + f (u - y)"
128       using fx [THEN bspec, of "u \<inter> y"] Int y u
129       by force
130     also have "... = f u"
131       by (metis fy u)
132     finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
133   qed
134 qed
136 lemma (in algebra) lambda_system_Un:
137   fixes f:: "'a set \<Rightarrow> ereal"
138   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
139   shows "x \<union> y \<in> lambda_system \<Omega> M f"
140 proof -
141   have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
142     by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
143   moreover
144   have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))"
145     by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
146   ultimately show ?thesis
147     by (metis lambda_system_Compl lambda_system_Int xl yl)
148 qed
150 lemma (in algebra) lambda_system_algebra:
151   "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
152   apply (auto simp add: algebra_iff_Un)
153   apply (metis lambda_system_sets set_mp sets_into_space)
154   apply (metis lambda_system_empty)
155   apply (metis lambda_system_Compl)
156   apply (metis lambda_system_Un)
157   done
159 lemma (in algebra) lambda_system_strong_additive:
160   assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
161       and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
162   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
163 proof -
164   have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
165   moreover
166   have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
167   moreover
168   have "(z \<inter> (x \<union> y)) \<in> M"
169     by (metis Int Un lambda_system_sets xl yl z)
170   ultimately show ?thesis using xl yl
171     by (simp add: lambda_system_eq)
172 qed
174 lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
176   fix x and y
177   assume disj: "x \<inter> y = {}"
178      and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
179   hence  "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+
180   thus "f (x \<union> y) = f x + f y"
181     using lambda_system_strong_additive [OF top disj xl yl]
182     by (simp add: Un)
183 qed
186   assumes f: "positive M f" and cs: "countably_subadditive M f"
187   shows  "subadditive M f"
189   fix x y
190   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
191   hence "disjoint_family (binaryset x y)"
192     by (auto simp add: disjoint_family_on_def binaryset_def)
193   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
194          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
195          f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
196     using cs by (auto simp add: countably_subadditive_def)
197   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
198          f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
199     by (simp add: range_binaryset_eq UN_binaryset_eq)
200   thus "f (x \<union> y) \<le>  f x + f y" using f x y
201     by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
202 qed
204 lemma lambda_system_increasing:
205  "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
206   by (simp add: increasing_def lambda_system_def)
208 lemma lambda_system_positive:
209   "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
210   by (simp add: positive_def lambda_system_def)
212 lemma (in algebra) lambda_system_strong_sum:
213   fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
214   assumes f: "positive M f" and a: "a \<in> M"
215       and A: "range A \<subseteq> lambda_system \<Omega> M f"
216       and disj: "disjoint_family A"
217   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
218 proof (induct n)
219   case 0 show ?case using f by (simp add: positive_def)
220 next
221   case (Suc n)
222   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
223     by (force simp add: disjoint_family_on_def neq_iff)
224   have 3: "A n \<in> lambda_system \<Omega> M f" using A
225     by blast
226   interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
227     using f by (rule lambda_system_algebra)
228   have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f"
229     using A l.UNION_in_sets by simp
230   from Suc.hyps show ?case
231     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
232 qed
234 lemma (in sigma_algebra) lambda_system_caratheodory:
235   assumes oms: "outer_measure_space M f"
236       and A: "range A \<subseteq> lambda_system \<Omega> M f"
237       and disj: "disjoint_family A"
238   shows  "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
239 proof -
240   have pos: "positive M f" and inc: "increasing M f"
241    and csa: "countably_subadditive M f"
242     by (metis oms outer_measure_space_def)+
243   have sa: "subadditive M f"
245   have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
246     by auto
247   interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
248     using pos by (rule lambda_system_algebra)
249   have A'': "range A \<subseteq> M"
250      by (metis A image_subset_iff lambda_system_sets)
252   have U_in: "(\<Union>i. A i) \<in> M"
253     by (metis A'' countable_UN)
254   have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
255   proof (rule antisym)
256     show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
257       using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
258     have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
259     have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
260     show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
261       using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis]
262       using A''
263       by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] countable_UN)
264   qed
265   {
266     fix a
267     assume a [iff]: "a \<in> M"
268     have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
269     proof -
270       show ?thesis
271       proof (rule antisym)
272         have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A''
273           by blast
274         moreover
275         have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
276           by (auto simp add: disjoint_family_on_def)
277         moreover
278         have "a \<inter> (\<Union>i. A i) \<in> M"
279           by (metis Int U_in a)
280         ultimately
281         have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
282           using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
283           by (simp add: o_def)
284         hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
285             (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
286           by (rule add_right_mono)
287         moreover
288         have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
289           proof (intro suminf_bound_add allI)
290             fix n
291             have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
292               by (metis A'' UNION_in_sets)
293             have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
294               by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
295             have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f"
296               using ls.UNION_in_sets by (simp add: A)
297             hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
298               by (simp add: lambda_system_eq UNION_in)
299             have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
300               by (blast intro: increasingD [OF inc] UNION_in U_in)
301             thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
302               by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
303           next
304             have "\<And>i. a \<inter> A i \<in> M" using A'' by auto
305             then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
306             have "\<And>i. a - (\<Union>i. A i) \<in> M" using A'' by auto
307             then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
308             then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
309           qed
310         ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
311           by (rule order_trans)
312       next
313         have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
314           by (blast intro:  increasingD [OF inc] U_in)
315         also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
316           by (blast intro: subadditiveD [OF sa] U_in)
317         finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
318         qed
319      qed
320   }
321   thus  ?thesis
322     by (simp add: lambda_system_eq sums_iff U_eq U_in)
323 qed
325 lemma (in sigma_algebra) caratheodory_lemma:
326   assumes oms: "outer_measure_space M f"
327   defines "L \<equiv> lambda_system \<Omega> M f"
328   shows "measure_space \<Omega> L f"
329 proof -
330   have pos: "positive M f"
331     by (metis oms outer_measure_space_def)
332   have alg: "algebra \<Omega> L"
333     using lambda_system_algebra [of f, OF pos]
334     by (simp add: algebra_iff_Un L_def)
335   then
336   have "sigma_algebra \<Omega> L"
337     using lambda_system_caratheodory [OF oms]
338     by (simp add: sigma_algebra_disjoint_iff L_def)
339   moreover
340   have "countably_additive L f" "positive L f"
341     using pos lambda_system_caratheodory [OF oms]
342     by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
343   ultimately
344   show ?thesis
345     using pos by (simp add: measure_space_def)
346 qed
348 lemma inf_measure_nonempty:
349   assumes f: "positive M f" and b: "b \<in> M" and a: "a \<subseteq> b" "{} \<in> M"
350   shows "f b \<in> measure_set M f a"
351 proof -
352   let ?A = "\<lambda>i::nat. (if i = 0 then b else {})"
353   have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))"
354     by (rule suminf_finite) (simp_all add: f[unfolded positive_def])
355   also have "... = f b"
356     by simp
357   finally show ?thesis using assms
358     by (auto intro!: exI [of _ ?A]
359              simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
360 qed
362 lemma (in ring_of_sets) inf_measure_agrees:
363   assumes posf: "positive M f" and ca: "countably_additive M f"
364       and s: "s \<in> M"
365   shows "Inf (measure_set M f s) = f s"
366 proof (intro Inf_eqI)
367   fix z
368   assume z: "z \<in> measure_set M f s"
369   from this obtain A where
370     A: "range A \<subseteq> M" and disj: "disjoint_family A"
371     and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z"
372     by (auto simp add: measure_set_def comp_def)
373   hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
374   have inc: "increasing M f"
376   have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
377     proof (rule ca[unfolded countably_additive_def, rule_format])
378       show "range (\<lambda>n. A n \<inter> s) \<subseteq> M" using A s
379         by blast
380       show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
381         by (auto simp add: disjoint_family_on_def)
382       show "(\<Union>i. A i \<inter> s) \<in> M" using A s
383         by (metis UN_extend_simps(4) s seq)
384     qed
385   hence "f s = (\<Sum>i. f (A i \<inter> s))"
386     using seq [symmetric] by (simp add: sums_iff)
387   also have "... \<le> (\<Sum>i. f (A i))"
388     proof (rule suminf_le_pos)
389       fix n show "f (A n \<inter> s) \<le> f (A n)" using A s
390         by (force intro: increasingD [OF inc])
391       fix N have "A N \<inter> s \<in> M"  using A s by auto
392       then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto
393     qed
394   also have "... = z" by (rule si)
395   finally show "f s \<le> z" .
396 qed (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl])
398 lemma measure_set_pos:
399   assumes posf: "positive M f" "r \<in> measure_set M f X"
400   shows "0 \<le> r"
401 proof -
402   obtain A where "range A \<subseteq> M" and r: "r = (\<Sum>i. f (A i))"
403     using `r \<in> measure_set M f X` unfolding measure_set_def by auto
404   then show "0 \<le> r" using posf unfolding r positive_def
405     by (intro suminf_0_le) auto
406 qed
408 lemma inf_measure_pos:
409   assumes posf: "positive M f"
410   shows "0 \<le> Inf (measure_set M f X)"
411 proof (rule complete_lattice_class.Inf_greatest)
412   fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r"
413     by (rule measure_set_pos)
414 qed
416 lemma inf_measure_empty:
417   assumes posf: "positive M f" and "{} \<in> M"
418   shows "Inf (measure_set M f {}) = 0"
419 proof (rule antisym)
420   show "Inf (measure_set M f {}) \<le> 0"
421     by (metis complete_lattice_class.Inf_lower `{} \<in> M`
422               inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
423 qed (rule inf_measure_pos[OF posf])
425 lemma (in ring_of_sets) inf_measure_positive:
426   assumes p: "positive M f" and "{} \<in> M"
427   shows "positive M (\<lambda>x. Inf (measure_set M f x))"
428 proof (unfold positive_def, intro conjI ballI)
429   show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto
430   fix A assume "A \<in> M"
431 qed (rule inf_measure_pos[OF p])
433 lemma (in ring_of_sets) inf_measure_increasing:
434   assumes posf: "positive M f"
435   shows "increasing (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
436 apply (clarsimp simp add: increasing_def)
437 apply (rule complete_lattice_class.Inf_greatest)
438 apply (rule complete_lattice_class.Inf_lower)
439 apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
440 done
442 lemma (in ring_of_sets) inf_measure_le:
443   assumes posf: "positive M f" and inc: "increasing M f"
444       and x: "x \<in> {r . \<exists>A. range A \<subseteq> M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}"
445   shows "Inf (measure_set M f s) \<le> x"
446 proof -
447   obtain A where A: "range A \<subseteq> M" and ss: "s \<subseteq> (\<Union>i. A i)"
448              and xeq: "(\<Sum>i. f (A i)) = x"
449     using x by auto
450   have dA: "range (disjointed A) \<subseteq> M"
451     by (metis A range_disjointed_sets)
452   have "\<forall>n. f (disjointed A n) \<le> f (A n)"
453     by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
454   moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
455     using posf dA unfolding positive_def by auto
456   ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
457     by (blast intro!: suminf_le_pos)
458   hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x"
459     by (metis xeq)
460   hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s"
461     apply (auto simp add: measure_set_def)
462     apply (rule_tac x="disjointed A" in exI)
463     apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
464     done
465   show ?thesis
466     by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower)
467 qed
469 lemma (in ring_of_sets) inf_measure_close:
470   fixes e :: ereal
471   assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (\<Omega>)" and "Inf (measure_set M f s) \<noteq> \<infinity>"
472   shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
473                (\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e"
474 proof -
475   from `Inf (measure_set M f s) \<noteq> \<infinity>` have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>"
476     using inf_measure_pos[OF posf, of s] by auto
477   obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
478     using Inf_ereal_close[OF fin e] by auto
479   thus ?thesis
480     by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
481 qed
483 lemma (in ring_of_sets) inf_measure_countably_subadditive:
484   assumes posf: "positive M f" and inc: "increasing M f"
485   shows "countably_subadditive (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
487   fix A :: "nat \<Rightarrow> 'a set"
488   let ?outer = "\<lambda>B. Inf (measure_set M f B)"
489   assume A: "range A \<subseteq> Pow (\<Omega>)"
490      and disj: "disjoint_family A"
491      and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
493   { fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
494     hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> M \<and> disjoint_family (BB n) \<and>
495         A n \<subseteq> (\<Union>i. BB n i) \<and> (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
496       apply (safe intro!: choice inf_measure_close [of f, OF posf])
497       using e sb by (auto simp: ereal_zero_less_0_iff one_ereal_def)
498     then obtain BB
499       where BB: "\<And>n. (range (BB n) \<subseteq> M)"
500       and disjBB: "\<And>n. disjoint_family (BB n)"
501       and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
502       and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)"
503       by auto blast
504     have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n)) + e"
505     proof -
506       have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e"
507         using suminf_half_series_ereal e
508         by (simp add: ereal_zero_le_0_iff zero_le_divide_ereal suminf_cmult_ereal)
509       have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto
510       then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le)
511       then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n) + e*(1/2) ^ Suc n)"
512         by (rule suminf_le_pos[OF BBle])
513       also have "... = (\<Sum>n. ?outer (A n)) + e"
514         using sum_eq_1 inf_measure_pos[OF posf] e
515         by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff)
516       finally show ?thesis .
517     qed
518     def C \<equiv> "(split BB) o prod_decode"
519     have C: "!!n. C n \<in> M"
520       apply (rule_tac p="prod_decode n" in PairE)
521       apply (simp add: C_def)
522       apply (metis BB subsetD rangeI)
523       done
524     have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
525     proof (auto simp add: C_def)
526       fix x i
527       assume x: "x \<in> A i"
528       with sbBB [of i] obtain j where "x \<in> BB i j"
529         by blast
530       thus "\<exists>i. x \<in> split BB (prod_decode i)"
531         by (metis prod_encode_inverse prod.case)
532     qed
533     have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
534       by (rule ext)  (auto simp add: C_def)
535     moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle
536       using BB posf[unfolded positive_def]
537       by (force intro!: suminf_ereal_2dimen simp: o_def)
538     ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def)
539     have "?outer (\<Union>i. A i) \<le> (\<Sum>n. \<Sum>i. f (BB n i))"
540       apply (rule inf_measure_le [OF posf(1) inc], auto)
541       apply (rule_tac x="C" in exI)
542       apply (auto simp add: C sbC Csums)
543       done
544     also have "... \<le> (\<Sum>n. ?outer (A n)) + e" using sll
545       by blast
546     finally have "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n)) + e" . }
547   note for_finite_Inf = this
549   show "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n))"
550   proof cases
551     assume "\<forall>i. ?outer (A i) \<noteq> \<infinity>"
552     with for_finite_Inf show ?thesis
553       by (intro ereal_le_epsilon) auto
554   next
555     assume "\<not> (\<forall>i. ?outer (A i) \<noteq> \<infinity>)"
556     then have "\<exists>i. ?outer (A i) = \<infinity>"
557       by auto
558     then have "(\<Sum>n. ?outer (A n)) = \<infinity>"
559       using suminf_PInfty[OF inf_measure_pos, OF posf]
560       by metis
561     then show ?thesis by simp
562   qed
563 qed
565 lemma (in ring_of_sets) inf_measure_outer:
566   "\<lbrakk> positive M f ; increasing M f \<rbrakk> \<Longrightarrow>
567     outer_measure_space (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
568   using inf_measure_pos[of M f]
569   by (simp add: outer_measure_space_def inf_measure_empty
570                 inf_measure_increasing inf_measure_countably_subadditive positive_def)
572 lemma (in ring_of_sets) algebra_subset_lambda_system:
573   assumes posf: "positive M f" and inc: "increasing M f"
575   shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (\<lambda>x. Inf (measure_set M f x))"
576 proof (auto dest: sets_into_space
577             simp add: algebra.lambda_system_eq [OF algebra_Pow])
578   fix x s
579   assume x: "x \<in> M"
580      and s: "s \<subseteq> \<Omega>"
581   have [simp]: "!!x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s-x" using s
582     by blast
583   have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
584         \<le> Inf (measure_set M f s)"
585   proof cases
586     assume "Inf (measure_set M f s) = \<infinity>" then show ?thesis by simp
587   next
588     assume fin: "Inf (measure_set M f s) \<noteq> \<infinity>"
589     then have "measure_set M f s \<noteq> {}"
590       by (auto simp: top_ereal_def)
591     show ?thesis
592     proof (rule complete_lattice_class.Inf_greatest)
593       fix r assume "r \<in> measure_set M f s"
594       then obtain A where A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)"
595         and r: "r = (\<Sum>i. f (A i))" unfolding measure_set_def by auto
596       have "Inf (measure_set M f (s \<inter> x)) \<le> (\<Sum>i. f (A i \<inter> x))"
597         unfolding measure_set_def
598       proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i \<inter> x"])
599         from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)"
600           by (rule disjoint_family_on_bisimulation) auto
601       qed (insert x A, auto)
602       moreover
603       have "Inf (measure_set M f (s - x)) \<le> (\<Sum>i. f (A i - x))"
604         unfolding measure_set_def
605       proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i - x"])
606         from A(1) show "disjoint_family (\<lambda>i. A i - x)"
607           by (rule disjoint_family_on_bisimulation) auto
608       qed (insert x A, auto)
609       ultimately have "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le>
610           (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
611       also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
612         using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def)
613       also have "\<dots> = (\<Sum>i. f (A i))"
614         using A x
616            (auto intro!: arg_cong[where f=suminf] arg_cong[where f=f])
617       finally show "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> r"
618         using r by simp
619     qed
620   qed
621   moreover
622   have "Inf (measure_set M f s)
623        \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
624   proof -
625     have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
626       by (metis Un_Diff_Int Un_commute)
627     also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
628       apply (rule subadditiveD)
630       apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf])
631       apply (rule inf_measure_countably_subadditive)
632       using s by (auto intro!: posf inc)
633     finally show ?thesis .
634   qed
635   ultimately
636   show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
637         = Inf (measure_set M f s)"
638     by (rule order_antisym)
639 qed
641 lemma measure_down:
642   "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
643   by (simp add: measure_space_def positive_def countably_additive_def)
644      blast
646 subsection {* Caratheodory's theorem *}
648 theorem (in ring_of_sets) caratheodory':
649   assumes posf: "positive M f" and ca: "countably_additive M f"
650   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
651 proof -
652   have inc: "increasing M f"
654   let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
655   def ls \<equiv> "lambda_system \<Omega> (Pow \<Omega>) ?infm"
656   have mls: "measure_space \<Omega> ls ?infm"
657     using sigma_algebra.caratheodory_lemma
658             [OF sigma_algebra_Pow  inf_measure_outer [OF posf inc]]
659     by (simp add: ls_def)
660   hence sls: "sigma_algebra \<Omega> ls"
661     by (simp add: measure_space_def)
662   have "M \<subseteq> ls"
663     by (simp add: ls_def)
664        (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
665   hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
666     using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
667     by simp
668   have "measure_space \<Omega> (sigma_sets \<Omega> M) ?infm"
669     by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
670        (simp_all add: sgs_sb space_closed)
671   thus ?thesis using inf_measure_agrees [OF posf ca]
672     by (intro exI[of _ ?infm]) auto
673 qed
675 lemma (in ring_of_sets) caratheodory_empty_continuous:
676   assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
677   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0"
678   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
679 proof (intro caratheodory' empty_continuous_imp_countably_additive f)
680   show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
681 qed (rule cont)
683 subsection {* Volumes *}
685 definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
686   "volume M f \<longleftrightarrow>
687   (f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
688   (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
690 lemma volumeI:
691   assumes "f {} = 0"
692   assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
693   assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)"
694   shows "volume M f"
695   using assms by (auto simp: volume_def)
697 lemma volume_positive:
698   "volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
699   by (auto simp: volume_def)
701 lemma volume_empty:
702   "volume M f \<Longrightarrow> f {} = 0"
703   by (auto simp: volume_def)
706   assumes "volume M f"
707   assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
708   shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
709 proof -
710   have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>(A`I) \<in> M"
711     using A unfolding SUP_def by (auto simp: disjoint_family_on_disjoint_image)
712   with `volume M f` have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
713     unfolding volume_def by blast
714   also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
715   proof (subst setsum.reindex_nontrivial)
716     fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
717     with `disjoint_family_on A I` have "A i = {}"
718       by (auto simp: disjoint_family_on_def)
719     then show "f (A i) = 0"
720       using volume_empty[OF `volume M f`] by simp
721   qed (auto intro: `finite I`)
722   finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
723     by simp
724 qed
726 lemma (in ring_of_sets) volume_additiveI:
727   assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
728   assumes [simp]: "\<mu> {} = 0"
729   assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
730   shows "volume M \<mu>"
731 proof (unfold volume_def, safe)
732   fix C assume "finite C" "C \<subseteq> M" "disjoint C"
733   then show "\<mu> (\<Union>C) = setsum \<mu> C"
734   proof (induct C)
735     case (insert c C)
736     from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
737       by (auto intro!: add simp: disjoint_def)
738     with insert show ?case
739       by (simp add: disjoint_def)
740   qed simp
741 qed fact+
743 lemma (in semiring_of_sets) extend_volume:
744   assumes "volume M \<mu>"
745   shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
746 proof -
747   let ?R = generated_ring
748   have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
749     by (auto simp: generated_ring_def)
750   from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
752   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
753     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
754     assume "\<Union>C = \<Union>D"
755     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
756     proof (intro setsum.cong refl)
757       fix d assume "d \<in> D"
758       have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
759         using `d \<in> D` `\<Union>C = \<Union>D` by auto
760       moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
761       proof (rule volume_finite_additive)
762         { fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
763             using C D `d \<in> D` by auto }
764         show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
765           unfolding Un_eq_d using `d \<in> D` D by auto
766         show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
767           using `disjoint C` by (auto simp: disjoint_family_on_def disjoint_def)
768       qed fact+
769       ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
770     qed }
771   note split_sum = this
773   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
774     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
775     assume "\<Union>C = \<Union>D"
776     with split_sum[OF C D] split_sum[OF D C]
777     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
778       by (simp, subst setsum.commute, simp add: ac_simps) }
779   note sum_eq = this
781   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
782     then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def)
783     with \<mu>'_spec[THEN bspec, of "\<Union>C"]
784     obtain D where
785       D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)"
786       by blast
787     with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp }
788   note \<mu>' = this
790   show ?thesis
791   proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI)
792     fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a"
793       by (simp add: disjoint_def)
794   next
795     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
796     with \<mu>'[of Ca] `volume M \<mu>`[THEN volume_positive]
797     show "0 \<le> \<mu>' a"
798       by (auto intro!: setsum_nonneg)
799   next
800     show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
801   next
802     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
803     fix b assume "b \<in> ?R" then guess Cb .. note Cb = this
804     assume "a \<inter> b = {}"
805     with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
806     then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
808     from `a \<inter> b = {}` have "\<mu>' (\<Union> (Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
809       using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
810     also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
811       using C_Int_cases volume_empty[OF `volume M \<mu>`] by (elim disjE) simp_all
812     also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
813       using Ca Cb by (simp add: setsum.union_inter)
814     also have "\<dots> = \<mu>' a + \<mu>' b"
815       using Ca Cb by (simp add: \<mu>')
816     finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
817       using Ca Cb by simp
818   qed
819 qed
821 subsubsection {* Caratheodory on semirings *}
823 theorem (in semiring_of_sets) caratheodory:
824   assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
825   shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
826 proof -
827   have "volume M \<mu>"
828   proof (rule volumeI)
829     { fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
830         using pos unfolding positive_def by auto }
831     note p = this
833     fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
834     have "\<exists>F'. bij_betw F' {..<card C} C"
835       by (rule finite_same_card_bij[OF _ `finite C`]) auto
836     then guess F' .. note F' = this
837     then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
838       by (auto simp: bij_betw_def)
839     { fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
840       with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
841         unfolding inj_on_def by auto
842       with `disjoint C`[THEN disjointD]
843       have "F' i \<inter> F' j = {}"
844         by auto }
845     note F'_disj = this
846     def F \<equiv> "\<lambda>i. if i < card C then F' i else {}"
847     then have "disjoint_family F"
848       using F'_disj by (auto simp: disjoint_family_on_def)
849     moreover from F' have "(\<Union>i. F i) = \<Union>C"
850       by (auto simp: F_def set_eq_iff split: split_if_asm)
851     moreover have sets_F: "\<And>i. F i \<in> M"
852       using F' sets_C by (auto simp: F_def)
853     moreover note sets_C
854     ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))"
855       using ca[unfolded countably_additive_def, THEN spec, of F] by auto
856     also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))"
857     proof -
858       have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))"
859         by (rule sums_If_finite_set) auto
860       also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))"
861         using pos by (auto simp: positive_def F_def)
862       finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))"
863         by (simp add: sums_iff)
864     qed
865     also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
866       using F'(2) by (subst (2) F') (simp add: setsum.reindex)
867     finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
868   next
869     show "\<mu> {} = 0"
870       using `positive M \<mu>` by (rule positiveD1)
871   qed
872   from extend_volume[OF this] obtain \<mu>_r where
873     V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
874     by auto
876   interpret G: ring_of_sets \<Omega> generated_ring
877     by (rule generating_ring)
879   have pos: "positive generated_ring \<mu>_r"
880     using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty)
882   have "countably_additive generated_ring \<mu>_r"
883   proof (rule countably_additiveI)
884     fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'"
885       and Un_A: "(\<Union>i. A' i) \<in> generated_ring"
887     from generated_ringE[OF Un_A] guess C' . note C' = this
889     { fix c assume "c \<in> C'"
890       moreover def A \<equiv> "\<lambda>i. A' i \<inter> c"
891       ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A"
892         and Un_A: "(\<Union>i. A i) \<in> generated_ring"
893         using A' C'
894         by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
895       from A C' `c \<in> C'` have UN_eq: "(\<Union>i. A i) = c"
896         by (auto simp: A_def)
898       have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)"
899         (is "\<forall>i. ?P i")
900       proof
901         fix i
902         from A have Ai: "A i \<in> generated_ring" by auto
903         from generated_ringE[OF this] guess C . note C = this
905         have "\<exists>F'. bij_betw F' {..<card C} C"
906           by (rule finite_same_card_bij[OF _ `finite C`]) auto
907         then guess F .. note F = this
908         def f \<equiv> "\<lambda>i. if i < card C then F i else {}"
909         then have f: "bij_betw f {..< card C} C"
910           by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto
911         with C have "\<forall>j. f j \<in> M"
912           by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset)
913         moreover
914         from f C have d_f: "disjoint_family_on f {..<card C}"
915           by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def)
916         then have "disjoint_family f"
917           by (auto simp: disjoint_family_on_def f_def)
918         moreover
919         have Ai_eq: "A i = (\<Union> x<card C. f x)"
920           using f C Ai unfolding bij_betw_def by (simp add: Union_image_eq[symmetric])
921         then have "\<Union>range f = A i"
922           using f C Ai unfolding bij_betw_def by (auto simp: f_def)
923         moreover
924         { have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
925             using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
926           also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
927             by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp
928           also have "\<dots> = \<mu>_r (A i)"
929             using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq
930             by (intro volume_finite_additive[OF V(1) _ d_f, symmetric])
931                (auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic)
932           finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. }
933         ultimately show "?P i"
934           by blast
935       qed
936       from choice[OF this] guess f .. note f = this
937       then have UN_f_eq: "(\<Union>i. split f (prod_decode i)) = (\<Union>i. A i)"
938         unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff)
940       have d: "disjoint_family (\<lambda>i. split f (prod_decode i))"
941         unfolding disjoint_family_on_def
942       proof (intro ballI impI)
943         fix m n :: nat assume "m \<noteq> n"
944         then have neq: "prod_decode m \<noteq> prod_decode n"
945           using inj_prod_decode[of UNIV] by (auto simp: inj_on_def)
946         show "split f (prod_decode m) \<inter> split f (prod_decode n) = {}"
947         proof cases
948           assume "fst (prod_decode m) = fst (prod_decode n)"
949           then show ?thesis
950             using neq f by (fastforce simp: disjoint_family_on_def)
951         next
952           assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)"
953           have "split f (prod_decode m) \<subseteq> A (fst (prod_decode m))"
954             "split f (prod_decode n) \<subseteq> A (fst (prod_decode n))"
955             using f[THEN spec, of "fst (prod_decode m)"]
956             using f[THEN spec, of "fst (prod_decode n)"]
957             by (auto simp: set_eq_iff)
958           with f A neq show ?thesis
959             by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff)
960         qed
961       qed
962       from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (split f (prod_decode n)))"
963         by (intro suminf_ereal_2dimen[symmetric] positiveD2[OF pos] generated_ringI_Basic)
964          (auto split: prod.split)
965       also have "\<dots> = (\<Sum>n. \<mu> (split f (prod_decode n)))"
966         using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
967       also have "\<dots> = \<mu> (\<Union>i. split f (prod_decode i))"
968         using f `c \<in> C'` C'
969         by (intro ca[unfolded countably_additive_def, rule_format])
970            (auto split: prod.split simp: UN_f_eq d UN_eq)
971       finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
972         using UN_f_eq UN_eq by (simp add: A_def) }
973     note eq = this
975     have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))"
976       using C' A'
977       by (subst volume_finite_additive[symmetric, OF V(1)])
978          (auto simp: disjoint_def disjoint_family_on_def Union_image_eq[symmetric] simp del: Sup_image_eq Union_image_eq
979                intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext
980                intro: generated_ringI_Basic)
981     also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
982       using C' A'
983       by (intro suminf_setsum_ereal positiveD2[OF pos] G.Int G.finite_Union)
984          (auto intro: generated_ringI_Basic)
985     also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
986       using eq V C' by (auto intro!: setsum.cong)
987     also have "\<dots> = \<mu>_r (\<Union>C')"
988       using C' Un_A
989       by (subst volume_finite_additive[symmetric, OF V(1)])
990          (auto simp: disjoint_family_on_def disjoint_def Union_image_eq[symmetric] simp del: Sup_image_eq Union_image_eq
991                intro: generated_ringI_Basic)
992     finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
993       using C' by simp
994   qed
995   from G.caratheodory'[OF `positive generated_ring \<mu>_r` `countably_additive generated_ring \<mu>_r`]
996   guess \<mu>' ..
997   with V show ?thesis
998     unfolding sigma_sets_generated_ring_eq
999     by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
1000 qed
1002 end