src/HOL/Probability/Radon_Nikodym.thy
 author hoelzl Mon Apr 23 12:14:35 2012 +0200 (2012-04-23) changeset 47694 05663f75964c parent 46731 5302e932d1e5 child 49778 bbbc0f492780 permissions -rw-r--r--
reworked Probability theory
```     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 "(\<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))
```
```   780     \<longleftrightarrow> (AE x in M. f x = g x)"
```
```   781     (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
```
```   782 proof (intro iffI ballI)
```
```   783   fix A assume eq: "AE x in M. f x = g x"
```
```   784   then show "?P f A = ?P g A"
```
```   785     by (auto intro: positive_integral_cong_AE)
```
```   786 next
```
```   787   assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   788   from this[THEN bspec, OF top] fin
```
```   789   have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
```
```   790   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   791       and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
```
```   792       and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   793     let ?N = "{x\<in>space M. g x < f x}"
```
```   794     have N: "?N \<in> sets M" using borel by simp
```
```   795     have "?P g ?N \<le> integral\<^isup>P M g" using pos
```
```   796       by (intro positive_integral_mono_AE) (auto split: split_indicator)
```
```   797     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
```
```   798     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
```
```   799       by (auto intro!: positive_integral_cong simp: indicator_def)
```
```   800     also have "\<dots> = ?P f ?N - ?P g ?N"
```
```   801     proof (rule positive_integral_diff)
```
```   802       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
```
```   803         using borel N by auto
```
```   804       show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
```
```   805            "AE x in M. 0 \<le> g x * indicator ?N x"
```
```   806         using pos by (auto split: split_indicator)
```
```   807     qed fact
```
```   808     also have "\<dots> = 0"
```
```   809       unfolding eq[THEN bspec, OF N] using positive_integral_positive[of M] Pg_fin by auto
```
```   810     finally have "AE x in M. f x \<le> g x"
```
```   811       using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
```
```   812       by (subst (asm) positive_integral_0_iff_AE)
```
```   813          (auto split: split_indicator simp: not_less ereal_minus_le_iff) }
```
```   814   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
```
```   815   show "AE x in M. f x = g x" by auto
```
```   816 qed
```
```   817
```
```   818 lemma (in finite_measure) density_unique_finite_measure:
```
```   819   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
```
```   820   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
```
```   821   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)"
```
```   822     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
```
```   823   shows "AE x in M. f x = f' x"
```
```   824 proof -
```
```   825   let ?D = "\<lambda>f. density M f"
```
```   826   let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
```
```   827   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
```
```   828
```
```   829   have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
```
```   830     using borel by (auto intro!: absolutely_continuousI_density)
```
```   831   from split_space_into_finite_sets_and_rest[OF this]
```
```   832   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
```
```   833     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
```
```   834     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
```
```   835     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>"
```
```   836     and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
```
```   837   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>"
```
```   838     and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
```
```   839
```
```   840   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
```
```   841   let ?D = "{x\<in>space M. f x \<noteq> f' x}"
```
```   842   have "?D \<in> sets M" using borel by auto
```
```   843   have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
```
```   844     unfolding indicator_def by auto
```
```   845   have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
```
```   846     by (intro finite_density_unique[THEN iffD1] allI)
```
```   847        (auto intro!: borel_measurable_ereal_times f Int simp: *)
```
```   848   moreover have "AE x in M. ?f Q0 x = ?f' Q0 x"
```
```   849   proof (rule AE_I')
```
```   850     { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
```
```   851         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   852       let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
```
```   853       have "(\<Union>i. ?A i) \<in> null_sets M"
```
```   854       proof (rule null_sets_UN)
```
```   855         fix i ::nat have "?A i \<in> sets M"
```
```   856           using borel Q0(1) by auto
```
```   857         have "?N (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
```
```   858           unfolding eq[OF `?A i \<in> sets M`]
```
```   859           by (auto intro!: positive_integral_mono simp: indicator_def)
```
```   860         also have "\<dots> = i * emeasure M (?A i)"
```
```   861           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
```
```   862         also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp
```
```   863         finally have "?N (?A i) \<noteq> \<infinity>" by simp
```
```   864         then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
```
```   865       qed
```
```   866       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
```
```   867         by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
```
```   868       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
```
```   869     from this[OF borel(1) refl] this[OF borel(2) f]
```
```   870     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
```
```   871     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)
```
```   872     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
```
```   873       (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)
```
```   874   qed
```
```   875   moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
```
```   876     ?f (space M) x = ?f' (space M) x"
```
```   877     by (auto simp: indicator_def Q0)
```
```   878   ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
```
```   879     unfolding AE_all_countable[symmetric]
```
```   880     by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def)
```
```   881   then show "AE x in M. f x = f' x" by auto
```
```   882 qed
```
```   883
```
```   884 lemma (in sigma_finite_measure) density_unique:
```
```   885   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```   886   assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x"
```
```   887   assumes density_eq: "density M f = density M f'"
```
```   888   shows "AE x in M. f x = f' x"
```
```   889 proof -
```
```   890   obtain h where h_borel: "h \<in> borel_measurable M"
```
```   891     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"
```
```   892     using Ex_finite_integrable_function by auto
```
```   893   then have h_nn: "AE x in M. 0 \<le> h x" by auto
```
```   894   let ?H = "density M h"
```
```   895   interpret h: finite_measure ?H
```
```   896     using fin h_borel pos
```
```   897     by (intro finite_measureI) (simp cong: positive_integral_cong emeasure_density add: fin)
```
```   898   let ?fM = "density M f"
```
```   899   let ?f'M = "density M f'"
```
```   900   { fix A assume "A \<in> sets M"
```
```   901     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
```
```   902       using pos(1) sets_into_space by (force simp: indicator_def)
```
```   903     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
```
```   904       using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
```
```   905   note h_null_sets = this
```
```   906   { fix A assume "A \<in> sets M"
```
```   907     have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
```
```   908       using `A \<in> sets M` h_borel h_nn f f'
```
```   909       by (intro positive_integral_density[symmetric]) auto
```
```   910     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
```
```   911       by (simp_all add: density_eq)
```
```   912     also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
```
```   913       using `A \<in> sets M` h_borel h_nn f f'
```
```   914       by (intro positive_integral_density) auto
```
```   915     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)"
```
```   916       by (simp add: ac_simps)
```
```   917     then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
```
```   918       using `A \<in> sets M` h_borel h_nn f f'
```
```   919       by (subst (asm) (1 2) positive_integral_density[symmetric]) auto }
```
```   920   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
```
```   921     by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M])
```
```   922        (auto simp add: AE_density)
```
```   923   then show "AE x in M. f x = f' x"
```
```   924     unfolding eventually_ae_filter using h_borel pos
```
```   925     by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric]
```
```   926                           AE_iff_null_sets[symmetric])
```
```   927        blast
```
```   928 qed
```
```   929
```
```   930 lemma (in sigma_finite_measure) density_unique_iff:
```
```   931   assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x"
```
```   932   assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x"
```
```   933   shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
```
```   934   using density_unique[OF assms] density_cong[OF f f'] by auto
```
```   935
```
```   936 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
```
```   937   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```   938   shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
```
```   939     (is "sigma_finite_measure ?N \<longleftrightarrow> _")
```
```   940 proof
```
```   941   assume "sigma_finite_measure ?N"
```
```   942   then interpret N: sigma_finite_measure ?N .
```
```   943   from N.Ex_finite_integrable_function obtain h where
```
```   944     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
```
```   945     h_nn: "\<And>x. 0 \<le> h x" and
```
```   946     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
```
```   947   have "AE x in M. f x * h x \<noteq> \<infinity>"
```
```   948   proof (rule AE_I')
```
```   949     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
```
```   950       by (auto intro!: positive_integral_density)
```
```   951     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
```
```   952       using h(2) by simp
```
```   953     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
```
```   954       using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
```
```   955   qed auto
```
```   956   then show "AE x in M. f x \<noteq> \<infinity>"
```
```   957     using fin by (auto elim!: AE_Ball_mp)
```
```   958 next
```
```   959   assume AE: "AE x in M. f x \<noteq> \<infinity>"
```
```   960   from sigma_finite guess Q .. note Q = this
```
```   961   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
```
```   962   { fix i j have "A i \<inter> Q j \<in> sets M"
```
```   963     unfolding A_def using f Q
```
```   964     apply (rule_tac Int)
```
```   965     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
```
```   966   note A_in_sets = this
```
```   967   let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
```
```   968   show "sigma_finite_measure ?N"
```
```   969   proof (default, intro exI conjI subsetI allI)
```
```   970     fix x assume "x \<in> range ?A"
```
```   971     then obtain n where n: "x = ?A n" by auto
```
```   972     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
```
```   973   next
```
```   974     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
```
```   975     proof safe
```
```   976       fix x i j assume "x \<in> A i" "x \<in> Q j"
```
```   977       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
```
```   978         by (intro UN_I[of "prod_encode (i,j)"]) auto
```
```   979     qed auto
```
```   980     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
```
```   981     also have "(\<Union>i. A i) = space M"
```
```   982     proof safe
```
```   983       fix x assume x: "x \<in> space M"
```
```   984       show "x \<in> (\<Union>i. A i)"
```
```   985       proof (cases "f x")
```
```   986         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
```
```   987       next
```
```   988         case (real r)
```
```   989         with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
```
```   990         then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
```
```   991       next
```
```   992         case MInf with x show ?thesis
```
```   993           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
```
```   994       qed
```
```   995     qed (auto simp: A_def)
```
```   996     finally show "(\<Union>i. ?A i) = space ?N" by simp
```
```   997   next
```
```   998     fix n obtain i j where
```
```   999       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
```
```  1000     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
```
```  1001     proof (cases i)
```
```  1002       case 0
```
```  1003       have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
```
```  1004         using AE by (auto simp: A_def `i = 0`)
```
```  1005       from positive_integral_cong_AE[OF this] show ?thesis by simp
```
```  1006     next
```
```  1007       case (Suc n)
```
```  1008       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
```
```  1009         (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
```
```  1010         by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
```
```  1011       also have "\<dots> = Suc n * emeasure M (Q j)"
```
```  1012         using Q by (auto intro!: positive_integral_cmult_indicator)
```
```  1013       also have "\<dots> < \<infinity>"
```
```  1014         using Q by (auto simp: real_eq_of_nat[symmetric])
```
```  1015       finally show ?thesis by simp
```
```  1016     qed
```
```  1017     then show "emeasure ?N (?A n) \<noteq> \<infinity>"
```
```  1018       using A_in_sets Q f by (auto simp: emeasure_density)
```
```  1019   qed
```
```  1020 qed
```
```  1021
```
```  1022 section "Radon-Nikodym derivative"
```
```  1023
```
```  1024 definition
```
```  1025   "RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
```
```  1026
```
```  1027 lemma
```
```  1028   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  1029   shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel)
```
```  1030     and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density)
```
```  1031     and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos)
```
```  1032 proof -
```
```  1033   let ?f = "\<lambda>x. max 0 (f x)"
```
```  1034   let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f"
```
```  1035   from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max)
```
```  1036   then have "?P (RN_deriv M (density M f))"
```
```  1037     unfolding RN_deriv_def by (rule someI[where P="?P"])
```
```  1038   then show ?borel ?density ?pos by auto
```
```  1039 qed
```
```  1040
```
```  1041 lemma (in sigma_finite_measure) RN_deriv:
```
```  1042   assumes "absolutely_continuous M N" "sets N = sets M"
```
```  1043   shows borel_measurable_RN_deriv: "RN_deriv M N \<in> borel_measurable M" (is ?borel)
```
```  1044     and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density)
```
```  1045     and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos)
```
```  1046 proof -
```
```  1047   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
```
```  1048   from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp
```
```  1049   from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp
```
```  1050   from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp
```
```  1051 qed
```
```  1052
```
```  1053 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
```
```  1054   assumes N: "absolutely_continuous M N" "sets N = sets M"
```
```  1055     and f: "f \<in> borel_measurable M"
```
```  1056   shows "integral\<^isup>P N f = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
```
```  1057 proof -
```
```  1058   have "integral\<^isup>P N f = integral\<^isup>P (density M (RN_deriv M N)) f"
```
```  1059     using N by (simp add: density_RN_deriv)
```
```  1060   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)"
```
```  1061     using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density)
```
```  1062   finally show ?thesis by simp
```
```  1063 qed
```
```  1064
```
```  1065 lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
```
```  1066   using AE_iff_null_sets[of N M] by auto
```
```  1067
```
```  1068 lemma (in sigma_finite_measure) RN_deriv_unique:
```
```  1069   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
```
```  1070   and eq: "density M f = N"
```
```  1071   shows "AE x in M. f x = RN_deriv M N x"
```
```  1072 proof (rule density_unique)
```
```  1073   have ac: "absolutely_continuous M N"
```
```  1074     using f(1) unfolding eq[symmetric] by (rule absolutely_continuousI_density)
```
```  1075   have eq2: "sets N = sets M"
```
```  1076     unfolding eq[symmetric] by simp
```
```  1077   show "RN_deriv M N \<in> borel_measurable M" "AE x in M. 0 \<le> RN_deriv M N x"
```
```  1078     "density M f = density M (RN_deriv M N)"
```
```  1079     using RN_deriv[OF ac eq2] eq by auto
```
```  1080 qed fact+
```
```  1081
```
```  1082 lemma (in sigma_finite_measure) RN_deriv_distr:
```
```  1083   fixes T :: "'a \<Rightarrow> 'b"
```
```  1084   assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
```
```  1085     and inv: "\<forall>x\<in>space M. T' (T x) = x"
```
```  1086   and ac: "absolutely_continuous (distr M M' T) (distr N M' T)"
```
```  1087   and N: "sets N = sets M"
```
```  1088   shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
```
```  1089 proof (rule RN_deriv_unique)
```
```  1090   have [simp]: "sets N = sets M" by fact
```
```  1091   note sets_eq_imp_space_eq[OF N, simp]
```
```  1092   have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
```
```  1093   { fix A assume "A \<in> sets M"
```
```  1094     with inv T T' sets_into_space[OF this]
```
```  1095     have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
```
```  1096       by (auto simp: measurable_def) }
```
```  1097   note eq = this[simp]
```
```  1098   { fix A assume "A \<in> sets M"
```
```  1099     with inv T T' sets_into_space[OF this]
```
```  1100     have "(T' \<circ> T) -` A \<inter> space M = A"
```
```  1101       by (auto simp: measurable_def) }
```
```  1102   note eq2 = this[simp]
```
```  1103   let ?M' = "distr M M' T" and ?N' = "distr N M' T"
```
```  1104   interpret M': sigma_finite_measure ?M'
```
```  1105   proof
```
```  1106     from sigma_finite guess F .. note F = this
```
```  1107     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>)"
```
```  1108     proof (intro exI conjI allI)
```
```  1109       show *: "range (\<lambda>i. T' -` F i \<inter> space ?M') \<subseteq> sets ?M'"
```
```  1110         using F T' by (auto simp: measurable_def)
```
```  1111       show "(\<Union>i. T' -` F i \<inter> space ?M') = space ?M'"
```
```  1112         using F T' by (force simp: measurable_def)
```
```  1113       fix i
```
```  1114       have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
```
```  1115       moreover
```
```  1116       have Fi: "F i \<in> sets M" using F by auto
```
```  1117       ultimately show "emeasure ?M' (T' -` F i \<inter> space ?M') \<noteq> \<infinity>"
```
```  1118         using F T T' by (simp add: emeasure_distr)
```
```  1119     qed
```
```  1120   qed
```
```  1121   have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
```
```  1122     using T ac by (intro measurable_comp[where b="?M'"] M'.borel_measurable_RN_deriv) simp_all
```
```  1123   then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
```
```  1124     by (simp add: comp_def)
```
```  1125   show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto
```
```  1126
```
```  1127   have "N = distr N M (T' \<circ> T)"
```
```  1128     by (subst measure_of_of_measure[of N, symmetric])
```
```  1129        (auto simp add: distr_def sigma_sets_eq intro!: measure_of_eq space_closed)
```
```  1130   also have "\<dots> = distr (distr N M' T) M T'"
```
```  1131     using T T' by (simp add: distr_distr)
```
```  1132   also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
```
```  1133     using ac by (simp add: M'.density_RN_deriv)
```
```  1134   also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
```
```  1135     using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv])
```
```  1136   finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
```
```  1137     by (simp add: comp_def)
```
```  1138 qed
```
```  1139
```
```  1140 lemma (in sigma_finite_measure) RN_deriv_finite:
```
```  1141   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1142   shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
```
```  1143 proof -
```
```  1144   interpret N: sigma_finite_measure N by fact
```
```  1145   from N show ?thesis
```
```  1146     using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp
```
```  1147 qed
```
```  1148
```
```  1149 lemma (in sigma_finite_measure)
```
```  1150   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1151     and f: "f \<in> borel_measurable M"
```
```  1152   shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
```
```  1153       integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable)
```
```  1154     and RN_deriv_integral: "integral\<^isup>L N f =
```
```  1155       (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
```
```  1156 proof -
```
```  1157   note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
```
```  1158   interpret N: sigma_finite_measure N by fact
```
```  1159   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
```
```  1160   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
```
```  1161   have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def)
```
```  1162   { fix f :: "'a \<Rightarrow> real"
```
```  1163     { fix x assume *: "RN_deriv M N x \<noteq> \<infinity>"
```
```  1164       have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
```
```  1165         by (simp add: mult_le_0_iff)
```
```  1166       then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
```
```  1167         using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) }
```
```  1168     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)"
```
```  1169               "(\<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)"
```
```  1170       using RN_deriv_finite[OF N ac] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
```
```  1171       by (auto intro!: positive_integral_cong_AE) }
```
```  1172   note * = this
```
```  1173   show ?integral ?integrable
```
```  1174     unfolding lebesgue_integral_def integrable_def *
```
```  1175     using Nf f RN_deriv(1)[OF ac]
```
```  1176     by (auto simp: RN_deriv_positive_integral[OF ac])
```
```  1177 qed
```
```  1178
```
```  1179 lemma (in sigma_finite_measure) real_RN_deriv:
```
```  1180   assumes "finite_measure N"
```
```  1181   assumes ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1182   obtains D where "D \<in> borel_measurable M"
```
```  1183     and "AE x in M. RN_deriv M N x = ereal (D x)"
```
```  1184     and "AE x in N. 0 < D x"
```
```  1185     and "\<And>x. 0 \<le> D x"
```
```  1186 proof
```
```  1187   interpret N: finite_measure N by fact
```
```  1188
```
```  1189   note RN = RN_deriv[OF ac]
```
```  1190
```
```  1191   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
```
```  1192
```
```  1193   show "(\<lambda>x. real (RN_deriv M N x)) \<in> borel_measurable M"
```
```  1194     using RN by auto
```
```  1195
```
```  1196   have "N (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1197     using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
```
```  1198   also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1199     by (intro positive_integral_cong) (auto simp: indicator_def)
```
```  1200   also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
```
```  1201     using RN by (intro positive_integral_cmult_indicator) auto
```
```  1202   finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
```
```  1203   moreover
```
```  1204   have "emeasure M (?RN \<infinity>) = 0"
```
```  1205   proof (rule ccontr)
```
```  1206     assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
```
```  1207     moreover from RN have "0 \<le> emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
```
```  1208     ultimately have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto
```
```  1209     with eq have "N (?RN \<infinity>) = \<infinity>" by simp
```
```  1210     with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
```
```  1211   qed
```
```  1212   ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
```
```  1213     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
```
```  1214   then show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
```
```  1215     using RN(3) by (auto simp: ereal_real)
```
```  1216   then have eq: "AE x in N. RN_deriv M N x = ereal (real (RN_deriv M N x))"
```
```  1217     using ac absolutely_continuous_AE by auto
```
```  1218
```
```  1219   show "\<And>x. 0 \<le> real (RN_deriv M N x)"
```
```  1220     using RN by (auto intro: real_of_ereal_pos)
```
```  1221
```
```  1222   have "N (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
```
```  1223     using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
```
```  1224   also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
```
```  1225     by (intro positive_integral_cong) (auto simp: indicator_def)
```
```  1226   finally have "AE x in N. RN_deriv M N x \<noteq> 0"
```
```  1227     using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
```
```  1228   with RN(3) eq show "AE x in N. 0 < real (RN_deriv M N x)"
```
```  1229     by (auto simp: zero_less_real_of_ereal le_less)
```
```  1230 qed
```
```  1231
```
```  1232 lemma (in sigma_finite_measure) RN_deriv_singleton:
```
```  1233   assumes ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1234   and x: "{x} \<in> sets M"
```
```  1235   shows "N {x} = RN_deriv M N x * emeasure M {x}"
```
```  1236 proof -
```
```  1237   note deriv = RN_deriv[OF ac]
```
```  1238   from deriv(1,3) `{x} \<in> sets M`
```
```  1239   have "density M (RN_deriv M N) {x} = (\<integral>\<^isup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
```
```  1240     by (auto simp: indicator_def emeasure_density intro!: positive_integral_cong)
```
```  1241   with x deriv show ?thesis
```
```  1242     by (auto simp: positive_integral_cmult_indicator)
```
```  1243 qed
```
```  1244
```
```  1245 end
```