src/HOL/Probability/Radon_Nikodym.thy
author huffman
Sun Apr 01 16:09:58 2012 +0200 (2012-04-01)
changeset 47255 30a1692557b0
parent 46731 5302e932d1e5
child 47694 05663f75964c
permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
     1 (*  Title:      HOL/Probability/Radon_Nikodym.thy
     2     Author:     Johannes Hölzl, TU München
     3 *)
     4 
     5 header {*Radon-Nikod{\'y}m derivative*}
     6 
     7 theory Radon_Nikodym
     8 imports Lebesgue_Integration
     9 begin
    10 
    11 lemma (in sigma_finite_measure) Ex_finite_integrable_function:
    12   shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)"
    13 proof -
    14   obtain A :: "nat \<Rightarrow> 'a set" where
    15     range: "range A \<subseteq> sets M" and
    16     space: "(\<Union>i. A i) = space M" and
    17     measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>" and
    18     disjoint: "disjoint_family A"
    19     using disjoint_sigma_finite by auto
    20   let ?B = "\<lambda>i. 2^Suc i * \<mu> (A i)"
    21   have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
    22   proof
    23     fix i have Ai: "A i \<in> sets M" using range by auto
    24     from measure positive_measure[OF this]
    25     show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
    26       by (auto intro!: ereal_dense simp: ereal_0_gt_inverse)
    27   qed
    28   from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
    29     "\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
    30   { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
    31   let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x"
    32   show ?thesis
    33   proof (safe intro!: bexI[of _ ?h] del: notI)
    34     have "\<And>i. A i \<in> sets M"
    35       using range by fastforce+
    36     then have "integral\<^isup>P M ?h = (\<Sum>i. n i * \<mu> (A i))" using pos
    37       by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
    38     also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
    39     proof (rule suminf_le_pos)
    40       fix N
    41       have "n N * \<mu> (A N) \<le> inverse (2^Suc N * \<mu> (A N)) * \<mu> (A N)"
    42         using positive_measure[OF `A N \<in> sets M`] n[of N]
    43         by (intro ereal_mult_right_mono) auto
    44       also have "\<dots> \<le> (1 / 2) ^ Suc N"
    45         using measure[of N] n[of N]
    46         by (cases rule: ereal2_cases[of "n N" "\<mu> (A N)"])
    47            (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
    48       finally show "n N * \<mu> (A N) \<le> (1 / 2) ^ Suc N" .
    49       show "0 \<le> n N * \<mu> (A N)" using n[of N] `A N \<in> sets M` by simp
    50     qed
    51     finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
    52   next
    53     { fix x assume "x \<in> space M"
    54       then obtain i where "x \<in> A i" using space[symmetric] by auto
    55       with disjoint n have "?h x = n i"
    56         by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
    57       then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto }
    58     note pos = this
    59     fix x show "0 \<le> ?h x"
    60     proof cases
    61       assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le)
    62     next
    63       assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto
    64       then show "0 \<le> ?h x" by auto
    65     qed
    66   next
    67     show "?h \<in> borel_measurable M" using range n
    68       by (auto intro!: borel_measurable_psuminf borel_measurable_ereal_times ereal_0_le_mult intro: less_imp_le)
    69   qed
    70 qed
    71 
    72 subsection "Absolutely continuous"
    73 
    74 definition (in measure_space)
    75   "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: ereal))"
    76 
    77 lemma (in measure_space) absolutely_continuous_AE:
    78   assumes "measure_space M'" and [simp]: "sets M' = sets M" "space M' = space M"
    79     and "absolutely_continuous (measure M')" "AE x. P x"
    80    shows "AE x in M'. P x"
    81 proof -
    82   interpret \<nu>: measure_space M' by fact
    83   from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
    84     unfolding almost_everywhere_def by auto
    85   show "AE x in M'. P x"
    86   proof (rule \<nu>.AE_I')
    87     show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
    88     from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
    89       using N unfolding absolutely_continuous_def by auto
    90   qed
    91 qed
    92 
    93 lemma (in finite_measure_space) absolutely_continuousI:
    94   assumes "finite_measure_space (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure_space ?\<nu>")
    95   assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
    96   shows "absolutely_continuous \<nu>"
    97 proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
    98   fix N assume "\<mu> N = 0" "N \<subseteq> space M"
    99   interpret v: finite_measure_space ?\<nu> by fact
   100   have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
   101   also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
   102   proof (rule v.measure_setsum[symmetric])
   103     show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
   104     show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
   105     fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
   106   qed
   107   also have "\<dots> = 0"
   108   proof (safe intro!: setsum_0')
   109     fix x assume "x \<in> N"
   110     hence "\<mu> {x} \<le> \<mu> N" "0 \<le> \<mu> {x}"
   111       using sets_eq_Pow `N \<subseteq> space M` positive_measure[of "{x}"]
   112       by (auto intro!: measure_mono)
   113     then have "\<mu> {x} = 0" using `\<mu> N = 0` by simp
   114     thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
   115   qed
   116   finally show "\<nu> N = 0" by simp
   117 qed
   118 
   119 lemma (in measure_space) density_is_absolutely_continuous:
   120   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   121   shows "absolutely_continuous \<nu>"
   122   using assms unfolding absolutely_continuous_def
   123   by (simp add: positive_integral_null_set)
   124 
   125 subsection "Existence of the Radon-Nikodym derivative"
   126 
   127 lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
   128   fixes e :: real assumes "0 < e"
   129   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
   130   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
   131                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
   132                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
   133 proof -
   134   interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
   135   let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
   136   let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
   137     then {}
   138     else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
   139   def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
   140   have A_simps[simp]:
   141     "A 0 = {}"
   142     "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
   143   { fix A assume "A \<in> sets M"
   144     have "?A A \<in> sets M"
   145       by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
   146   note A'_in_sets = this
   147   { fix n have "A n \<in> sets M"
   148     proof (induct n)
   149       case (Suc n) thus "A (Suc n) \<in> sets M"
   150         using A'_in_sets[of "A n"] by (auto split: split_if_asm)
   151     qed (simp add: A_def) }
   152   note A_in_sets = this
   153   hence "range A \<subseteq> sets M" by auto
   154   { fix n B
   155     assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
   156     hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
   157     have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
   158     proof (rule someI2_ex[OF Ex])
   159       fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
   160       hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
   161       hence "?d (A n \<union> B) = ?d (A n) + ?d B"
   162         using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by simp
   163       also have "\<dots> \<le> ?d (A n) - e" using dB by simp
   164       finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
   165     qed }
   166   note dA_epsilon = this
   167   { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
   168     proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
   169       case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
   170     next
   171       case False
   172       hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
   173       thus ?thesis by simp
   174     qed }
   175   note dA_mono = this
   176   show ?thesis
   177   proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
   178     case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast
   179     show ?thesis
   180     proof (safe intro!: bexI[of _ "space M - A n"])
   181       fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
   182       from B[OF this] show "-e < ?d B" .
   183     next
   184       show "space M - A n \<in> sets M" by (rule compl_sets) fact
   185     next
   186       show "?d (space M) \<le> ?d (space M - A n)"
   187       proof (induct n)
   188         fix n assume "?d (space M) \<le> ?d (space M - A n)"
   189         also have "\<dots> \<le> ?d (space M - A (Suc n))"
   190           using A_in_sets sets_into_space dA_mono[of n]
   191           by (simp del: A_simps add: finite_measure_Diff M'.finite_measure_Diff)
   192         finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
   193       qed simp
   194     qed
   195   next
   196     case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
   197       by (auto simp add: not_less)
   198     { fix n have "?d (A n) \<le> - real n * e"
   199       proof (induct n)
   200         case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
   201       next
   202         case 0 with M'.empty_measure show ?case by (simp add: zero_ereal_def)
   203       qed } note dA_less = this
   204     have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
   205     proof (rule incseq_SucI)
   206       fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
   207     qed
   208     have A: "incseq A" by (auto intro!: incseq_SucI)
   209     from finite_continuity_from_below[OF _ A] `range A \<subseteq> sets M`
   210       M'.finite_continuity_from_below[OF _ A]
   211     have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
   212       by (auto intro!: tendsto_diff)
   213     obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
   214     moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
   215     have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
   216     ultimately show ?thesis by auto
   217   qed
   218 qed
   219 
   220 lemma (in finite_measure) restricted_measure_subset:
   221   assumes S: "S \<in> sets M" and X: "X \<subseteq> S"
   222   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
   223 proof cases
   224   note r = restricted_finite_measure[OF S]
   225   { assume "X \<in> sets M" with S X show ?thesis
   226       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def by auto }
   227   { assume "X \<notin> sets M"
   228     moreover then have "S \<inter> X \<notin> sets M"
   229       using X by (simp add: Int_absorb1)
   230     ultimately show ?thesis
   231       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def using S by auto }
   232 qed
   233 
   234 lemma (in finite_measure) restricted_measure:
   235   assumes X: "S \<in> sets M" "X \<in> sets (restricted_space S)"
   236   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
   237 proof -
   238   from X have "S \<in> sets M" "X \<subseteq> S" by auto
   239   from restricted_measure_subset[OF this] show ?thesis .
   240 qed
   241 
   242 lemma (in finite_measure) Radon_Nikodym_aux:
   243   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
   244   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
   245                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
   246                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
   247 proof -
   248   interpret M': finite_measure ?M' where
   249     "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
   250   let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
   251   let ?P = "\<lambda>A B n. A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
   252   let ?r = "\<lambda>S. restricted_space S"
   253   { fix S n assume S: "S \<in> sets M"
   254     note r = M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
   255     then have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
   256       "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
   257     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
   258     have "?P X S n"
   259     proof (intro conjI ballI impI)
   260       show "X \<in> sets M" "X \<subseteq> S" using X(1) `S \<in> sets M` by auto
   261       have "S \<in> op \<inter> S ` sets M" using `S \<in> sets M` by auto
   262       then show "?d S \<le> ?d X"
   263         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by simp
   264       fix C assume "C \<in> sets M" "C \<subseteq> X"
   265       then have "C \<in> sets (restricted_space S)" "C \<subseteq> X"
   266         using `S \<in> sets M` `X \<subseteq> S` by auto
   267       with X(2) show "- 1 / real (Suc n) < ?d C"
   268         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by auto
   269     qed
   270     hence "\<exists>A. ?P A S n" by auto }
   271   note Ex_P = this
   272   def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
   273   have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
   274   have A_0[simp]: "A 0 = space M" unfolding A_def by simp
   275   { fix i have "A i \<in> sets M" unfolding A_def
   276     proof (induct i)
   277       case (Suc i)
   278       from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
   279         by (rule someI2_ex) simp
   280     qed simp }
   281   note A_in_sets = this
   282   { fix n have "?P (A (Suc n)) (A n) n"
   283       using Ex_P[OF A_in_sets] unfolding A_Suc
   284       by (rule someI2_ex) simp }
   285   note P_A = this
   286   have "range A \<subseteq> sets M" using A_in_sets by auto
   287   have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
   288   have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
   289   have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
   290       using P_A by auto
   291   show ?thesis
   292   proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
   293     show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
   294     have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
   295     from
   296       finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
   297       M'.finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
   298     have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (intro tendsto_diff)
   299     thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
   300       by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
   301   next
   302     fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
   303     show "0 \<le> ?d B"
   304     proof (rule ccontr)
   305       assume "\<not> 0 \<le> ?d B"
   306       hence "0 < - ?d B" by auto
   307       from ex_inverse_of_nat_Suc_less[OF this]
   308       obtain n where *: "?d B < - 1 / real (Suc n)"
   309         by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
   310       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
   311       from epsilon[OF B(1) this] *
   312       show False by auto
   313     qed
   314   qed
   315 qed
   316 
   317 lemma (in finite_measure) Radon_Nikodym_finite_measure:
   318   assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
   319   assumes "absolutely_continuous \<nu>"
   320   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   321 proof -
   322   interpret M': finite_measure ?M'
   323     where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
   324     using assms(1) by auto
   325   def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A)}"
   326   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
   327   hence "G \<noteq> {}" by auto
   328   { fix f g assume f: "f \<in> G" and g: "g \<in> G"
   329     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
   330     proof safe
   331       show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
   332       let ?A = "{x \<in> space M. f x \<le> g x}"
   333       have "?A \<in> sets M" using f g unfolding G_def by auto
   334       fix A assume "A \<in> sets M"
   335       hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
   336       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
   337         using sets_into_space[OF `A \<in> sets M`] by auto
   338       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
   339         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
   340         by (auto simp: indicator_def max_def)
   341       hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
   342         (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
   343         (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
   344         using f g sets unfolding G_def
   345         by (auto cong: positive_integral_cong intro!: positive_integral_add)
   346       also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
   347         using f g sets unfolding G_def by (auto intro!: add_mono)
   348       also have "\<dots> = \<nu> A"
   349         using M'.measure_additive[OF sets] union by auto
   350       finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
   351     next
   352       fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
   353     qed }
   354   note max_in_G = this
   355   { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
   356     have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
   357     proof safe
   358       show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
   359         using f by (auto simp: G_def)
   360       { fix x show "0 \<le> (SUP i. f i x)"
   361           using f by (auto simp: G_def intro: SUP_upper2) }
   362     next
   363       fix A assume "A \<in> sets M"
   364       have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
   365         (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
   366         by (intro positive_integral_cong) (simp split: split_indicator)
   367       also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
   368         using `incseq f` f `A \<in> sets M`
   369         by (intro positive_integral_monotone_convergence_SUP)
   370            (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
   371       finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> \<nu> A"
   372         using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
   373     qed }
   374   note SUP_in_G = this
   375   let ?y = "SUP g : G. integral\<^isup>P M g"
   376   have "?y \<le> \<nu> (space M)" unfolding G_def
   377   proof (safe intro!: SUP_least)
   378     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
   379     from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
   380       by (simp cong: positive_integral_cong)
   381   qed
   382   from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
   383   then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
   384   proof safe
   385     fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
   386     hence "ys n \<in> integral\<^isup>P M ` G" by auto
   387     thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
   388   qed
   389   from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto
   390   hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
   391   let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
   392   def f \<equiv> "\<lambda>x. SUP i. ?g i x"
   393   let ?F = "\<lambda>A x. f x * indicator A x"
   394   have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
   395   { fix i have "?g i \<in> G"
   396     proof (induct i)
   397       case 0 thus ?case by simp fact
   398     next
   399       case (Suc i)
   400       with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
   401         by (auto simp add: atMost_Suc intro!: max_in_G)
   402     qed }
   403   note g_in_G = this
   404   have "incseq ?g" using gs_not_empty
   405     by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
   406   from SUP_in_G[OF this g_in_G] have "f \<in> G" unfolding f_def .
   407   then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
   408   have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def
   409     using g_in_G `incseq ?g`
   410     by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def)
   411   also have "\<dots> = ?y"
   412   proof (rule antisym)
   413     show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
   414       using g_in_G
   415       by (auto intro!: exI Sup_mono simp: SUP_def)
   416     show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
   417       by (auto intro!: SUP_mono positive_integral_mono Max_ge)
   418   qed
   419   finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
   420   have "\<And>x. 0 \<le> f x"
   421     unfolding f_def using `\<And>i. gs i \<in> G`
   422     by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
   423   let ?t = "\<lambda>A. \<nu> A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
   424   let ?M = "M\<lparr> measure := ?t\<rparr>"
   425   interpret M: sigma_algebra ?M
   426     by (intro sigma_algebra_cong) auto
   427   have f_le_\<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> \<nu> A"
   428     using `f \<in> G` unfolding G_def by auto
   429   have fmM: "finite_measure ?M"
   430   proof
   431     show "measure_space ?M"
   432     proof (default, simp_all add: countably_additive_def positive_def, safe del: notI)
   433       fix A :: "nat \<Rightarrow> 'a set"  assume A: "range A \<subseteq> sets M" "disjoint_family A"
   434       have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. (\<Sum>n. ?F (A n) x) \<partial>M)"
   435         using `range A \<subseteq> sets M` `\<And>x. 0 \<le> f x`
   436         by (intro positive_integral_suminf[symmetric]) auto
   437       also have "\<dots> = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)"
   438         using `\<And>x. 0 \<le> f x`
   439         by (intro positive_integral_cong) (simp add: suminf_cmult_ereal suminf_indicator[OF `disjoint_family A`])
   440       finally have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)" .
   441       moreover have "(\<Sum>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
   442         using M'.measure_countably_additive A by (simp add: comp_def)
   443       moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<infinity>" using M'.finite_measure A by (simp add: countable_UN)
   444       moreover {
   445         have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
   446           using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
   447         also have "\<nu> (\<Union>i. A i) < \<infinity>" using v_fin by simp
   448         finally have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<noteq> \<infinity>" by simp }
   449       moreover have "\<And>i. (\<integral>\<^isup>+x. ?F (A i) x \<partial>M) \<le> \<nu> (A i)"
   450         using A by (intro f_le_\<nu>) auto
   451       ultimately
   452       show "(\<Sum>n. ?t (A n)) = ?t (\<Union>i. A i)"
   453         by (subst suminf_ereal_minus) (simp_all add: positive_integral_positive)
   454     next
   455       fix A assume A: "A \<in> sets M" show "0 \<le> \<nu> A - \<integral>\<^isup>+ x. ?F A x \<partial>M"
   456         using f_le_\<nu>[OF A] `f \<in> G` M'.finite_measure[OF A] by (auto simp: G_def ereal_le_minus_iff)
   457     qed
   458   next
   459     show "measure ?M (space ?M) \<noteq> \<infinity>"
   460       using positive_integral_positive[of "?F (space M)"]
   461       by (cases rule: ereal2_cases[of "\<nu> (space M)" "\<integral>\<^isup>+ x. ?F (space M) x \<partial>M"]) auto
   462   qed
   463   then interpret M: finite_measure ?M
   464     where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
   465     by (simp_all add: fmM)
   466   have ac: "absolutely_continuous ?t" unfolding absolutely_continuous_def
   467   proof
   468     fix N assume N: "N \<in> null_sets"
   469     with `absolutely_continuous \<nu>` have "\<nu> N = 0" unfolding absolutely_continuous_def by auto
   470     moreover with N have "(\<integral>\<^isup>+ x. ?F N x \<partial>M) \<le> \<nu> N" using `f \<in> G` by (auto simp: G_def)
   471     ultimately show "\<nu> N - (\<integral>\<^isup>+ x. ?F N x \<partial>M) = 0"
   472       using positive_integral_positive by (auto intro!: antisym)
   473   qed
   474   have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
   475   proof (rule ccontr)
   476     assume "\<not> ?thesis"
   477     then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
   478       by (auto simp: not_le)
   479     note pos
   480     also have "?t A \<le> ?t (space M)"
   481       using M.measure_mono[of A "space M"] A sets_into_space by simp
   482     finally have pos_t: "0 < ?t (space M)" by simp
   483     moreover
   484     then have "\<mu> (space M) \<noteq> 0"
   485       using ac unfolding absolutely_continuous_def by auto
   486     then have pos_M: "0 < \<mu> (space M)"
   487       using positive_measure[OF top] by (simp add: le_less)
   488     moreover
   489     have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
   490       using `f \<in> G` unfolding G_def by auto
   491     hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
   492       using M'.finite_measure_of_space by auto
   493     moreover
   494     def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
   495     ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
   496       using M'.finite_measure_of_space positive_integral_positive[of "?F (space M)"]
   497       by (cases rule: ereal3_cases[of "integral\<^isup>P M (?F (space M))" "\<nu> (space M)" "\<mu> (space M)"])
   498          (simp_all add: field_simps)
   499     then have b: "b \<noteq> 0" "0 \<le> b" "0 < b"  "b \<noteq> \<infinity>" by auto
   500     let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
   501     interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
   502     have Mb: "finite_measure ?Mb"
   503     proof
   504       show "measure_space ?Mb"
   505       proof
   506         show "positive ?Mb (measure ?Mb)"
   507           using `0 \<le> b` by (auto simp: positive_def)
   508         show "countably_additive ?Mb (measure ?Mb)"
   509           using `0 \<le> b` measure_countably_additive
   510           by (auto simp: countably_additive_def suminf_cmult_ereal subset_eq)
   511       qed
   512       show "measure ?Mb (space ?Mb) \<noteq> \<infinity>"
   513         using b by auto
   514     qed
   515     from M.Radon_Nikodym_aux[OF this]
   516     obtain A0 where "A0 \<in> sets M" and
   517       space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
   518       *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)"
   519       unfolding M.\<mu>'_def finite_measure.\<mu>'_def[OF Mb] by auto
   520     { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
   521       with *[OF this] have "b * \<mu> B \<le> ?t B"
   522         using M'.finite_measure b finite_measure M.positive_measure[OF B(1)]
   523         by (cases rule: ereal2_cases[of "?t B" "b * \<mu> B"]) auto }
   524     note bM_le_t = this
   525     let ?f0 = "\<lambda>x. f x + b * indicator A0 x"
   526     { fix A assume A: "A \<in> sets M"
   527       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
   528       have "(\<integral>\<^isup>+x. ?f0 x  * indicator A x \<partial>M) =
   529         (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
   530         by (auto intro!: positive_integral_cong split: split_indicator)
   531       hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
   532           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
   533         using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
   534         by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
   535     note f0_eq = this
   536     { fix A assume A: "A \<in> sets M"
   537       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
   538       have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
   539         using `f \<in> G` A unfolding G_def by auto
   540       note f0_eq[OF A]
   541       also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
   542           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
   543         using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
   544         by (auto intro!: add_left_mono)
   545       also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
   546         using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
   547         by (auto intro!: add_left_mono)
   548       also have "\<dots> \<le> \<nu> A"
   549         using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] positive_integral_positive[of "?F A"]
   550         by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "\<nu> A") auto
   551       finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
   552     hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
   553       by (auto intro!: ereal_add_nonneg_nonneg)
   554     have real: "?t (space M) \<noteq> \<infinity>" "?t A0 \<noteq> \<infinity>"
   555       "b * \<mu> (space M) \<noteq> \<infinity>" "b * \<mu> A0 \<noteq> \<infinity>"
   556       using `A0 \<in> sets M` b
   557         finite_measure[of A0] M.finite_measure[of A0]
   558         finite_measure_of_space M.finite_measure_of_space
   559       by auto
   560     have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
   561       using M'.finite_measure_of_space pos_t unfolding ereal_less_minus_iff
   562       by (auto cong: positive_integral_cong)
   563     have  "0 < ?t (space M) - b * \<mu> (space M)" unfolding b_def
   564       using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
   565       using positive_integral_positive[of "?F (space M)"]
   566       by (cases rule: ereal3_cases[of "\<mu> (space M)" "\<nu> (space M)" "integral\<^isup>P M (?F (space M))"])
   567          (auto simp: field_simps mult_less_cancel_left)
   568     also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
   569       using space_less_A0 b
   570       using
   571         `A0 \<in> sets M`[THEN M.real_measure]
   572         top[THEN M.real_measure]
   573       apply safe
   574       apply simp
   575       using
   576         `A0 \<in> sets M`[THEN real_measure]
   577         `A0 \<in> sets M`[THEN M'.real_measure]
   578         top[THEN real_measure]
   579         top[THEN M'.real_measure]
   580       by (cases b) auto
   581     finally have 1: "b * \<mu> A0 < ?t A0"
   582       using
   583         `A0 \<in> sets M`[THEN M.real_measure]
   584       apply safe
   585       apply simp
   586       using
   587         `A0 \<in> sets M`[THEN real_measure]
   588         `A0 \<in> sets M`[THEN M'.real_measure]
   589       by (cases b) auto
   590     have "0 < ?t A0"
   591       using b `A0 \<in> sets M` by (auto intro!: le_less_trans[OF _ 1])
   592     then have "\<mu> A0 \<noteq> 0" using ac unfolding absolutely_continuous_def
   593       using `A0 \<in> sets M` by auto
   594     then have "0 < \<mu> A0" using positive_measure[OF `A0 \<in> sets M`] by auto
   595     hence "0 < b * \<mu> A0" using b by (auto simp: ereal_zero_less_0_iff)
   596     with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
   597       using `f \<in> G`
   598       by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive)
   599     also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
   600       by (simp cong: positive_integral_cong)
   601     finally have "?y < integral\<^isup>P M ?f0" by simp
   602     moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
   603     ultimately show False by auto
   604   qed
   605   show ?thesis
   606   proof (safe intro!: bexI[of _ f])
   607     fix A assume A: "A\<in>sets M"
   608     show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   609     proof (rule antisym)
   610       show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
   611         using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
   612       show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   613         using upper_bound[THEN bspec, OF `A \<in> sets M`]
   614         using M'.real_measure[OF A]
   615         by (cases "integral\<^isup>P M (?F A)") auto
   616     qed
   617   qed simp
   618 qed
   619 
   620 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
   621   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
   622   assumes ac: "absolutely_continuous \<nu>"
   623   shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and>
   624     (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<infinity>)) \<and>
   625     (\<forall>i. \<nu> (B i) \<noteq> \<infinity>)"
   626 proof -
   627   interpret v: measure_space ?N
   628     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
   629     by fact auto
   630   let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<infinity>}"
   631   let ?a = "SUP Q:?Q. \<mu> Q"
   632   have "{} \<in> ?Q" using v.empty_measure by auto
   633   then have Q_not_empty: "?Q \<noteq> {}" by blast
   634   have "?a \<le> \<mu> (space M)" using sets_into_space
   635     by (auto intro!: SUP_least measure_mono)
   636   then have "?a \<noteq> \<infinity>" using finite_measure_of_space
   637     by auto
   638   from SUPR_countable_SUPR[OF Q_not_empty, of \<mu>]
   639   obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
   640     by auto
   641   then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
   642   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
   643     by auto
   644   then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
   645   let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
   646   have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
   647   proof (rule continuity_from_below[of ?O])
   648     show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
   649     show "incseq ?O" by (fastforce intro!: incseq_SucI)
   650   qed
   651   have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
   652   have O_sets: "\<And>i. ?O i \<in> sets M"
   653      using Q' by (auto intro!: finite_UN Un)
   654   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
   655   proof (safe del: notI)
   656     fix i have "Q' ` {..i} \<subseteq> sets M"
   657       using Q' by (auto intro: finite_UN)
   658     with v.measure_finitely_subadditive[of "{.. i}" Q']
   659     have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
   660     also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
   661     finally show "\<nu> (?O i) \<noteq> \<infinity>" by simp
   662   qed auto
   663   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
   664   have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
   665   proof (rule antisym)
   666     show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
   667       using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
   668     show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUP_def
   669     proof (safe intro!: Sup_mono, unfold bex_simps)
   670       fix i
   671       have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
   672       then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<infinity>) \<and>
   673         \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
   674         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
   675     qed
   676   qed
   677   let ?O_0 = "(\<Union>i. ?O i)"
   678   have "?O_0 \<in> sets M" using Q' by auto
   679   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
   680   { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
   681   note Q_sets = this
   682   show ?thesis
   683   proof (intro bexI exI conjI ballI impI allI)
   684     show "disjoint_family Q"
   685       by (fastforce simp: disjoint_family_on_def Q_def
   686         split: nat.split_asm)
   687     show "range Q \<subseteq> sets M"
   688       using Q_sets by auto
   689     { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
   690       show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
   691       proof (rule disjCI, simp)
   692         assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<infinity>"
   693         show "\<mu> A = 0 \<and> \<nu> A = 0"
   694         proof cases
   695           assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
   696             unfolding absolutely_continuous_def by auto
   697           ultimately show ?thesis by simp
   698         next
   699           assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<infinity>" using positive_measure[OF A(1)] by auto
   700           with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
   701             using Q' by (auto intro!: measure_additive countable_UN)
   702           also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
   703           proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
   704             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
   705               using `\<nu> A \<noteq> \<infinity>` O_sets A by auto
   706           qed (fastforce intro!: incseq_SucI)
   707           also have "\<dots> \<le> ?a"
   708           proof (safe intro!: SUP_least)
   709             fix i have "?O i \<union> A \<in> ?Q"
   710             proof (safe del: notI)
   711               show "?O i \<union> A \<in> sets M" using O_sets A by auto
   712               from O_in_G[of i] have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
   713                 using v.measure_subadditive[of "?O i" A] A O_sets by auto
   714               with O_in_G[of i] show "\<nu> (?O i \<union> A) \<noteq> \<infinity>"
   715                 using `\<nu> A \<noteq> \<infinity>` by auto
   716             qed
   717             then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
   718           qed
   719           finally have "\<mu> A = 0"
   720             unfolding a_eq using real_measure[OF `?O_0 \<in> sets M`] real_measure[OF A(1)] by auto
   721           with `\<mu> A \<noteq> 0` show ?thesis by auto
   722         qed
   723       qed }
   724     { fix i show "\<nu> (Q i) \<noteq> \<infinity>"
   725       proof (cases i)
   726         case 0 then show ?thesis
   727           unfolding Q_def using Q'[of 0] by simp
   728       next
   729         case (Suc n)
   730         then show ?thesis unfolding Q_def
   731           using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
   732           using v.measure_mono[OF O_mono, of n] v.positive_measure[of "?O n"] v.positive_measure[of "?O (Suc n)"]
   733           using v.measure_Diff[of "?O n" "?O (Suc n)", OF _ _ _ O_mono]
   734           by (cases rule: ereal2_cases[of "\<nu> (\<Union> x\<le>Suc n. Q' x)" "\<nu> (\<Union> i\<le>n. Q' i)"]) auto
   735       qed }
   736     show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
   737     { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
   738       proof (induct j)
   739         case 0 then show ?case by (simp add: Q_def)
   740       next
   741         case (Suc j)
   742         have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce
   743         have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
   744         then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
   745           by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
   746         then show ?case using Suc by (auto simp add: eq atMost_Suc)
   747       qed }
   748     then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
   749     then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce
   750   qed
   751 qed
   752 
   753 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
   754   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
   755   assumes "absolutely_continuous \<nu>"
   756   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
   757 proof -
   758   interpret v: measure_space ?N
   759     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
   760     by fact auto
   761   from split_space_into_finite_sets_and_rest[OF assms]
   762   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
   763     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
   764     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
   765     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
   766     and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<infinity>" by force
   767   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
   768   have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
   769     \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
   770   proof
   771     fix i
   772     have indicator_eq: "\<And>f x A. (f x :: ereal) * indicator (Q i \<inter> A) x * indicator (Q i) x
   773       = (f x * indicator (Q i) x) * indicator A x"
   774       unfolding indicator_def by auto
   775     have fm: "finite_measure (restricted_space (Q i))"
   776       (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
   777     then interpret R: finite_measure ?R .
   778     have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
   779     proof
   780       show "measure_space ?Q"
   781         using v.restricted_measure_space Q_sets[of i] by auto
   782       show "measure ?Q (space ?Q) \<noteq> \<infinity>" using Q_fin by simp
   783     qed
   784     have "R.absolutely_continuous \<nu>"
   785       using `absolutely_continuous \<nu>` `Q i \<in> sets M`
   786       by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
   787     from R.Radon_Nikodym_finite_measure[OF fmv this]
   788     obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
   789       and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x \<partial>M)"
   790       unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
   791         positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
   792     then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
   793       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
   794       by (auto intro!: exI[of _ "\<lambda>x. max 0 (f x * indicator (Q i) x)"] positive_integral_cong_pos
   795         split: split_indicator split_if_asm simp: max_def)
   796   qed
   797   from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
   798     and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
   799       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
   800     by auto
   801   let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
   802   show ?thesis
   803   proof (safe intro!: bexI[of _ ?f])
   804     show "?f \<in> borel_measurable M" using Q0 borel Q_sets
   805       by (auto intro!: measurable_If)
   806     show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
   807     fix A assume "A \<in> sets M"
   808     have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
   809     have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
   810       "\<And>i. AE x. 0 \<le> f i x * indicator (Q i \<inter> A) x"
   811       using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
   812     have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
   813       using borel by (intro positive_integral_cong) (auto simp: indicator_def)
   814     also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * \<mu> (Q0 \<inter> A)"
   815       using borel Qi Q0(1) `A \<in> sets M`
   816       by (subst positive_integral_add) (auto simp del: ereal_infty_mult
   817           simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
   818     also have "\<dots> = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)"
   819       by (subst f[OF `A \<in> sets M`], subst positive_integral_suminf) auto
   820     finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)" .
   821     moreover have "(\<Sum>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
   822       using Q Q_sets `A \<in> sets M`
   823       by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
   824          (auto simp: disjoint_family_on_def)
   825     moreover have "\<infinity> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
   826     proof -
   827       have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
   828       from in_Q0[OF this] show ?thesis by auto
   829     qed
   830     moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
   831       using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
   832     moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
   833       using `A \<in> sets M` sets_into_space Q0 by auto
   834     ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
   835       using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
   836       by simp
   837   qed
   838 qed
   839 
   840 lemma (in sigma_finite_measure) Radon_Nikodym:
   841   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
   842   assumes ac: "absolutely_continuous \<nu>"
   843   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
   844 proof -
   845   from Ex_finite_integrable_function
   846   obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
   847     borel: "h \<in> borel_measurable M" and
   848     nn: "\<And>x. 0 \<le> h x" and
   849     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
   850     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
   851   let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
   852   let ?MT = "M\<lparr> measure := ?T \<rparr>"
   853   interpret T: finite_measure ?MT
   854     where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
   855     using borel finite nn
   856     by (auto intro!: measure_space_density finite_measureI cong: positive_integral_cong)
   857   have "T.absolutely_continuous \<nu>"
   858   proof (unfold T.absolutely_continuous_def, safe)
   859     fix N assume "N \<in> sets M" "(\<integral>\<^isup>+x. h x * indicator N x \<partial>M) = 0"
   860     with borel ac pos have "AE x. x \<notin> N"
   861       by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_le)
   862     then have "N \<in> null_sets" using `N \<in> sets M` sets_into_space
   863       by (subst (asm) AE_iff_measurable[OF `N \<in> sets M`]) auto
   864     then show "\<nu> N = 0" using ac by (auto simp: absolutely_continuous_def)
   865   qed
   866   from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
   867   obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and
   868     fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
   869     by (auto simp: measurable_def)
   870   show ?thesis
   871   proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
   872     show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
   873       using borel f_borel by auto
   874     show "\<And>x. 0 \<le> h x * f x" using nn f_borel by auto
   875     fix A assume "A \<in> sets M"
   876     then show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
   877       unfolding fT[OF `A \<in> sets M`] mult_assoc using nn borel f_borel
   878       by (intro positive_integral_translated_density) auto
   879   qed
   880 qed
   881 
   882 section "Uniqueness of densities"
   883 
   884 lemma (in measure_space) finite_density_unique:
   885   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   886   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
   887   and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
   888   shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator A x \<partial>M))
   889     \<longleftrightarrow> (AE x. f x = g x)"
   890     (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
   891 proof (intro iffI ballI)
   892   fix A assume eq: "AE x. f x = g x"
   893   then show "?P f A = ?P g A"
   894     by (auto intro: positive_integral_cong_AE)
   895 next
   896   assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
   897   from this[THEN bspec, OF top] fin
   898   have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
   899   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   900       and pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
   901       and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
   902     let ?N = "{x\<in>space M. g x < f x}"
   903     have N: "?N \<in> sets M" using borel by simp
   904     have "?P g ?N \<le> integral\<^isup>P M g" using pos
   905       by (intro positive_integral_mono_AE) (auto split: split_indicator)
   906     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
   907     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
   908       by (auto intro!: positive_integral_cong simp: indicator_def)
   909     also have "\<dots> = ?P f ?N - ?P g ?N"
   910     proof (rule positive_integral_diff)
   911       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
   912         using borel N by auto
   913       show "AE x. g x * indicator ?N x \<le> f x * indicator ?N x"
   914            "AE x. 0 \<le> g x * indicator ?N x"
   915         using pos by (auto split: split_indicator)
   916     qed fact
   917     also have "\<dots> = 0"
   918       unfolding eq[THEN bspec, OF N] using positive_integral_positive Pg_fin by auto
   919     finally have "AE x. f x \<le> g x"
   920       using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
   921       by (subst (asm) positive_integral_0_iff_AE)
   922          (auto split: split_indicator simp: not_less ereal_minus_le_iff) }
   923   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
   924   show "AE x. f x = g x" by auto
   925 qed
   926 
   927 lemma (in finite_measure) density_unique_finite_measure:
   928   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
   929   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> f' x"
   930   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
   931     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
   932   shows "AE x. f x = f' x"
   933 proof -
   934   let ?\<nu> = "\<lambda>A. ?P f A" and ?\<nu>' = "\<lambda>A. ?P f' A"
   935   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
   936   interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
   937     using borel(1) pos(1) by (rule measure_space_density) simp
   938   have ac: "absolutely_continuous ?\<nu>"
   939     using f by (rule density_is_absolutely_continuous)
   940   from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
   941   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
   942     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
   943     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
   944     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<infinity>"
   945     and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<infinity>" by force
   946   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
   947   let ?N = "{x\<in>space M. f x \<noteq> f' x}"
   948   have "?N \<in> sets M" using borel by auto
   949   have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
   950     unfolding indicator_def by auto
   951   have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
   952     by (intro finite_density_unique[THEN iffD1] allI)
   953        (auto intro!: borel_measurable_ereal_times f Int simp: *)
   954   moreover have "AE x. ?f Q0 x = ?f' Q0 x"
   955   proof (rule AE_I')
   956     { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
   957         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   958       let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
   959       have "(\<Union>i. ?A i) \<in> null_sets"
   960       proof (rule null_sets_UN)
   961         fix i ::nat have "?A i \<in> sets M"
   962           using borel Q0(1) by auto
   963         have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
   964           unfolding eq[OF `?A i \<in> sets M`]
   965           by (auto intro!: positive_integral_mono simp: indicator_def)
   966         also have "\<dots> = i * \<mu> (?A i)"
   967           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
   968         also have "\<dots> < \<infinity>"
   969           using `?A i \<in> sets M`[THEN finite_measure] by auto
   970         finally have "?\<nu> (?A i) \<noteq> \<infinity>" by simp
   971         then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
   972       qed
   973       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
   974         by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
   975       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" by simp }
   976     from this[OF borel(1) refl] this[OF borel(2) f]
   977     have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets" by simp_all
   978     then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets" by (rule nullsets.Un)
   979     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
   980       (Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
   981   qed
   982   moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
   983     ?f (space M) x = ?f' (space M) x"
   984     by (auto simp: indicator_def Q0)
   985   ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
   986     by (auto simp: AE_all_countable[symmetric])
   987   then show "AE x. f x = f' x" by auto
   988 qed
   989 
   990 lemma (in sigma_finite_measure) density_unique:
   991   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
   992   assumes f': "f' \<in> borel_measurable M" "AE x. 0 \<le> f' x"
   993   assumes eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
   994     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
   995   shows "AE x. f x = f' x"
   996 proof -
   997   obtain h where h_borel: "h \<in> borel_measurable M"
   998     and fin: "integral\<^isup>P M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
   999     using Ex_finite_integrable_function by auto
  1000   then have h_nn: "AE x. 0 \<le> h x" by auto
  1001   let ?H = "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
  1002   have H: "measure_space ?H"
  1003     using h_borel h_nn by (rule measure_space_density) simp
  1004   then interpret h: measure_space ?H .
  1005   interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
  1006     by (intro H finite_measureI) (simp cong: positive_integral_cong add: fin)
  1007   let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
  1008   interpret f: measure_space ?fM
  1009     using f by (rule measure_space_density) simp
  1010   let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
  1011   interpret f': measure_space ?f'M
  1012     using f' by (rule measure_space_density) simp
  1013   { fix A assume "A \<in> sets M"
  1014     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
  1015       using pos(1) sets_into_space by (force simp: indicator_def)
  1016     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
  1017       using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
  1018   note h_null_sets = this
  1019   { fix A assume "A \<in> sets M"
  1020     have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
  1021       using `A \<in> sets M` h_borel h_nn f f'
  1022       by (intro positive_integral_translated_density[symmetric]) auto
  1023     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
  1024       by (rule f'.positive_integral_cong_measure) (simp_all add: eq)
  1025     also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
  1026       using `A \<in> sets M` h_borel h_nn f f'
  1027       by (intro positive_integral_translated_density) auto
  1028     finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
  1029       by (simp add: ac_simps)
  1030     then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
  1031       using `A \<in> sets M` h_borel h_nn f f'
  1032       by (subst (asm) (1 2) positive_integral_translated_density[symmetric]) auto }
  1033   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
  1034     by (intro h.density_unique_finite_measure absolutely_continuous_AE[OF H] density_is_absolutely_continuous)
  1035        simp_all
  1036   then show "AE x. f x = f' x"
  1037     unfolding h.almost_everywhere_def almost_everywhere_def
  1038     by (auto simp add: h_null_sets)
  1039 qed
  1040 
  1041 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
  1042   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
  1043     and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
  1044     and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
  1045   shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<infinity>)"
  1046 proof
  1047   assume "sigma_finite_measure ?N"
  1048   then interpret \<nu>: sigma_finite_measure ?N
  1049     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
  1050     and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
  1051   from \<nu>.Ex_finite_integrable_function obtain h where
  1052     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
  1053     h_nn: "\<And>x. 0 \<le> h x" and
  1054     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
  1055   have "AE x. f x * h x \<noteq> \<infinity>"
  1056   proof (rule AE_I')
  1057     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
  1058       by (subst \<nu>.positive_integral_cong_measure[symmetric,
  1059           of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
  1060          (auto intro!: positive_integral_translated_density simp: eq)
  1061     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
  1062       using h(2) by simp
  1063     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets"
  1064       using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
  1065   qed auto
  1066   then show "AE x. f x \<noteq> \<infinity>"
  1067     using fin by (auto elim!: AE_Ball_mp)
  1068 next
  1069   assume AE: "AE x. f x \<noteq> \<infinity>"
  1070   from sigma_finite guess Q .. note Q = this
  1071   interpret \<nu>: measure_space ?N
  1072     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
  1073     and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
  1074   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
  1075   { fix i j have "A i \<inter> Q j \<in> sets M"
  1076     unfolding A_def using f Q
  1077     apply (rule_tac Int)
  1078     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
  1079   note A_in_sets = this
  1080   let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
  1081   show "sigma_finite_measure ?N"
  1082   proof (default, intro exI conjI subsetI allI)
  1083     fix x assume "x \<in> range ?A"
  1084     then obtain n where n: "x = ?A n" by auto
  1085     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
  1086   next
  1087     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
  1088     proof safe
  1089       fix x i j assume "x \<in> A i" "x \<in> Q j"
  1090       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
  1091         by (intro UN_I[of "prod_encode (i,j)"]) auto
  1092     qed auto
  1093     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
  1094     also have "(\<Union>i. A i) = space M"
  1095     proof safe
  1096       fix x assume x: "x \<in> space M"
  1097       show "x \<in> (\<Union>i. A i)"
  1098       proof (cases "f x")
  1099         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
  1100       next
  1101         case (real r)
  1102         with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
  1103         then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
  1104       next
  1105         case MInf with x show ?thesis
  1106           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
  1107       qed
  1108     qed (auto simp: A_def)
  1109     finally show "(\<Union>i. ?A i) = space ?N" by simp
  1110   next
  1111     fix n obtain i j where
  1112       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
  1113     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
  1114     proof (cases i)
  1115       case 0
  1116       have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
  1117         using AE by (auto simp: A_def `i = 0`)
  1118       from positive_integral_cong_AE[OF this] show ?thesis by simp
  1119     next
  1120       case (Suc n)
  1121       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
  1122         (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
  1123         by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
  1124       also have "\<dots> = Suc n * \<mu> (Q j)"
  1125         using Q by (auto intro!: positive_integral_cmult_indicator)
  1126       also have "\<dots> < \<infinity>"
  1127         using Q by (auto simp: real_eq_of_nat[symmetric])
  1128       finally show ?thesis by simp
  1129     qed
  1130     then show "measure ?N (?A n) \<noteq> \<infinity>"
  1131       using A_in_sets Q eq by auto
  1132   qed
  1133 qed
  1134 
  1135 section "Radon-Nikodym derivative"
  1136 
  1137 definition
  1138   "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and>
  1139     (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
  1140 
  1141 lemma (in sigma_finite_measure) RN_deriv_cong:
  1142   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
  1143     and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
  1144   shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
  1145 proof -
  1146   interpret \<mu>': sigma_finite_measure M'
  1147     using cong by (rule sigma_finite_measure_cong)
  1148   show ?thesis
  1149     unfolding RN_deriv_def
  1150     by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
  1151 qed
  1152 
  1153 lemma (in sigma_finite_measure) RN_deriv:
  1154   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
  1155   assumes "absolutely_continuous \<nu>"
  1156   shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
  1157   and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
  1158     (is "\<And>A. _ \<Longrightarrow> ?int A")
  1159   and "0 \<le> RN_deriv M \<nu> x"
  1160 proof -
  1161   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
  1162   then show ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
  1163   from Ex show "0 \<le> RN_deriv M \<nu> x" unfolding RN_deriv_def
  1164     by (rule someI2_ex) simp
  1165   fix A assume "A \<in> sets M"
  1166   from Ex show "?int A" unfolding RN_deriv_def
  1167     by (rule someI2_ex) (simp add: `A \<in> sets M`)
  1168 qed
  1169 
  1170 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
  1171   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
  1172     and f: "f \<in> borel_measurable M"
  1173   shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
  1174 proof -
  1175   interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
  1176   note RN = RN_deriv[OF \<nu>]
  1177   have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>)"
  1178     unfolding positive_integral_max_0 ..
  1179   also have "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>) =
  1180     (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>)"
  1181     by (intro \<nu>.positive_integral_cong_measure[symmetric]) (simp_all add: RN(2))
  1182   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * max 0 (f x) \<partial>M)"
  1183     by (intro positive_integral_translated_density) (auto simp add: RN f)
  1184   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
  1185     using RN_deriv(3)[OF \<nu>]
  1186     by (auto intro!: positive_integral_cong_pos split: split_if_asm
  1187              simp: max_def ereal_mult_le_0_iff)
  1188   finally show ?thesis .
  1189 qed
  1190 
  1191 lemma (in sigma_finite_measure) RN_deriv_unique:
  1192   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
  1193   and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
  1194   and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
  1195   shows "AE x. f x = RN_deriv M \<nu> x"
  1196 proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
  1197   show "AE x. 0 \<le> RN_deriv M \<nu> x" using RN_deriv[OF \<nu>] by auto
  1198   fix A assume A: "A \<in> sets M"
  1199   show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
  1200     unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
  1201 qed
  1202 
  1203 lemma (in sigma_finite_measure) RN_deriv_vimage:
  1204   assumes T: "T \<in> measure_preserving M M'"
  1205     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
  1206     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
  1207   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
  1208   shows "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
  1209 proof (rule RN_deriv_unique)
  1210   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
  1211   show "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
  1212     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
  1213   interpret M': measure_space M'
  1214   proof (rule measure_space_vimage)
  1215     have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
  1216     then show "sigma_algebra M'" by simp
  1217   qed fact
  1218   show "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
  1219   proof safe
  1220     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
  1221     then have N': "T' -` N \<inter> space M' \<in> sets M'"
  1222       using T' by (auto simp: measurable_def measure_preserving_def)
  1223     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
  1224       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
  1225     then have "measure M' (T' -` N \<inter> space M') = 0"
  1226       using measure_preservingD[OF T N'] N_0 by auto
  1227     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
  1228       unfolding M'.absolutely_continuous_def measurable_def by auto
  1229   qed
  1230   interpret M': sigma_finite_measure M'
  1231   proof
  1232     from sigma_finite guess F .. note F = this
  1233     show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<infinity>)"
  1234     proof (intro exI conjI allI)
  1235       show *: "range (\<lambda>i. T' -` F i \<inter> space M') \<subseteq> sets M'"
  1236         using F T' by (auto simp: measurable_def measure_preserving_def)
  1237       show "(\<Union>i. T' -` F i \<inter> space M') = space M'"
  1238         using F T' by (force simp: measurable_def measure_preserving_def)
  1239       fix i
  1240       have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
  1241       note measure_preservingD[OF T this, symmetric]
  1242       moreover
  1243       have Fi: "F i \<in> sets M" using F by auto
  1244       then have "T -` (T' -` F i \<inter> space M') \<inter> space M = F i"
  1245         using T inv sets_into_space[OF Fi]
  1246         by (auto simp: measurable_def measure_preserving_def)
  1247       ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<infinity>"
  1248         using F by simp
  1249     qed
  1250   qed
  1251   have "(RN_deriv M' \<nu>') \<circ> T \<in> borel_measurable M"
  1252     by (intro measurable_comp[where b=M'] M'.RN_deriv(1) measure_preservingD2[OF T]) fact+
  1253   then show "(\<lambda>x. RN_deriv M' \<nu>' (T x)) \<in> borel_measurable M"
  1254     by (simp add: comp_def)
  1255   show "AE x. 0 \<le> RN_deriv M' \<nu>' (T x)" using M'.RN_deriv(3)[OF \<nu>'] by auto
  1256   fix A let ?A = "T' -` A \<inter> space M'"
  1257   assume A: "A \<in> sets M"
  1258   then have A': "?A \<in> sets M'" using T' unfolding measurable_def measure_preserving_def
  1259     by auto
  1260   from A have "\<nu> A = \<nu>' ?A" using T'[THEN measure_preservingD] by simp
  1261   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' x * indicator ?A x \<partial>M'"
  1262     using A' by (rule M'.RN_deriv(2)[OF \<nu>'])
  1263   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator ?A (T x) \<partial>M"
  1264   proof (rule positive_integral_vimage)
  1265     show "sigma_algebra M'" by default
  1266     show "(\<lambda>x. RN_deriv M' \<nu>' x * indicator (T' -` A \<inter> space M') x) \<in> borel_measurable M'"
  1267       by (auto intro!: A' M'.RN_deriv(1)[OF \<nu>'])
  1268   qed fact
  1269   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M"
  1270     using T inv by (auto intro!: positive_integral_cong simp: measure_preserving_def measurable_def indicator_def)
  1271   finally show "\<nu> A = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M" .
  1272 qed
  1273 
  1274 lemma (in sigma_finite_measure) RN_deriv_finite:
  1275   assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
  1276   shows "AE x. RN_deriv M \<nu> x \<noteq> \<infinity>"
  1277 proof -
  1278   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
  1279   have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
  1280   from sfm show ?thesis
  1281     using sigma_finite_iff_density_finite[OF \<nu> RN_deriv(1)[OF \<nu> ac]] RN_deriv(2,3)[OF \<nu> ac] by simp
  1282 qed
  1283 
  1284 lemma (in sigma_finite_measure)
  1285   assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
  1286     and f: "f \<in> borel_measurable M"
  1287   shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
  1288       integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
  1289     and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
  1290       (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
  1291 proof -
  1292   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
  1293   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
  1294   have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
  1295   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
  1296   have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f by simp
  1297   { fix f :: "'a \<Rightarrow> real"
  1298     { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<infinity>"
  1299       have "ereal (real (RN_deriv M \<nu> x)) * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
  1300         by (simp add: mult_le_0_iff)
  1301       then have "RN_deriv M \<nu> x * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
  1302         using RN_deriv(3)[OF ms \<nu>(2)] * by (auto simp add: ereal_real split: split_if_asm) }
  1303     then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M \<nu> x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (f x) \<partial>M)"
  1304               "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M \<nu> x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (- f x) \<partial>M)"
  1305       using RN_deriv_finite[OF \<nu>] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
  1306       by (auto intro!: positive_integral_cong_AE) }
  1307   note * = this
  1308   show ?integral ?integrable
  1309     unfolding lebesgue_integral_def integrable_def *
  1310     using f RN_deriv(1)[OF ms \<nu>(2)]
  1311     by (auto simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
  1312 qed
  1313 
  1314 lemma (in sigma_finite_measure) real_RN_deriv:
  1315   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?\<nu>")
  1316   assumes ac: "absolutely_continuous \<nu>"
  1317   obtains D where "D \<in> borel_measurable M"
  1318     and "AE x. RN_deriv M \<nu> x = ereal (D x)"
  1319     and "AE x in M\<lparr>measure := \<nu>\<rparr>. 0 < D x"
  1320     and "\<And>x. 0 \<le> D x"
  1321 proof
  1322   interpret \<nu>: finite_measure ?\<nu> by fact
  1323   have ms: "measure_space ?\<nu>" by default
  1324   note RN = RN_deriv[OF ms ac]
  1325 
  1326   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M \<nu> x = t}"
  1327 
  1328   show "(\<lambda>x. real (RN_deriv M \<nu> x)) \<in> borel_measurable M"
  1329     using RN by auto
  1330 
  1331   have "\<nu> (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN \<infinity>) x \<partial>M)"
  1332     using RN by auto
  1333   also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
  1334     by (intro positive_integral_cong) (auto simp: indicator_def)
  1335   also have "\<dots> = \<infinity> * \<mu> (?RN \<infinity>)"
  1336     using RN by (intro positive_integral_cmult_indicator) auto
  1337   finally have eq: "\<nu> (?RN \<infinity>) = \<infinity> * \<mu> (?RN \<infinity>)" .
  1338   moreover
  1339   have "\<mu> (?RN \<infinity>) = 0"
  1340   proof (rule ccontr)
  1341     assume "\<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>} \<noteq> 0"
  1342     moreover from RN have "0 \<le> \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
  1343     ultimately have "0 < \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
  1344     with eq have "\<nu> (?RN \<infinity>) = \<infinity>" by simp
  1345     with \<nu>.finite_measure[of "?RN \<infinity>"] RN show False by auto
  1346   qed
  1347   ultimately have "AE x. RN_deriv M \<nu> x < \<infinity>"
  1348     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
  1349   then show "AE x. RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
  1350     using RN(3) by (auto simp: ereal_real)
  1351   then have eq: "AE x in (M\<lparr>measure := \<nu>\<rparr>) . RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
  1352     using ac absolutely_continuous_AE[OF ms] by auto
  1353 
  1354   show "\<And>x. 0 \<le> real (RN_deriv M \<nu> x)"
  1355     using RN by (auto intro: real_of_ereal_pos)
  1356 
  1357   have "\<nu> (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN 0) x \<partial>M)"
  1358     using RN by auto
  1359   also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
  1360     by (intro positive_integral_cong) (auto simp: indicator_def)
  1361   finally have "AE x in (M\<lparr>measure := \<nu>\<rparr>). RN_deriv M \<nu> x \<noteq> 0"
  1362     using RN by (intro \<nu>.AE_iff_measurable[THEN iffD2]) auto
  1363   with RN(3) eq show "AE x in (M\<lparr>measure := \<nu>\<rparr>). 0 < real (RN_deriv M \<nu> x)"
  1364     by (auto simp: zero_less_real_of_ereal le_less)
  1365 qed
  1366 
  1367 lemma (in sigma_finite_measure) RN_deriv_singleton:
  1368   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
  1369   and ac: "absolutely_continuous \<nu>"
  1370   and "{x} \<in> sets M"
  1371   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
  1372 proof -
  1373   note deriv = RN_deriv[OF assms(1, 2)]
  1374   from deriv(2)[OF `{x} \<in> sets M`]
  1375   have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
  1376     by (auto simp: indicator_def intro!: positive_integral_cong)
  1377   thus ?thesis using positive_integral_cmult_indicator[OF _ `{x} \<in> sets M`] deriv(3)
  1378     by auto
  1379 qed
  1380 
  1381 theorem (in finite_measure_space) RN_deriv_finite_measure:
  1382   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
  1383   and ac: "absolutely_continuous \<nu>"
  1384   and "x \<in> space M"
  1385   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
  1386 proof -
  1387   have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
  1388   from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
  1389 qed
  1390 
  1391 end