src/HOL/Probability/Radon_Nikodym.thy
author hoelzl
Wed Oct 10 12:12:24 2012 +0200 (2012-10-10)
changeset 49785 0a8adca22974
parent 49778 bbbc0f492780
child 50003 8c213922ed49
permissions -rw-r--r--
simplified entropy_uniform
     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 definition "diff_measure M N =
    12   measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"
    13 
    14 lemma 
    15   shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
    16     and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
    17   by (auto simp: diff_measure_def)
    18 
    19 lemma emeasure_diff_measure:
    20   assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N"
    21   assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M"
    22   shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A")
    23   unfolding diff_measure_def
    24 proof (rule emeasure_measure_of_sigma)
    25   show "sigma_algebra (space M) (sets M)" ..
    26   show "positive (sets M) ?\<mu>"
    27     using pos by (simp add: positive_def ereal_diff_positive)
    28   show "countably_additive (sets M) ?\<mu>"
    29   proof (rule countably_additiveI)
    30     fix A :: "nat \<Rightarrow> _"  assume A: "range A \<subseteq> sets M" and "disjoint_family A"
    31     then have suminf:
    32       "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
    33       "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)"
    34       by (simp_all add: suminf_emeasure sets_eq)
    35     with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
    36       (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))"
    37       using fin
    38       by (intro suminf_ereal_minus pos emeasure_nonneg)
    39          (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure)
    40     then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
    41       emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) "
    42       by (simp add: suminf)
    43   qed
    44 qed fact
    45 
    46 lemma (in sigma_finite_measure) Ex_finite_integrable_function:
    47   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)"
    48 proof -
    49   obtain A :: "nat \<Rightarrow> 'a set" where
    50     range: "range A \<subseteq> sets M" and
    51     space: "(\<Union>i. A i) = space M" and
    52     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and
    53     disjoint: "disjoint_family A"
    54     using sigma_finite_disjoint by auto
    55   let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)"
    56   have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
    57   proof
    58     fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
    59       using measure[of i] emeasure_nonneg[of M "A i"]
    60       by (auto intro!: ereal_dense simp: ereal_0_gt_inverse ereal_zero_le_0_iff)
    61   qed
    62   from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
    63     "\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto
    64   { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
    65   let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x"
    66   show ?thesis
    67   proof (safe intro!: bexI[of _ ?h] del: notI)
    68     have "\<And>i. A i \<in> sets M"
    69       using range by fastforce+
    70     then have "integral\<^isup>P M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos
    71       by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
    72     also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
    73     proof (rule suminf_le_pos)
    74       fix N
    75       have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)"
    76         using n[of N]
    77         by (intro ereal_mult_right_mono) auto
    78       also have "\<dots> \<le> (1 / 2) ^ Suc N"
    79         using measure[of N] n[of N]
    80         by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"])
    81            (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
    82       finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" .
    83       show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg)
    84     qed
    85     finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
    86   next
    87     { fix x assume "x \<in> space M"
    88       then obtain i where "x \<in> A i" using space[symmetric] by auto
    89       with disjoint n have "?h x = n i"
    90         by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
    91       then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto }
    92     note pos = this
    93     fix x show "0 \<le> ?h x"
    94     proof cases
    95       assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le)
    96     next
    97       assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto
    98       then show "0 \<le> ?h x" by auto
    99     qed
   100   next
   101     show "?h \<in> borel_measurable M" using range n
   102       by (auto intro!: borel_measurable_psuminf borel_measurable_ereal_times ereal_0_le_mult intro: less_imp_le)
   103   qed
   104 qed
   105 
   106 subsection "Absolutely continuous"
   107 
   108 definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
   109   "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N"
   110 
   111 lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M"
   112   unfolding absolutely_continuous_def by (auto simp: null_sets_count_space)
   113 
   114 lemma absolutely_continuousI_density:
   115   "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)"
   116   by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in)
   117 
   118 lemma absolutely_continuousI_point_measure_finite:
   119   "(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)"
   120   unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff)
   121 
   122 lemma absolutely_continuous_AE:
   123   assumes sets_eq: "sets M' = sets M"
   124     and "absolutely_continuous M M'" "AE x in M. P x"
   125    shows "AE x in M'. P x"
   126 proof -
   127   from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
   128     unfolding eventually_ae_filter by auto
   129   show "AE x in M'. P x"
   130   proof (rule AE_I')
   131     show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
   132     from `absolutely_continuous M M'` show "N \<in> null_sets M'"
   133       using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
   134   qed
   135 qed
   136 
   137 subsection "Existence of the Radon-Nikodym derivative"
   138 
   139 lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
   140   fixes e :: real assumes "0 < e"
   141   assumes "finite_measure N" and sets_eq: "sets N = sets M"
   142   shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> measure M A - measure N A \<and>
   143                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < measure M B - measure N B)"
   144 proof -
   145   interpret M': finite_measure N by fact
   146   let ?d = "\<lambda>A. measure M A - measure N A"
   147   let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
   148     then {}
   149     else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
   150   def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
   151   have A_simps[simp]:
   152     "A 0 = {}"
   153     "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
   154   { fix A assume "A \<in> sets M"
   155     have "?A A \<in> sets M"
   156       by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
   157   note A'_in_sets = this
   158   { fix n have "A n \<in> sets M"
   159     proof (induct n)
   160       case (Suc n) thus "A (Suc n) \<in> sets M"
   161         using A'_in_sets[of "A n"] by (auto split: split_if_asm)
   162     qed (simp add: A_def) }
   163   note A_in_sets = this
   164   hence "range A \<subseteq> sets M" by auto
   165   { fix n B
   166     assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
   167     hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
   168     have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
   169     proof (rule someI2_ex[OF Ex])
   170       fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
   171       hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
   172       hence "?d (A n \<union> B) = ?d (A n) + ?d B"
   173         using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq)
   174       also have "\<dots> \<le> ?d (A n) - e" using dB by simp
   175       finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
   176     qed }
   177   note dA_epsilon = this
   178   { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
   179     proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
   180       case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
   181     next
   182       case False
   183       hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
   184       thus ?thesis by simp
   185     qed }
   186   note dA_mono = this
   187   show ?thesis
   188   proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
   189     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
   190     show ?thesis
   191     proof (safe intro!: bexI[of _ "space M - A n"])
   192       fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
   193       from B[OF this] show "-e < ?d B" .
   194     next
   195       show "space M - A n \<in> sets M" by (rule compl_sets) fact
   196     next
   197       show "?d (space M) \<le> ?d (space M - A n)"
   198       proof (induct n)
   199         fix n assume "?d (space M) \<le> ?d (space M - A n)"
   200         also have "\<dots> \<le> ?d (space M - A (Suc n))"
   201           using A_in_sets sets_into_space dA_mono[of n] finite_measure_compl M'.finite_measure_compl
   202           by (simp del: A_simps add: sets_eq sets_eq_imp_space_eq[OF sets_eq])
   203         finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
   204       qed simp
   205     qed
   206   next
   207     case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
   208       by (auto simp add: not_less)
   209     { fix n have "?d (A n) \<le> - real n * e"
   210       proof (induct n)
   211         case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
   212       next
   213         case 0 with measure_empty show ?case by (simp add: zero_ereal_def)
   214       qed } note dA_less = this
   215     have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
   216     proof (rule incseq_SucI)
   217       fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
   218     qed
   219     have A: "incseq A" by (auto intro!: incseq_SucI)
   220     from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M`
   221       M'.finite_Lim_measure_incseq[OF _ A]
   222     have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
   223       by (auto intro!: tendsto_diff simp: sets_eq)
   224     obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
   225     moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
   226     have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
   227     ultimately show ?thesis by auto
   228   qed
   229 qed
   230 
   231 lemma (in finite_measure) Radon_Nikodym_aux:
   232   assumes "finite_measure N" and sets_eq: "sets N = sets M"
   233   shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le>
   234                     measure M A - measure N A \<and>
   235                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> measure M B - measure N B)"
   236 proof -
   237   interpret N: finite_measure N by fact
   238   let ?d = "\<lambda>A. measure M A - measure N A"
   239   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)"
   240   let ?r = "\<lambda>S. restricted_space S"
   241   { fix S n assume S: "S \<in> sets M"
   242     then have "finite_measure (density M (indicator S))" "0 < 1 / real (Suc n)"
   243          "finite_measure (density N (indicator S))" "sets (density N (indicator S)) = sets (density M (indicator S))"
   244       by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq)
   245     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
   246     with S have "?P (S \<inter> X) S n"
   247       by (simp add: measure_restricted sets_eq Int) (metis inf_absorb2)
   248     hence "\<exists>A. ?P A S n" .. }
   249   note Ex_P = this
   250   def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
   251   have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
   252   have A_0[simp]: "A 0 = space M" unfolding A_def by simp
   253   { fix i have "A i \<in> sets M" unfolding A_def
   254     proof (induct i)
   255       case (Suc i)
   256       from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
   257         by (rule someI2_ex) simp
   258     qed simp }
   259   note A_in_sets = this
   260   { fix n have "?P (A (Suc n)) (A n) n"
   261       using Ex_P[OF A_in_sets] unfolding A_Suc
   262       by (rule someI2_ex) simp }
   263   note P_A = this
   264   have "range A \<subseteq> sets M" using A_in_sets by auto
   265   have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
   266   have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
   267   have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
   268       using P_A by auto
   269   show ?thesis
   270   proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
   271     show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
   272     have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
   273     from `range A \<subseteq> sets M`
   274       finite_Lim_measure_decseq[OF _ A] N.finite_Lim_measure_decseq[OF _ A]
   275     have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq)
   276     thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
   277       by (rule_tac LIMSEQ_le_const) auto
   278   next
   279     fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
   280     show "0 \<le> ?d B"
   281     proof (rule ccontr)
   282       assume "\<not> 0 \<le> ?d B"
   283       hence "0 < - ?d B" by auto
   284       from ex_inverse_of_nat_Suc_less[OF this]
   285       obtain n where *: "?d B < - 1 / real (Suc n)"
   286         by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
   287       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
   288       from epsilon[OF B(1) this] *
   289       show False by auto
   290     qed
   291   qed
   292 qed
   293 
   294 lemma (in finite_measure) Radon_Nikodym_finite_measure:
   295   assumes "finite_measure N" and sets_eq: "sets N = sets M"
   296   assumes "absolutely_continuous M N"
   297   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
   298 proof -
   299   interpret N: finite_measure N by fact
   300   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> N A)}"
   301   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
   302   hence "G \<noteq> {}" by auto
   303   { fix f g assume f: "f \<in> G" and g: "g \<in> G"
   304     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
   305     proof safe
   306       show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
   307       let ?A = "{x \<in> space M. f x \<le> g x}"
   308       have "?A \<in> sets M" using f g unfolding G_def by auto
   309       fix A assume "A \<in> sets M"
   310       hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
   311       hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq)
   312       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
   313         using sets_into_space[OF `A \<in> sets M`] by auto
   314       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
   315         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
   316         by (auto simp: indicator_def max_def)
   317       hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
   318         (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
   319         (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
   320         using f g sets unfolding G_def
   321         by (auto cong: positive_integral_cong intro!: positive_integral_add)
   322       also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)"
   323         using f g sets unfolding G_def by (auto intro!: add_mono)
   324       also have "\<dots> = N A"
   325         using plus_emeasure[OF sets'] union by auto
   326       finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" .
   327     next
   328       fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
   329     qed }
   330   note max_in_G = this
   331   { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
   332     have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
   333     proof safe
   334       show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
   335         using f by (auto simp: G_def)
   336       { fix x show "0 \<le> (SUP i. f i x)"
   337           using f by (auto simp: G_def intro: SUP_upper2) }
   338     next
   339       fix A assume "A \<in> sets M"
   340       have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
   341         (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
   342         by (intro positive_integral_cong) (simp split: split_indicator)
   343       also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
   344         using `incseq f` f `A \<in> sets M`
   345         by (intro positive_integral_monotone_convergence_SUP)
   346            (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
   347       finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
   348         using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
   349     qed }
   350   note SUP_in_G = this
   351   let ?y = "SUP g : G. integral\<^isup>P M g"
   352   have y_le: "?y \<le> N (space M)" unfolding G_def
   353   proof (safe intro!: SUP_least)
   354     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A"
   355     from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> N (space M)"
   356       by (simp cong: positive_integral_cong)
   357   qed
   358   from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
   359   then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
   360   proof safe
   361     fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
   362     hence "ys n \<in> integral\<^isup>P M ` G" by auto
   363     thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
   364   qed
   365   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
   366   hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
   367   let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
   368   def f \<equiv> "\<lambda>x. SUP i. ?g i x"
   369   let ?F = "\<lambda>A x. f x * indicator A x"
   370   have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
   371   { fix i have "?g i \<in> G"
   372     proof (induct i)
   373       case 0 thus ?case by simp fact
   374     next
   375       case (Suc i)
   376       with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
   377         by (auto simp add: atMost_Suc intro!: max_in_G)
   378     qed }
   379   note g_in_G = this
   380   have "incseq ?g" using gs_not_empty
   381     by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
   382   from SUP_in_G[OF this g_in_G] have "f \<in> G" unfolding f_def .
   383   then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
   384   have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def
   385     using g_in_G `incseq ?g`
   386     by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def)
   387   also have "\<dots> = ?y"
   388   proof (rule antisym)
   389     show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
   390       using g_in_G by (auto intro: Sup_mono simp: SUP_def)
   391     show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
   392       by (auto intro!: SUP_mono positive_integral_mono Max_ge)
   393   qed
   394   finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
   395   have "\<And>x. 0 \<le> f x"
   396     unfolding f_def using `\<And>i. gs i \<in> G`
   397     by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
   398   let ?t = "\<lambda>A. N A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
   399   let ?M = "diff_measure N (density M f)"
   400   have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A"
   401     using `f \<in> G` unfolding G_def by auto
   402   have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A"
   403   proof (subst emeasure_diff_measure)
   404     from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)"
   405       by (auto intro!: finite_measureI simp: emeasure_density cong: positive_integral_cong)
   406   next
   407     fix B assume "B \<in> sets N" with f_le_N[of B] show "emeasure (density M f) B \<le> emeasure N B"
   408       by (auto simp: sets_eq emeasure_density cong: positive_integral_cong)
   409   qed (auto simp: sets_eq emeasure_density)
   410   from emeasure_M[of "space M"] N.finite_emeasure_space positive_integral_positive[of M "?F (space M)"]
   411   interpret M': finite_measure ?M
   412     by (auto intro!: finite_measureI simp: sets_eq_imp_space_eq[OF sets_eq] N.emeasure_eq_measure ereal_minus_eq_PInfty_iff)
   413 
   414   have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def
   415   proof
   416     fix A assume A: "A \<in> null_sets M"
   417     with `absolutely_continuous M N` have "A \<in> null_sets N"
   418       unfolding absolutely_continuous_def by auto
   419     moreover with A have "(\<integral>\<^isup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def)
   420     ultimately have "N A - (\<integral>\<^isup>+ x. ?F A x \<partial>M) = 0"
   421       using positive_integral_positive[of M] by (auto intro!: antisym)
   422     then show "A \<in> null_sets ?M"
   423       using A by (simp add: emeasure_M null_sets_def sets_eq)
   424   qed
   425   have upper_bound: "\<forall>A\<in>sets M. ?M A \<le> 0"
   426   proof (rule ccontr)
   427     assume "\<not> ?thesis"
   428     then obtain A where A: "A \<in> sets M" and pos: "0 < ?M A"
   429       by (auto simp: not_le)
   430     note pos
   431     also have "?M A \<le> ?M (space M)"
   432       using emeasure_space[of ?M A] by (simp add: sets_eq[THEN sets_eq_imp_space_eq])
   433     finally have pos_t: "0 < ?M (space M)" by simp
   434     moreover
   435     then have "emeasure M (space M) \<noteq> 0"
   436       using ac unfolding absolutely_continuous_def by (auto simp: null_sets_def)
   437     then have pos_M: "0 < emeasure M (space M)"
   438       using emeasure_nonneg[of M "space M"] by (simp add: le_less)
   439     moreover
   440     have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)"
   441       using `f \<in> G` unfolding G_def by auto
   442     hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
   443       using M'.finite_emeasure_space by auto
   444     moreover
   445     def b \<equiv> "?M (space M) / emeasure M (space M) / 2"
   446     ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
   447       by (auto simp: ereal_divide_eq)
   448     then have b: "b \<noteq> 0" "0 \<le> b" "0 < b"  "b \<noteq> \<infinity>" by auto
   449     let ?Mb = "density M (\<lambda>_. b)"
   450     have Mb: "finite_measure ?Mb" "sets ?Mb = sets ?M"
   451         using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI)
   452     from M'.Radon_Nikodym_aux[OF this] guess A0 ..
   453     then have "A0 \<in> sets M"
   454       and space_less_A0: "measure ?M (space M) - real b * measure M (space M) \<le> measure ?M A0 - real b * measure M A0"
   455       and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real b * measure M B"
   456       using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq)
   457     { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
   458       with *[OF this] have "b * emeasure M B \<le> ?M B"
   459         using b unfolding M'.emeasure_eq_measure emeasure_eq_measure by (cases b) auto }
   460     note bM_le_t = this
   461     let ?f0 = "\<lambda>x. f x + b * indicator A0 x"
   462     { fix A assume A: "A \<in> sets M"
   463       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
   464       have "(\<integral>\<^isup>+x. ?f0 x  * indicator A x \<partial>M) =
   465         (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
   466         by (auto intro!: positive_integral_cong split: split_indicator)
   467       hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
   468           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)"
   469         using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
   470         by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
   471     note f0_eq = this
   472     { fix A assume A: "A \<in> sets M"
   473       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
   474       have f_le_v: "(\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto
   475       note f0_eq[OF A]
   476       also have "(\<integral>\<^isup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^isup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)"
   477         using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
   478         by (auto intro!: add_left_mono)
   479       also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?M A"
   480         using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M`
   481         by (auto intro!: add_left_mono simp: sets_eq)
   482       also have "\<dots> \<le> N A"
   483         unfolding emeasure_M[OF `A \<in> sets M`]
   484         using f_le_v N.emeasure_eq_measure[of A] positive_integral_positive[of M "?F A"]
   485         by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "N A") auto
   486       finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . }
   487     hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
   488       by (auto intro!: ereal_add_nonneg_nonneg)
   489     have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
   490       by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le)
   491     have  "0 < ?M (space M) - emeasure ?Mb (space M)"
   492       using pos_t
   493       by (simp add: b emeasure_density_const)
   494          (simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def)
   495     also have "\<dots> \<le> ?M A0 - b * emeasure M A0"
   496       using space_less_A0 `A0 \<in> sets M` b
   497       by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure)
   498     finally have 1: "b * emeasure M A0 < ?M A0"
   499       by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1)
   500                 less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def)
   501     with b have "0 < ?M A0"
   502       by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times
   503                ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def)
   504     then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M`
   505       by (auto simp: absolutely_continuous_def null_sets_def)
   506     then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto
   507     hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff)
   508     with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * emeasure M A0" unfolding int_f_eq_y
   509       using `f \<in> G`
   510       by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive)
   511     also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
   512       by (simp cong: positive_integral_cong)
   513     finally have "?y < integral\<^isup>P M ?f0" by simp
   514     moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
   515     ultimately show False by auto
   516   qed
   517   let ?f = "\<lambda>x. max 0 (f x)"
   518   show ?thesis
   519   proof (intro bexI[of _ ?f] measure_eqI conjI)
   520     show "sets (density M ?f) = sets N"
   521       by (simp add: sets_eq)
   522     fix A assume A: "A\<in>sets (density M ?f)"
   523     then show "emeasure (density M ?f) A = emeasure N A"
   524       using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A]
   525       by (cases "integral\<^isup>P M (?F A)")
   526          (auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric])
   527   qed auto
   528 qed
   529 
   530 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
   531   assumes ac: "absolutely_continuous M N" and sets_eq: "sets N = sets M"
   532   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>
   533     (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>)) \<and>
   534     (\<forall>i. N (B i) \<noteq> \<infinity>)"
   535 proof -
   536   let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}"
   537   let ?a = "SUP Q:?Q. emeasure M Q"
   538   have "{} \<in> ?Q" by auto
   539   then have Q_not_empty: "?Q \<noteq> {}" by blast
   540   have "?a \<le> emeasure M (space M)" using sets_into_space
   541     by (auto intro!: SUP_least emeasure_mono)
   542   then have "?a \<noteq> \<infinity>" using finite_emeasure_space
   543     by auto
   544   from SUPR_countable_SUPR[OF Q_not_empty, of "emeasure M"]
   545   obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
   546     by auto
   547   then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto
   548   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q"
   549     by auto
   550   then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp
   551   let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
   552   have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)"
   553   proof (rule SUP_emeasure_incseq[of ?O])
   554     show "range ?O \<subseteq> sets M" using Q' by auto
   555     show "incseq ?O" by (fastforce intro!: incseq_SucI)
   556   qed
   557   have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
   558   have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto
   559   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
   560   proof (safe del: notI)
   561     fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto
   562     then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))"
   563       by (simp add: sets_eq emeasure_subadditive_finite)
   564     also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
   565     finally show "N (?O i) \<noteq> \<infinity>" by simp
   566   qed auto
   567   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
   568   have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric]
   569   proof (rule antisym)
   570     show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim
   571       using Q' by (auto intro!: SUP_mono emeasure_mono)
   572     show "(SUP i. emeasure M (?O i)) \<le> ?a" unfolding SUP_def
   573     proof (safe intro!: Sup_mono, unfold bex_simps)
   574       fix i
   575       have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
   576       then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and>
   577         emeasure M (\<Union>Q' ` {..i}) \<le> emeasure M x"
   578         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
   579     qed
   580   qed
   581   let ?O_0 = "(\<Union>i. ?O i)"
   582   have "?O_0 \<in> sets M" using Q' by auto
   583   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
   584   { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
   585   note Q_sets = this
   586   show ?thesis
   587   proof (intro bexI exI conjI ballI impI allI)
   588     show "disjoint_family Q"
   589       by (fastforce simp: disjoint_family_on_def Q_def
   590         split: nat.split_asm)
   591     show "range Q \<subseteq> sets M"
   592       using Q_sets by auto
   593     { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
   594       show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
   595       proof (rule disjCI, simp)
   596         assume *: "0 < emeasure M A \<longrightarrow> N A \<noteq> \<infinity>"
   597         show "emeasure M A = 0 \<and> N A = 0"
   598         proof cases
   599           assume "emeasure M A = 0" moreover with ac A have "N A = 0"
   600             unfolding absolutely_continuous_def by auto
   601           ultimately show ?thesis by simp
   602         next
   603           assume "emeasure M A \<noteq> 0" with * have "N A \<noteq> \<infinity>" using emeasure_nonneg[of M A] by auto
   604           with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
   605             using Q' by (auto intro!: plus_emeasure countable_UN)
   606           also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
   607           proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
   608             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
   609               using `N A \<noteq> \<infinity>` O_sets A by auto
   610           qed (fastforce intro!: incseq_SucI)
   611           also have "\<dots> \<le> ?a"
   612           proof (safe intro!: SUP_least)
   613             fix i have "?O i \<union> A \<in> ?Q"
   614             proof (safe del: notI)
   615               show "?O i \<union> A \<in> sets M" using O_sets A by auto
   616               from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
   617                 using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq)
   618               with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
   619                 using `N A \<noteq> \<infinity>` by auto
   620             qed
   621             then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
   622           qed
   623           finally have "emeasure M A = 0"
   624             unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
   625           with `emeasure M A \<noteq> 0` show ?thesis by auto
   626         qed
   627       qed }
   628     { fix i show "N (Q i) \<noteq> \<infinity>"
   629       proof (cases i)
   630         case 0 then show ?thesis
   631           unfolding Q_def using Q'[of 0] by simp
   632       next
   633         case (Suc n)
   634         with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
   635             emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union> x\<le>n. Q' x)"]
   636         show ?thesis
   637           by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def)
   638       qed }
   639     show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
   640     { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
   641       proof (induct j)
   642         case 0 then show ?case by (simp add: Q_def)
   643       next
   644         case (Suc j)
   645         have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce
   646         have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
   647         then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
   648           by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
   649         then show ?case using Suc by (auto simp add: eq atMost_Suc)
   650       qed }
   651     then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
   652     then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce
   653   qed
   654 qed
   655 
   656 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
   657   assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
   658   shows "\<exists>f\<in>borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
   659 proof -
   660   from split_space_into_finite_sets_and_rest[OF assms]
   661   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
   662     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
   663     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
   664     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
   665     and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force
   666   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
   667   let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))"
   668   have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). (\<forall>x. 0 \<le> f x) \<and> density (?M i) f = ?N i"
   669   proof (intro allI finite_measure.Radon_Nikodym_finite_measure)
   670     fix i
   671     from Q show "finite_measure (?M i)"
   672       by (auto intro!: finite_measureI cong: positive_integral_cong
   673                simp add: emeasure_density subset_eq sets_eq)
   674     from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
   675       by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: positive_integral_cong)
   676     with Q_fin show "finite_measure (?N i)"
   677       by (auto intro!: finite_measureI)
   678     show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
   679     show "absolutely_continuous (?M i) (?N i)"
   680       using `absolutely_continuous M N` `Q i \<in> sets M`
   681       by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
   682                intro!: absolutely_continuous_AE[OF sets_eq])
   683   qed
   684   from choice[OF this[unfolded Bex_def]]
   685   obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
   686     and f_density: "\<And>i. density (?M i) (f i) = ?N i"
   687     by auto
   688   { fix A i assume A: "A \<in> sets M"
   689     with Q borel have "(\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A"
   690       by (auto simp add: emeasure_density positive_integral_density subset_eq
   691                intro!: positive_integral_cong split: split_indicator)
   692     also have "\<dots> = emeasure N (Q i \<inter> A)"
   693       using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
   694     finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. }
   695   note integral_eq = this
   696   let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
   697   show ?thesis
   698   proof (safe intro!: bexI[of _ ?f])
   699     show "?f \<in> borel_measurable M" using Q0 borel Q_sets
   700       by (auto intro!: measurable_If)
   701     show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
   702     show "density M ?f = N"
   703     proof (rule measure_eqI)
   704       fix A assume "A \<in> sets (density M ?f)"
   705       then have "A \<in> sets M" by simp
   706       have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
   707       have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
   708         "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
   709         using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
   710       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)"
   711         using borel by (intro positive_integral_cong) (auto simp: indicator_def)
   712       also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)"
   713         using borel Qi Q0(1) `A \<in> sets M`
   714         by (subst positive_integral_add) (auto simp del: ereal_infty_mult
   715             simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
   716       also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)"
   717         by (subst integral_eq[OF `A \<in> sets M`], subst positive_integral_suminf) auto
   718       finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" .
   719       moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
   720         using Q Q_sets `A \<in> sets M`
   721         by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
   722       moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)"
   723       proof -
   724         have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
   725         from in_Q0[OF this] show ?thesis by auto
   726       qed
   727       moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
   728         using Q_sets `A \<in> sets M` Q0(1) by auto
   729       moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
   730         using `A \<in> sets M` sets_into_space Q0 by auto
   731       ultimately have "N A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
   732         using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq)
   733       with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A"
   734         by (subst emeasure_density)
   735            (auto intro!: borel_measurable_ereal_add borel_measurable_psuminf measurable_If
   736                  simp: subset_eq)
   737     qed (simp add: sets_eq)
   738   qed
   739 qed
   740 
   741 lemma (in sigma_finite_measure) Radon_Nikodym:
   742   assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
   743   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N"
   744 proof -
   745   from Ex_finite_integrable_function
   746   obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
   747     borel: "h \<in> borel_measurable M" and
   748     nn: "\<And>x. 0 \<le> h x" and
   749     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
   750     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
   751   let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
   752   let ?MT = "density M h"
   753   from borel finite nn interpret T: finite_measure ?MT
   754     by (auto intro!: finite_measureI cong: positive_integral_cong simp: emeasure_density)
   755   have "absolutely_continuous ?MT N" "sets N = sets ?MT"
   756   proof (unfold absolutely_continuous_def, safe)
   757     fix A assume "A \<in> null_sets ?MT"
   758     with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0"
   759       by (auto simp add: null_sets_density_iff)
   760     with pos sets_into_space have "AE x in M. x \<notin> A"
   761       by (elim eventually_elim1) (auto simp: not_le[symmetric])
   762     then have "A \<in> null_sets M"
   763       using `A \<in> sets M` by (simp add: AE_iff_null_sets)
   764     with ac show "A \<in> null_sets N"
   765       by (auto simp: absolutely_continuous_def)
   766   qed (auto simp add: sets_eq)
   767   from T.Radon_Nikodym_finite_measure_infinite[OF this]
   768   obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density ?MT f = N" by auto
   769   with nn borel show ?thesis
   770     by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
   771 qed
   772 
   773 section "Uniqueness of densities"
   774 
   775 lemma finite_density_unique:
   776   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   777   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
   778   and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
   779   shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
   780 proof (intro iffI ballI)
   781   fix A assume eq: "AE x in M. f x = g x"
   782   with borel show "density M f = density M g"
   783     by (auto intro: density_cong)
   784 next
   785   let ?P = "\<lambda>f A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M"
   786   assume "density M f = density M g"
   787   with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
   788     by (simp add: emeasure_density[symmetric])
   789   from this[THEN bspec, OF top] fin
   790   have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
   791   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   792       and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
   793       and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
   794     let ?N = "{x\<in>space M. g x < f x}"
   795     have N: "?N \<in> sets M" using borel by simp
   796     have "?P g ?N \<le> integral\<^isup>P M g" using pos
   797       by (intro positive_integral_mono_AE) (auto split: split_indicator)
   798     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
   799     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
   800       by (auto intro!: positive_integral_cong simp: indicator_def)
   801     also have "\<dots> = ?P f ?N - ?P g ?N"
   802     proof (rule positive_integral_diff)
   803       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
   804         using borel N by auto
   805       show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
   806            "AE x in M. 0 \<le> g x * indicator ?N x"
   807         using pos by (auto split: split_indicator)
   808     qed fact
   809     also have "\<dots> = 0"
   810       unfolding eq[THEN bspec, OF N] using positive_integral_positive[of M] Pg_fin by auto
   811     finally have "AE x in M. f x \<le> g x"
   812       using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
   813       by (subst (asm) positive_integral_0_iff_AE)
   814          (auto split: split_indicator simp: not_less ereal_minus_le_iff) }
   815   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
   816   show "AE x in M. f x = g x" by auto
   817 qed
   818 
   819 lemma (in finite_measure) density_unique_finite_measure:
   820   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
   821   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
   822   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)"
   823     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
   824   shows "AE x in M. f x = f' x"
   825 proof -
   826   let ?D = "\<lambda>f. density M f"
   827   let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
   828   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
   829 
   830   have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
   831     using borel by (auto intro!: absolutely_continuousI_density) 
   832   from split_space_into_finite_sets_and_rest[OF this]
   833   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
   834     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
   835     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
   836     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>"
   837     and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
   838   with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>"
   839     and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
   840 
   841   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
   842   let ?D = "{x\<in>space M. f x \<noteq> f' x}"
   843   have "?D \<in> sets M" using borel by auto
   844   have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
   845     unfolding indicator_def by auto
   846   have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
   847     by (intro finite_density_unique[THEN iffD1] allI)
   848        (auto intro!: borel_measurable_ereal_times f measure_eqI Int simp: emeasure_density * subset_eq)
   849   moreover have "AE x in M. ?f Q0 x = ?f' Q0 x"
   850   proof (rule AE_I')
   851     { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
   852         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
   853       let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
   854       have "(\<Union>i. ?A i) \<in> null_sets M"
   855       proof (rule null_sets_UN)
   856         fix i ::nat have "?A i \<in> sets M"
   857           using borel Q0(1) by auto
   858         have "?N (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
   859           unfolding eq[OF `?A i \<in> sets M`]
   860           by (auto intro!: positive_integral_mono simp: indicator_def)
   861         also have "\<dots> = i * emeasure M (?A i)"
   862           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
   863         also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp
   864         finally have "?N (?A i) \<noteq> \<infinity>" by simp
   865         then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
   866       qed
   867       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
   868         by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
   869       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
   870     from this[OF borel(1) refl] this[OF borel(2) f]
   871     have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
   872     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 M" by (rule null_sets.Un)
   873     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
   874       (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)
   875   qed
   876   moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
   877     ?f (space M) x = ?f' (space M) x"
   878     by (auto simp: indicator_def Q0)
   879   ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
   880     unfolding AE_all_countable[symmetric]
   881     by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def)
   882   then show "AE x in M. f x = f' x" by auto
   883 qed
   884 
   885 lemma (in sigma_finite_measure) density_unique:
   886   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
   887   assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x"
   888   assumes density_eq: "density M f = density M f'"
   889   shows "AE x in M. f x = f' x"
   890 proof -
   891   obtain h where h_borel: "h \<in> borel_measurable M"
   892     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"
   893     using Ex_finite_integrable_function by auto
   894   then have h_nn: "AE x in M. 0 \<le> h x" by auto
   895   let ?H = "density M h"
   896   interpret h: finite_measure ?H
   897     using fin h_borel pos
   898     by (intro finite_measureI) (simp cong: positive_integral_cong emeasure_density add: fin)
   899   let ?fM = "density M f"
   900   let ?f'M = "density M f'"
   901   { fix A assume "A \<in> sets M"
   902     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
   903       using pos(1) sets_into_space by (force simp: indicator_def)
   904     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
   905       using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
   906   note h_null_sets = this
   907   { fix A assume "A \<in> sets M"
   908     have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
   909       using `A \<in> sets M` h_borel h_nn f f'
   910       by (intro positive_integral_density[symmetric]) auto
   911     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
   912       by (simp_all add: density_eq)
   913     also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
   914       using `A \<in> sets M` h_borel h_nn f f'
   915       by (intro positive_integral_density) auto
   916     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)"
   917       by (simp add: ac_simps)
   918     then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
   919       using `A \<in> sets M` h_borel h_nn f f'
   920       by (subst (asm) (1 2) positive_integral_density[symmetric]) auto }
   921   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
   922     by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M])
   923        (auto simp add: AE_density)
   924   then show "AE x in M. f x = f' x"
   925     unfolding eventually_ae_filter using h_borel pos
   926     by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric]
   927                           AE_iff_null_sets[symmetric])
   928        blast
   929 qed
   930 
   931 lemma (in sigma_finite_measure) density_unique_iff:
   932   assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x"
   933   assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x"
   934   shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
   935   using density_unique[OF assms] density_cong[OF f f'] by auto
   936 
   937 lemma sigma_finite_density_unique:
   938   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
   939   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
   940   and fin: "sigma_finite_measure (density M f)"
   941   shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
   942 proof
   943   assume "AE x in M. f x = g x" with borel show "density M f = density M g" 
   944     by (auto intro: density_cong)
   945 next
   946   assume eq: "density M f = density M g"
   947   interpret f!: sigma_finite_measure "density M f" by fact
   948   from f.sigma_finite_incseq guess A . note cover = this
   949 
   950   have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x"
   951     unfolding AE_all_countable
   952   proof
   953     fix i
   954     have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))"
   955       unfolding eq ..
   956     moreover have "(\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>"
   957       using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq)
   958     ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x"
   959       using borel pos cover(1) pos
   960       by (intro finite_density_unique[THEN iffD1])
   961          (auto simp: density_density_eq subset_eq)
   962     then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x"
   963       by auto
   964   qed
   965   with AE_space show "AE x in M. f x = g x"
   966     apply eventually_elim
   967     using cover(2)[symmetric]
   968     apply auto
   969     done
   970 qed
   971 
   972 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite':
   973   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
   974   shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
   975     (is "sigma_finite_measure ?N \<longleftrightarrow> _")
   976 proof
   977   assume "sigma_finite_measure ?N"
   978   then interpret N: sigma_finite_measure ?N .
   979   from N.Ex_finite_integrable_function obtain h where
   980     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
   981     h_nn: "\<And>x. 0 \<le> h x" and
   982     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
   983   have "AE x in M. f x * h x \<noteq> \<infinity>"
   984   proof (rule AE_I')
   985     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
   986       by (auto intro!: positive_integral_density)
   987     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
   988       using h(2) by simp
   989     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
   990       using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
   991   qed auto
   992   then show "AE x in M. f x \<noteq> \<infinity>"
   993     using fin by (auto elim!: AE_Ball_mp)
   994 next
   995   assume AE: "AE x in M. f x \<noteq> \<infinity>"
   996   from sigma_finite guess Q .. note Q = this
   997   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
   998   { fix i j have "A i \<inter> Q j \<in> sets M"
   999     unfolding A_def using f Q
  1000     apply (rule_tac Int)
  1001     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
  1002   note A_in_sets = this
  1003   let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
  1004   show "sigma_finite_measure ?N"
  1005   proof (default, intro exI conjI subsetI allI)
  1006     fix x assume "x \<in> range ?A"
  1007     then obtain n where n: "x = ?A n" by auto
  1008     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
  1009   next
  1010     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
  1011     proof safe
  1012       fix x i j assume "x \<in> A i" "x \<in> Q j"
  1013       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
  1014         by (intro UN_I[of "prod_encode (i,j)"]) auto
  1015     qed auto
  1016     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
  1017     also have "(\<Union>i. A i) = space M"
  1018     proof safe
  1019       fix x assume x: "x \<in> space M"
  1020       show "x \<in> (\<Union>i. A i)"
  1021       proof (cases "f x")
  1022         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
  1023       next
  1024         case (real r)
  1025         with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
  1026         then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
  1027       next
  1028         case MInf with x show ?thesis
  1029           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
  1030       qed
  1031     qed (auto simp: A_def)
  1032     finally show "(\<Union>i. ?A i) = space ?N" by simp
  1033   next
  1034     fix n obtain i j where
  1035       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
  1036     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
  1037     proof (cases i)
  1038       case 0
  1039       have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
  1040         using AE by (auto simp: A_def `i = 0`)
  1041       from positive_integral_cong_AE[OF this] show ?thesis by simp
  1042     next
  1043       case (Suc n)
  1044       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
  1045         (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
  1046         by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
  1047       also have "\<dots> = Suc n * emeasure M (Q j)"
  1048         using Q by (auto intro!: positive_integral_cmult_indicator)
  1049       also have "\<dots> < \<infinity>"
  1050         using Q by (auto simp: real_eq_of_nat[symmetric])
  1051       finally show ?thesis by simp
  1052     qed
  1053     then show "emeasure ?N (?A n) \<noteq> \<infinity>"
  1054       using A_in_sets Q f by (auto simp: emeasure_density)
  1055   qed
  1056 qed
  1057 
  1058 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
  1059   "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
  1060   apply (subst density_max_0)
  1061   apply (subst sigma_finite_iff_density_finite')
  1062   apply (auto simp: max_def intro!: measurable_If)
  1063   done
  1064 
  1065 section "Radon-Nikodym derivative"
  1066 
  1067 definition
  1068   "RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
  1069 
  1070 lemma
  1071   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
  1072   shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel)
  1073     and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density)
  1074     and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos)
  1075 proof -
  1076   let ?f = "\<lambda>x. max 0 (f x)"
  1077   let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f"
  1078   from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max)
  1079   then have "?P (RN_deriv M (density M f))"
  1080     unfolding RN_deriv_def by (rule someI[where P="?P"])
  1081   then show ?borel ?density ?pos by auto
  1082 qed
  1083 
  1084 lemma (in sigma_finite_measure) RN_deriv:
  1085   assumes "absolutely_continuous M N" "sets N = sets M"
  1086   shows borel_measurable_RN_deriv: "RN_deriv M N \<in> borel_measurable M" (is ?borel)
  1087     and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density)
  1088     and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos)
  1089 proof -
  1090   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
  1091   from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp
  1092   from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp
  1093   from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp
  1094 qed
  1095 
  1096 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
  1097   assumes N: "absolutely_continuous M N" "sets N = sets M"
  1098     and f: "f \<in> borel_measurable M"
  1099   shows "integral\<^isup>P N f = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
  1100 proof -
  1101   have "integral\<^isup>P N f = integral\<^isup>P (density M (RN_deriv M N)) f"
  1102     using N by (simp add: density_RN_deriv)
  1103   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
  1104     using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density)
  1105   finally show ?thesis by simp
  1106 qed
  1107 
  1108 lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
  1109   using AE_iff_null_sets[of N M] by auto
  1110 
  1111 lemma (in sigma_finite_measure) RN_deriv_unique:
  1112   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
  1113   and eq: "density M f = N"
  1114   shows "AE x in M. f x = RN_deriv M N x"
  1115   unfolding eq[symmetric]
  1116   by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv_density
  1117             RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
  1118 
  1119 lemma RN_deriv_unique_sigma_finite:
  1120   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
  1121   and eq: "density M f = N" and fin: "sigma_finite_measure N"
  1122   shows "AE x in M. f x = RN_deriv M N x"
  1123   using fin unfolding eq[symmetric]
  1124   by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv_density
  1125             RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
  1126 
  1127 lemma (in sigma_finite_measure) RN_deriv_distr:
  1128   fixes T :: "'a \<Rightarrow> 'b"
  1129   assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
  1130     and inv: "\<forall>x\<in>space M. T' (T x) = x"
  1131   and ac: "absolutely_continuous (distr M M' T) (distr N M' T)"
  1132   and N: "sets N = sets M"
  1133   shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
  1134 proof (rule RN_deriv_unique)
  1135   have [simp]: "sets N = sets M" by fact
  1136   note sets_eq_imp_space_eq[OF N, simp]
  1137   have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
  1138   { fix A assume "A \<in> sets M"
  1139     with inv T T' sets_into_space[OF this]
  1140     have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
  1141       by (auto simp: measurable_def) }
  1142   note eq = this[simp]
  1143   { fix A assume "A \<in> sets M"
  1144     with inv T T' sets_into_space[OF this]
  1145     have "(T' \<circ> T) -` A \<inter> space M = A"
  1146       by (auto simp: measurable_def) }
  1147   note eq2 = this[simp]
  1148   let ?M' = "distr M M' T" and ?N' = "distr N M' T"
  1149   interpret M': sigma_finite_measure ?M'
  1150   proof
  1151     from sigma_finite guess F .. note F = this
  1152     show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets ?M' \<and> (\<Union>i. A i) = space ?M' \<and> (\<forall>i. emeasure ?M' (A i) \<noteq> \<infinity>)"
  1153     proof (intro exI conjI allI)
  1154       show *: "range (\<lambda>i. T' -` F i \<inter> space ?M') \<subseteq> sets ?M'"
  1155         using F T' by (auto simp: measurable_def)
  1156       show "(\<Union>i. T' -` F i \<inter> space ?M') = space ?M'"
  1157         using F T' by (force simp: measurable_def)
  1158       fix i
  1159       have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
  1160       moreover
  1161       have Fi: "F i \<in> sets M" using F by auto
  1162       ultimately show "emeasure ?M' (T' -` F i \<inter> space ?M') \<noteq> \<infinity>"
  1163         using F T T' by (simp add: emeasure_distr)
  1164     qed
  1165   qed
  1166   have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
  1167     using T ac by (intro measurable_comp[where b="?M'"] M'.borel_measurable_RN_deriv) simp_all
  1168   then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
  1169     by (simp add: comp_def)
  1170   show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto
  1171 
  1172   have "N = distr N M (T' \<circ> T)"
  1173     by (subst measure_of_of_measure[of N, symmetric])
  1174        (auto simp add: distr_def sigma_sets_eq intro!: measure_of_eq space_closed)
  1175   also have "\<dots> = distr (distr N M' T) M T'"
  1176     using T T' by (simp add: distr_distr)
  1177   also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
  1178     using ac by (simp add: M'.density_RN_deriv)
  1179   also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
  1180     using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv])
  1181   finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
  1182     by (simp add: comp_def)
  1183 qed
  1184 
  1185 lemma (in sigma_finite_measure) RN_deriv_finite:
  1186   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
  1187   shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
  1188 proof -
  1189   interpret N: sigma_finite_measure N by fact
  1190   from N show ?thesis
  1191     using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp
  1192 qed
  1193 
  1194 lemma (in sigma_finite_measure)
  1195   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
  1196     and f: "f \<in> borel_measurable M"
  1197   shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
  1198       integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable)
  1199     and RN_deriv_integral: "integral\<^isup>L N f =
  1200       (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
  1201 proof -
  1202   note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
  1203   interpret N: sigma_finite_measure N by fact
  1204   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
  1205   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
  1206   have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def)
  1207   { fix f :: "'a \<Rightarrow> real"
  1208     { fix x assume *: "RN_deriv M N x \<noteq> \<infinity>"
  1209       have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
  1210         by (simp add: mult_le_0_iff)
  1211       then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
  1212         using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) }
  1213     then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M N x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (f x) \<partial>M)"
  1214               "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M N x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M N x * ereal (- f x) \<partial>M)"
  1215       using RN_deriv_finite[OF N ac] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
  1216       by (auto intro!: positive_integral_cong_AE) }
  1217   note * = this
  1218   show ?integral ?integrable
  1219     unfolding lebesgue_integral_def integrable_def *
  1220     using Nf f RN_deriv(1)[OF ac]
  1221     by (auto simp: RN_deriv_positive_integral[OF ac])
  1222 qed
  1223 
  1224 lemma (in sigma_finite_measure) real_RN_deriv:
  1225   assumes "finite_measure N"
  1226   assumes ac: "absolutely_continuous M N" "sets N = sets M"
  1227   obtains D where "D \<in> borel_measurable M"
  1228     and "AE x in M. RN_deriv M N x = ereal (D x)"
  1229     and "AE x in N. 0 < D x"
  1230     and "\<And>x. 0 \<le> D x"
  1231 proof
  1232   interpret N: finite_measure N by fact
  1233   
  1234   note RN = RN_deriv[OF ac]
  1235 
  1236   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
  1237 
  1238   show "(\<lambda>x. real (RN_deriv M N x)) \<in> borel_measurable M"
  1239     using RN by auto
  1240 
  1241   have "N (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
  1242     using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
  1243   also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
  1244     by (intro positive_integral_cong) (auto simp: indicator_def)
  1245   also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
  1246     using RN by (intro positive_integral_cmult_indicator) auto
  1247   finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
  1248   moreover
  1249   have "emeasure M (?RN \<infinity>) = 0"
  1250   proof (rule ccontr)
  1251     assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
  1252     moreover from RN have "0 \<le> emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
  1253     ultimately have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
  1254     with eq have "N (?RN \<infinity>) = \<infinity>" by simp
  1255     with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
  1256   qed
  1257   ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
  1258     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
  1259   then show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
  1260     using RN(3) by (auto simp: ereal_real)
  1261   then have eq: "AE x in N. RN_deriv M N x = ereal (real (RN_deriv M N x))"
  1262     using ac absolutely_continuous_AE by auto
  1263 
  1264   show "\<And>x. 0 \<le> real (RN_deriv M N x)"
  1265     using RN by (auto intro: real_of_ereal_pos)
  1266 
  1267   have "N (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
  1268     using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
  1269   also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
  1270     by (intro positive_integral_cong) (auto simp: indicator_def)
  1271   finally have "AE x in N. RN_deriv M N x \<noteq> 0"
  1272     using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
  1273   with RN(3) eq show "AE x in N. 0 < real (RN_deriv M N x)"
  1274     by (auto simp: zero_less_real_of_ereal le_less)
  1275 qed
  1276 
  1277 lemma (in sigma_finite_measure) RN_deriv_singleton:
  1278   assumes ac: "absolutely_continuous M N" "sets N = sets M"
  1279   and x: "{x} \<in> sets M"
  1280   shows "N {x} = RN_deriv M N x * emeasure M {x}"
  1281 proof -
  1282   note deriv = RN_deriv[OF ac]
  1283   from deriv(1,3) `{x} \<in> sets M`
  1284   have "density M (RN_deriv M N) {x} = (\<integral>\<^isup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
  1285     by (auto simp: indicator_def emeasure_density intro!: positive_integral_cong)
  1286   with x deriv show ?thesis
  1287     by (auto simp: positive_integral_cmult_indicator)
  1288 qed
  1289 
  1290 end