src/HOL/Probability/Radon_Nikodym.thy
 author hoelzl Wed Dec 07 15:10:29 2011 +0100 (2011-12-07) changeset 45777 c36637603821 parent 45769 2d5b1af2426a child 46731 5302e932d1e5 permissions -rw-r--r--
remove unnecessary sublocale instantiations in HOL-Probability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
```     1 (*  Title:      HOL/Probability/Radon_Nikodym.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 header {*Radon-Nikod{\'y}m derivative*}
```
```     6
```
```     7 theory Radon_Nikodym
```
```     8 imports Lebesgue_Integration
```
```     9 begin
```
```    10
```
```    11 lemma (in sigma_finite_measure) Ex_finite_integrable_function:
```
```    12   shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)"
```
```    13 proof -
```
```    14   obtain A :: "nat \<Rightarrow> 'a set" where
```
```    15     range: "range A \<subseteq> sets M" and
```
```    16     space: "(\<Union>i. A i) = space M" and
```
```    17     measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>" and
```
```    18     disjoint: "disjoint_family A"
```
```    19     using disjoint_sigma_finite by auto
```
```    20   let "?B i" = "2^Suc i * \<mu> (A i)"
```
```    21   have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
```
```    22   proof
```
```    23     fix i have Ai: "A i \<in> sets M" using range by auto
```
```    24     from measure positive_measure[OF this]
```
```    25     show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
```
```    26       by (auto intro!: ereal_dense simp: ereal_0_gt_inverse)
```
```    27   qed
```
```    28   from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
```
```    29     "\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
```
```    30   { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
```
```    31   let "?h x" = "\<Sum>i. n i * indicator (A i) x"
```
```    32   show ?thesis
```
```    33   proof (safe intro!: bexI[of _ ?h] del: notI)
```
```    34     have "\<And>i. A i \<in> sets M"
```
```    35       using range by fastforce+
```
```    36     then have "integral\<^isup>P M ?h = (\<Sum>i. n i * \<mu> (A i))" using pos
```
```    37       by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
```
```    38     also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
```
```    39     proof (rule suminf_le_pos)
```
```    40       fix N
```
```    41       have "n N * \<mu> (A N) \<le> inverse (2^Suc N * \<mu> (A N)) * \<mu> (A N)"
```
```    42         using positive_measure[OF `A N \<in> sets M`] n[of N]
```
```    43         by (intro ereal_mult_right_mono) auto
```
```    44       also have "\<dots> \<le> (1 / 2) ^ Suc N"
```
```    45         using measure[of N] n[of N]
```
```    46         by (cases rule: ereal2_cases[of "n N" "\<mu> (A N)"])
```
```    47            (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
```
```    48       finally show "n N * \<mu> (A N) \<le> (1 / 2) ^ Suc N" .
```
```    49       show "0 \<le> n N * \<mu> (A N)" using n[of N] `A N \<in> sets M` by simp
```
```    50     qed
```
```    51     finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
```
```    52   next
```
```    53     { fix x assume "x \<in> space M"
```
```    54       then obtain i where "x \<in> A i" using space[symmetric] by auto
```
```    55       with disjoint n have "?h x = n i"
```
```    56         by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
```
```    57       then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto }
```
```    58     note pos = this
```
```    59     fix x show "0 \<le> ?h x"
```
```    60     proof cases
```
```    61       assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le)
```
```    62     next
```
```    63       assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto
```
```    64       then show "0 \<le> ?h x" by auto
```
```    65     qed
```
```    66   next
```
```    67     show "?h \<in> borel_measurable M" using range n
```
```    68       by (auto intro!: borel_measurable_psuminf borel_measurable_ereal_times ereal_0_le_mult intro: less_imp_le)
```
```    69   qed
```
```    70 qed
```
```    71
```
```    72 subsection "Absolutely continuous"
```
```    73
```
```    74 definition (in measure_space)
```
```    75   "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: ereal))"
```
```    76
```
```    77 lemma (in measure_space) absolutely_continuous_AE:
```
```    78   assumes "measure_space M'" and [simp]: "sets M' = sets M" "space M' = space M"
```
```    79     and "absolutely_continuous (measure M')" "AE x. P x"
```
```    80    shows "AE x in M'. P x"
```
```    81 proof -
```
```    82   interpret \<nu>: measure_space M' by fact
```
```    83   from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
```
```    84     unfolding almost_everywhere_def by auto
```
```    85   show "AE x in M'. P x"
```
```    86   proof (rule \<nu>.AE_I')
```
```    87     show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
```
```    88     from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
```
```    89       using N unfolding absolutely_continuous_def by auto
```
```    90   qed
```
```    91 qed
```
```    92
```
```    93 lemma (in finite_measure_space) absolutely_continuousI:
```
```    94   assumes "finite_measure_space (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure_space ?\<nu>")
```
```    95   assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
```
```    96   shows "absolutely_continuous \<nu>"
```
```    97 proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
```
```    98   fix N assume "\<mu> N = 0" "N \<subseteq> space M"
```
```    99   interpret v: finite_measure_space ?\<nu> by fact
```
```   100   have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
```
```   101   also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
```
```   102   proof (rule v.measure_setsum[symmetric])
```
```   103     show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
```
```   104     show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
```
```   105     fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
```
```   106   qed
```
```   107   also have "\<dots> = 0"
```
```   108   proof (safe intro!: setsum_0')
```
```   109     fix x assume "x \<in> N"
```
```   110     hence "\<mu> {x} \<le> \<mu> N" "0 \<le> \<mu> {x}"
```
```   111       using sets_eq_Pow `N \<subseteq> space M` positive_measure[of "{x}"]
```
```   112       by (auto intro!: measure_mono)
```
```   113     then have "\<mu> {x} = 0" using `\<mu> N = 0` by simp
```
```   114     thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
```
```   115   qed
```
```   116   finally show "\<nu> N = 0" by simp
```
```   117 qed
```
```   118
```
```   119 lemma (in measure_space) density_is_absolutely_continuous:
```
```   120   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   121   shows "absolutely_continuous \<nu>"
```
```   122   using assms unfolding absolutely_continuous_def
```
```   123   by (simp add: positive_integral_null_set)
```
```   124
```
```   125 subsection "Existence of the Radon-Nikodym derivative"
```
```   126
```
```   127 lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
```
```   128   fixes e :: real assumes "0 < e"
```
```   129   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
```
```   130   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
```
```   131                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
```
```   132                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
```
```   133 proof -
```
```   134   interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```   135   let "?d A" = "\<mu>' A - M'.\<mu>' A"
```
```   136   let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
```
```   137     then {}
```
```   138     else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
```
```   139   def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
```
```   140   have A_simps[simp]:
```
```   141     "A 0 = {}"
```
```   142     "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
```
```   143   { fix A assume "A \<in> sets M"
```
```   144     have "?A A \<in> sets M"
```
```   145       by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
```
```   146   note A'_in_sets = this
```
```   147   { fix n have "A n \<in> sets M"
```
```   148     proof (induct n)
```
```   149       case (Suc n) thus "A (Suc n) \<in> sets M"
```
```   150         using A'_in_sets[of "A n"] by (auto split: split_if_asm)
```
```   151     qed (simp add: A_def) }
```
```   152   note A_in_sets = this
```
```   153   hence "range A \<subseteq> sets M" by auto
```
```   154   { fix n B
```
```   155     assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
```
```   156     hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
```
```   157     have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
```
```   158     proof (rule someI2_ex[OF Ex])
```
```   159       fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
```
```   160       hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
```
```   161       hence "?d (A n \<union> B) = ?d (A n) + ?d B"
```
```   162         using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by simp
```
```   163       also have "\<dots> \<le> ?d (A n) - e" using dB by simp
```
```   164       finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
```
```   165     qed }
```
```   166   note dA_epsilon = this
```
```   167   { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
```
```   168     proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
```
```   169       case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
```
```   170     next
```
```   171       case False
```
```   172       hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
```
```   173       thus ?thesis by simp
```
```   174     qed }
```
```   175   note dA_mono = this
```
```   176   show ?thesis
```
```   177   proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
```
```   178     case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast
```
```   179     show ?thesis
```
```   180     proof (safe intro!: bexI[of _ "space M - A n"])
```
```   181       fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
```
```   182       from B[OF this] show "-e < ?d B" .
```
```   183     next
```
```   184       show "space M - A n \<in> sets M" by (rule compl_sets) fact
```
```   185     next
```
```   186       show "?d (space M) \<le> ?d (space M - A n)"
```
```   187       proof (induct n)
```
```   188         fix n assume "?d (space M) \<le> ?d (space M - A n)"
```
```   189         also have "\<dots> \<le> ?d (space M - A (Suc n))"
```
```   190           using A_in_sets sets_into_space dA_mono[of n]
```
```   191           by (simp del: A_simps add: finite_measure_Diff M'.finite_measure_Diff)
```
```   192         finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
```
```   193       qed simp
```
```   194     qed
```
```   195   next
```
```   196     case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
```
```   197       by (auto simp add: not_less)
```
```   198     { fix n have "?d (A n) \<le> - real n * e"
```
```   199       proof (induct n)
```
```   200         case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
```
```   201       next
```
```   202         case 0 with M'.empty_measure show ?case by (simp add: zero_ereal_def)
```
```   203       qed } note dA_less = this
```
```   204     have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
```
```   205     proof (rule incseq_SucI)
```
```   206       fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
```
```   207     qed
```
```   208     have A: "incseq A" by (auto intro!: incseq_SucI)
```
```   209     from finite_continuity_from_below[OF _ A] `range A \<subseteq> sets M`
```
```   210       M'.finite_continuity_from_below[OF _ A]
```
```   211     have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
```
```   212       by (auto intro!: tendsto_diff)
```
```   213     obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
```
```   214     moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
```
```   215     have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
```
```   216     ultimately show ?thesis by auto
```
```   217   qed
```
```   218 qed
```
```   219
```
```   220 lemma (in finite_measure) restricted_measure_subset:
```
```   221   assumes S: "S \<in> sets M" and X: "X \<subseteq> S"
```
```   222   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
```
```   223 proof cases
```
```   224   note r = restricted_finite_measure[OF S]
```
```   225   { assume "X \<in> sets M" with S X show ?thesis
```
```   226       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def by auto }
```
```   227   { assume "X \<notin> sets M"
```
```   228     moreover then have "S \<inter> X \<notin> sets M"
```
```   229       using X by (simp add: Int_absorb1)
```
```   230     ultimately show ?thesis
```
```   231       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def using S by auto }
```
```   232 qed
```
```   233
```
```   234 lemma (in finite_measure) restricted_measure:
```
```   235   assumes X: "S \<in> sets M" "X \<in> sets (restricted_space S)"
```
```   236   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
```
```   237 proof -
```
```   238   from X have "S \<in> sets M" "X \<subseteq> S" by auto
```
```   239   from restricted_measure_subset[OF this] show ?thesis .
```
```   240 qed
```
```   241
```
```   242 lemma (in finite_measure) Radon_Nikodym_aux:
```
```   243   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
```
```   244   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
```
```   245                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
```
```   246                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
```
```   247 proof -
```
```   248   interpret M': finite_measure ?M' where
```
```   249     "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
```
```   250   let "?d A" = "\<mu>' A - M'.\<mu>' A"
```
```   251   let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)"
```
```   252   let "?r S" = "restricted_space S"
```
```   253   { fix S n assume S: "S \<in> sets M"
```
```   254     note r = M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
```
```   255     then have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
```
```   256       "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
```
```   257     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
```
```   258     have "?P X S n"
```
```   259     proof (intro conjI ballI impI)
```
```   260       show "X \<in> sets M" "X \<subseteq> S" using X(1) `S \<in> sets M` by auto
```
```   261       have "S \<in> op \<inter> S ` sets M" using `S \<in> sets M` by auto
```
```   262       then show "?d S \<le> ?d X"
```
```   263         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by simp
```
```   264       fix C assume "C \<in> sets M" "C \<subseteq> X"
```
```   265       then have "C \<in> sets (restricted_space S)" "C \<subseteq> X"
```
```   266         using `S \<in> sets M` `X \<subseteq> S` by auto
```
```   267       with X(2) show "- 1 / real (Suc n) < ?d C"
```
```   268         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by auto
```
```   269     qed
```
```   270     hence "\<exists>A. ?P A S n" by auto }
```
```   271   note Ex_P = this
```
```   272   def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
```
```   273   have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
```
```   274   have A_0[simp]: "A 0 = space M" unfolding A_def by simp
```
```   275   { fix i have "A i \<in> sets M" unfolding A_def
```
```   276     proof (induct i)
```
```   277       case (Suc i)
```
```   278       from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
```
```   279         by (rule someI2_ex) simp
```
```   280     qed simp }
```
```   281   note A_in_sets = this
```
```   282   { fix n have "?P (A (Suc n)) (A n) n"
```
```   283       using Ex_P[OF A_in_sets] unfolding A_Suc
```
```   284       by (rule someI2_ex) simp }
```
```   285   note P_A = this
```
```   286   have "range A \<subseteq> sets M" using A_in_sets by auto
```
```   287   have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
```
```   288   have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
```
```   289   have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
```
```   290       using P_A by auto
```
```   291   show ?thesis
```
```   292   proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
```
```   293     show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
```
```   294     have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
```
```   295     from
```
```   296       finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
```
```   297       M'.finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
```
```   298     have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (intro tendsto_diff)
```
```   299     thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
```
```   300       by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
```
```   301   next
```
```   302     fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
```
```   303     show "0 \<le> ?d B"
```
```   304     proof (rule ccontr)
```
```   305       assume "\<not> 0 \<le> ?d B"
```
```   306       hence "0 < - ?d B" by auto
```
```   307       from ex_inverse_of_nat_Suc_less[OF this]
```
```   308       obtain n where *: "?d B < - 1 / real (Suc n)"
```
```   309         by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
```
```   310       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
```
```   311       from epsilon[OF B(1) this] *
```
```   312       show False by auto
```
```   313     qed
```
```   314   qed
```
```   315 qed
```
```   316
```
```   317 lemma (in finite_measure) Radon_Nikodym_finite_measure:
```
```   318   assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
```
```   319   assumes "absolutely_continuous \<nu>"
```
```   320   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   321 proof -
```
```   322   interpret M': finite_measure ?M'
```
```   323     where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
```
```   324     using assms(1) by auto
```
```   325   def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A)}"
```
```   326   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
```
```   327   hence "G \<noteq> {}" by auto
```
```   328   { fix f g assume f: "f \<in> G" and g: "g \<in> G"
```
```   329     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
```
```   330     proof safe
```
```   331       show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
```
```   332       let ?A = "{x \<in> space M. f x \<le> g x}"
```
```   333       have "?A \<in> sets M" using f g unfolding G_def by auto
```
```   334       fix A assume "A \<in> sets M"
```
```   335       hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
```
```   336       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
```
```   337         using sets_into_space[OF `A \<in> sets M`] by auto
```
```   338       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
```
```   339         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
```
```   340         by (auto simp: indicator_def max_def)
```
```   341       hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
```
```   342         (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
```
```   343         (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
```
```   344         using f g sets unfolding G_def
```
```   345         by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator)
```
```   346       also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
```
```   347         using f g sets unfolding G_def by (auto intro!: add_mono)
```
```   348       also have "\<dots> = \<nu> A"
```
```   349         using M'.measure_additive[OF sets] union by auto
```
```   350       finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
```
```   351     next
```
```   352       fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
```
```   353     qed }
```
```   354   note max_in_G = this
```
```   355   { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
```
```   356     have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
```
```   357     proof safe
```
```   358       show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
```
```   359         using f by (auto simp: G_def)
```
```   360       { fix x show "0 \<le> (SUP i. f i x)"
```
```   361           using f by (auto simp: G_def intro: SUP_upper2) }
```
```   362     next
```
```   363       fix A assume "A \<in> sets M"
```
```   364       have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
```
```   365         (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
```
```   366         by (intro positive_integral_cong) (simp split: split_indicator)
```
```   367       also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
```
```   368         using `incseq f` f `A \<in> sets M`
```
```   369         by (intro positive_integral_monotone_convergence_SUP)
```
```   370            (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
```
```   371       finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> \<nu> A"
```
```   372         using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
```
```   373     qed }
```
```   374   note SUP_in_G = this
```
```   375   let ?y = "SUP g : G. integral\<^isup>P M g"
```
```   376   have "?y \<le> \<nu> (space M)" unfolding G_def
```
```   377   proof (safe intro!: SUP_least)
```
```   378     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
```
```   379     from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
```
```   380       by (simp cong: positive_integral_cong)
```
```   381   qed
```
```   382   from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
```
```   383   then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
```
```   384   proof safe
```
```   385     fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
```
```   386     hence "ys n \<in> integral\<^isup>P M ` G" by auto
```
```   387     thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
```
```   388   qed
```
```   389   from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto
```
```   390   hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
```
```   391   let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})"
```
```   392   def f \<equiv> "\<lambda>x. SUP i. ?g i x"
```
```   393   let "?F A x" = "f x * indicator A x"
```
```   394   have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
```
```   395   { fix i have "?g i \<in> G"
```
```   396     proof (induct i)
```
```   397       case 0 thus ?case by simp fact
```
```   398     next
```
```   399       case (Suc i)
```
```   400       with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
```
```   401         by (auto simp add: atMost_Suc intro!: max_in_G)
```
```   402     qed }
```
```   403   note g_in_G = this
```
```   404   have "incseq ?g" using gs_not_empty
```
```   405     by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
```
```   406   from SUP_in_G[OF this g_in_G] have "f \<in> G" unfolding f_def .
```
```   407   then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
```
```   408   have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def
```
```   409     using g_in_G `incseq ?g`
```
```   410     by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def)
```
```   411   also have "\<dots> = ?y"
```
```   412   proof (rule antisym)
```
```   413     show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
```
```   414       using g_in_G
```
```   415       by (auto intro!: exI Sup_mono simp: SUP_def)
```
```   416     show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
```
```   417       by (auto intro!: SUP_mono positive_integral_mono Max_ge)
```
```   418   qed
```
```   419   finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
```
```   420   have "\<And>x. 0 \<le> f x"
```
```   421     unfolding f_def using `\<And>i. gs i \<in> G`
```
```   422     by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
```
```   423   let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
```
```   424   let ?M = "M\<lparr> measure := ?t\<rparr>"
```
```   425   interpret M: sigma_algebra ?M
```
```   426     by (intro sigma_algebra_cong) auto
```
```   427   have f_le_\<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> \<nu> A"
```
```   428     using `f \<in> G` unfolding G_def by auto
```
```   429   have fmM: "finite_measure ?M"
```
```   430   proof
```
```   431     show "measure_space ?M"
```
```   432     proof (default, simp_all add: countably_additive_def positive_def, safe del: notI)
```
```   433       fix A :: "nat \<Rightarrow> 'a set"  assume A: "range A \<subseteq> sets M" "disjoint_family A"
```
```   434       have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. (\<Sum>n. ?F (A n) x) \<partial>M)"
```
```   435         using `range A \<subseteq> sets M` `\<And>x. 0 \<le> f x`
```
```   436         by (intro positive_integral_suminf[symmetric]) auto
```
```   437       also have "\<dots> = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)"
```
```   438         using `\<And>x. 0 \<le> f x`
```
```   439         by (intro positive_integral_cong) (simp add: suminf_cmult_ereal suminf_indicator[OF `disjoint_family A`])
```
```   440       finally have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)" .
```
```   441       moreover have "(\<Sum>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
```
```   442         using M'.measure_countably_additive A by (simp add: comp_def)
```
```   443       moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<infinity>" using M'.finite_measure A by (simp add: countable_UN)
```
```   444       moreover {
```
```   445         have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
```
```   446           using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
```
```   447         also have "\<nu> (\<Union>i. A i) < \<infinity>" using v_fin by simp
```
```   448         finally have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<noteq> \<infinity>" by simp }
```
```   449       moreover have "\<And>i. (\<integral>\<^isup>+x. ?F (A i) x \<partial>M) \<le> \<nu> (A i)"
```
```   450         using A by (intro f_le_\<nu>) auto
```
```   451       ultimately
```
```   452       show "(\<Sum>n. ?t (A n)) = ?t (\<Union>i. A i)"
```
```   453         by (subst suminf_ereal_minus) (simp_all add: positive_integral_positive)
```
```   454     next
```
```   455       fix A assume A: "A \<in> sets M" show "0 \<le> \<nu> A - \<integral>\<^isup>+ x. ?F A x \<partial>M"
```
```   456         using f_le_\<nu>[OF A] `f \<in> G` M'.finite_measure[OF A] by (auto simp: G_def ereal_le_minus_iff)
```
```   457     qed
```
```   458   next
```
```   459     show "measure ?M (space ?M) \<noteq> \<infinity>"
```
```   460       using positive_integral_positive[of "?F (space M)"]
```
```   461       by (cases rule: ereal2_cases[of "\<nu> (space M)" "\<integral>\<^isup>+ x. ?F (space M) x \<partial>M"]) auto
```
```   462   qed
```
```   463   then interpret M: finite_measure ?M
```
```   464     where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
```
```   465     by (simp_all add: fmM)
```
```   466   have ac: "absolutely_continuous ?t" unfolding absolutely_continuous_def
```
```   467   proof
```
```   468     fix N assume N: "N \<in> null_sets"
```
```   469     with `absolutely_continuous \<nu>` have "\<nu> N = 0" unfolding absolutely_continuous_def by auto
```
```   470     moreover with N have "(\<integral>\<^isup>+ x. ?F N x \<partial>M) \<le> \<nu> N" using `f \<in> G` by (auto simp: G_def)
```
```   471     ultimately show "\<nu> N - (\<integral>\<^isup>+ x. ?F N x \<partial>M) = 0"
```
```   472       using positive_integral_positive by (auto intro!: antisym)
```
```   473   qed
```
```   474   have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
```
```   475   proof (rule ccontr)
```
```   476     assume "\<not> ?thesis"
```
```   477     then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
```
```   478       by (auto simp: not_le)
```
```   479     note pos
```
```   480     also have "?t A \<le> ?t (space M)"
```
```   481       using M.measure_mono[of A "space M"] A sets_into_space by simp
```
```   482     finally have pos_t: "0 < ?t (space M)" by simp
```
```   483     moreover
```
```   484     then have "\<mu> (space M) \<noteq> 0"
```
```   485       using ac unfolding absolutely_continuous_def by auto
```
```   486     then have pos_M: "0 < \<mu> (space M)"
```
```   487       using positive_measure[OF top] by (simp add: le_less)
```
```   488     moreover
```
```   489     have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
```
```   490       using `f \<in> G` unfolding G_def by auto
```
```   491     hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
```
```   492       using M'.finite_measure_of_space by auto
```
```   493     moreover
```
```   494     def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
```
```   495     ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
```
```   496       using M'.finite_measure_of_space positive_integral_positive[of "?F (space M)"]
```
```   497       by (cases rule: ereal3_cases[of "integral\<^isup>P M (?F (space M))" "\<nu> (space M)" "\<mu> (space M)"])
```
```   498          (simp_all add: field_simps)
```
```   499     then have b: "b \<noteq> 0" "0 \<le> b" "0 < b"  "b \<noteq> \<infinity>" by auto
```
```   500     let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
```
```   501     interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
```
```   502     have Mb: "finite_measure ?Mb"
```
```   503     proof
```
```   504       show "measure_space ?Mb"
```
```   505       proof
```
```   506         show "positive ?Mb (measure ?Mb)"
```
```   507           using `0 \<le> b` by (auto simp: positive_def)
```
```   508         show "countably_additive ?Mb (measure ?Mb)"
```
```   509           using `0 \<le> b` measure_countably_additive
```
```   510           by (auto simp: countably_additive_def suminf_cmult_ereal subset_eq)
```
```   511       qed
```
```   512       show "measure ?Mb (space ?Mb) \<noteq> \<infinity>"
```
```   513         using b by auto
```
```   514     qed
```
```   515     from M.Radon_Nikodym_aux[OF this]
```
```   516     obtain A0 where "A0 \<in> sets M" and
```
```   517       space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
```
```   518       *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)"
```
```   519       unfolding M.\<mu>'_def finite_measure.\<mu>'_def[OF Mb] by auto
```
```   520     { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
```
```   521       with *[OF this] have "b * \<mu> B \<le> ?t B"
```
```   522         using M'.finite_measure b finite_measure M.positive_measure[OF B(1)]
```
```   523         by (cases rule: ereal2_cases[of "?t B" "b * \<mu> B"]) auto }
```
```   524     note bM_le_t = this
```
```   525     let "?f0 x" = "f x + b * indicator A0 x"
```
```   526     { fix A assume A: "A \<in> sets M"
```
```   527       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
```
```   528       have "(\<integral>\<^isup>+x. ?f0 x  * indicator A x \<partial>M) =
```
```   529         (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
```
```   530         by (auto intro!: positive_integral_cong split: split_indicator)
```
```   531       hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
```
```   532           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
```
```   533         using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
```
```   534         by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
```
```   535     note f0_eq = this
```
```   536     { fix A assume A: "A \<in> sets M"
```
```   537       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
```
```   538       have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
```
```   539         using `f \<in> G` A unfolding G_def by auto
```
```   540       note f0_eq[OF A]
```
```   541       also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
```
```   542           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
```
```   543         using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
```
```   544         by (auto intro!: add_left_mono)
```
```   545       also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
```
```   546         using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
```
```   547         by (auto intro!: add_left_mono)
```
```   548       also have "\<dots> \<le> \<nu> A"
```
```   549         using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] positive_integral_positive[of "?F A"]
```
```   550         by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "\<nu> A") auto
```
```   551       finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
```
```   552     hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
```
```   553       by (auto intro!: borel_measurable_indicator borel_measurable_ereal_add
```
```   554                        borel_measurable_ereal_times ereal_add_nonneg_nonneg)
```
```   555     have real: "?t (space M) \<noteq> \<infinity>" "?t A0 \<noteq> \<infinity>"
```
```   556       "b * \<mu> (space M) \<noteq> \<infinity>" "b * \<mu> A0 \<noteq> \<infinity>"
```
```   557       using `A0 \<in> sets M` b
```
```   558         finite_measure[of A0] M.finite_measure[of A0]
```
```   559         finite_measure_of_space M.finite_measure_of_space
```
```   560       by auto
```
```   561     have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
```
```   562       using M'.finite_measure_of_space pos_t unfolding ereal_less_minus_iff
```
```   563       by (auto cong: positive_integral_cong)
```
```   564     have  "0 < ?t (space M) - b * \<mu> (space M)" unfolding b_def
```
```   565       using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
```
```   566       using positive_integral_positive[of "?F (space M)"]
```
```   567       by (cases rule: ereal3_cases[of "\<mu> (space M)" "\<nu> (space M)" "integral\<^isup>P M (?F (space M))"])
```
```   568          (auto simp: field_simps mult_less_cancel_left)
```
```   569     also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
```
```   570       using space_less_A0 b
```
```   571       using
```
```   572         `A0 \<in> sets M`[THEN M.real_measure]
```
```   573         top[THEN M.real_measure]
```
```   574       apply safe
```
```   575       apply simp
```
```   576       using
```
```   577         `A0 \<in> sets M`[THEN real_measure]
```
```   578         `A0 \<in> sets M`[THEN M'.real_measure]
```
```   579         top[THEN real_measure]
```
```   580         top[THEN M'.real_measure]
```
```   581       by (cases b) auto
```
```   582     finally have 1: "b * \<mu> A0 < ?t A0"
```
```   583       using
```
```   584         `A0 \<in> sets M`[THEN M.real_measure]
```
```   585       apply safe
```
```   586       apply simp
```
```   587       using
```
```   588         `A0 \<in> sets M`[THEN real_measure]
```
```   589         `A0 \<in> sets M`[THEN M'.real_measure]
```
```   590       by (cases b) auto
```
```   591     have "0 < ?t A0"
```
```   592       using b `A0 \<in> sets M` by (auto intro!: le_less_trans[OF _ 1])
```
```   593     then have "\<mu> A0 \<noteq> 0" using ac unfolding absolutely_continuous_def
```
```   594       using `A0 \<in> sets M` by auto
```
```   595     then have "0 < \<mu> A0" using positive_measure[OF `A0 \<in> sets M`] by auto
```
```   596     hence "0 < b * \<mu> A0" using b by (auto simp: ereal_zero_less_0_iff)
```
```   597     with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
```
```   598       using `f \<in> G`
```
```   599       by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive)
```
```   600     also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
```
```   601       by (simp cong: positive_integral_cong)
```
```   602     finally have "?y < integral\<^isup>P M ?f0" by simp
```
```   603     moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
```
```   604     ultimately show False by auto
```
```   605   qed
```
```   606   show ?thesis
```
```   607   proof (safe intro!: bexI[of _ f])
```
```   608     fix A assume A: "A\<in>sets M"
```
```   609     show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   610     proof (rule antisym)
```
```   611       show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
```
```   612         using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
```
```   613       show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   614         using upper_bound[THEN bspec, OF `A \<in> sets M`]
```
```   615         using M'.real_measure[OF A]
```
```   616         by (cases "integral\<^isup>P M (?F A)") auto
```
```   617     qed
```
```   618   qed simp
```
```   619 qed
```
```   620
```
```   621 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
```
```   622   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
```
```   623   assumes ac: "absolutely_continuous \<nu>"
```
```   624   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>
```
```   625     (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<infinity>)) \<and>
```
```   626     (\<forall>i. \<nu> (B i) \<noteq> \<infinity>)"
```
```   627 proof -
```
```   628   interpret v: measure_space ?N
```
```   629     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
```
```   630     by fact auto
```
```   631   let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<infinity>}"
```
```   632   let ?a = "SUP Q:?Q. \<mu> Q"
```
```   633   have "{} \<in> ?Q" using v.empty_measure by auto
```
```   634   then have Q_not_empty: "?Q \<noteq> {}" by blast
```
```   635   have "?a \<le> \<mu> (space M)" using sets_into_space
```
```   636     by (auto intro!: SUP_least measure_mono top)
```
```   637   then have "?a \<noteq> \<infinity>" using finite_measure_of_space
```
```   638     by auto
```
```   639   from SUPR_countable_SUPR[OF Q_not_empty, of \<mu>]
```
```   640   obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
```
```   641     by auto
```
```   642   then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
```
```   643   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
```
```   644     by auto
```
```   645   then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
```
```   646   let "?O n" = "\<Union>i\<le>n. Q' i"
```
```   647   have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
```
```   648   proof (rule continuity_from_below[of ?O])
```
```   649     show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
```
```   650     show "incseq ?O" by (fastforce intro!: incseq_SucI)
```
```   651   qed
```
```   652   have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
```
```   653   have O_sets: "\<And>i. ?O i \<in> sets M"
```
```   654      using Q' by (auto intro!: finite_UN Un)
```
```   655   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
```
```   656   proof (safe del: notI)
```
```   657     fix i have "Q' ` {..i} \<subseteq> sets M"
```
```   658       using Q' by (auto intro: finite_UN)
```
```   659     with v.measure_finitely_subadditive[of "{.. i}" Q']
```
```   660     have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
```
```   661     also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
```
```   662     finally show "\<nu> (?O i) \<noteq> \<infinity>" by simp
```
```   663   qed auto
```
```   664   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
```
```   665   have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
```
```   666   proof (rule antisym)
```
```   667     show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
```
```   668       using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
```
```   669     show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUP_def
```
```   670     proof (safe intro!: Sup_mono, unfold bex_simps)
```
```   671       fix i
```
```   672       have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
```
```   673       then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<infinity>) \<and>
```
```   674         \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
```
```   675         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
```
```   676     qed
```
```   677   qed
```
```   678   let "?O_0" = "(\<Union>i. ?O i)"
```
```   679   have "?O_0 \<in> sets M" using Q' by auto
```
```   680   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
```
```   681   { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
```
```   682   note Q_sets = this
```
```   683   show ?thesis
```
```   684   proof (intro bexI exI conjI ballI impI allI)
```
```   685     show "disjoint_family Q"
```
```   686       by (fastforce simp: disjoint_family_on_def Q_def
```
```   687         split: nat.split_asm)
```
```   688     show "range Q \<subseteq> sets M"
```
```   689       using Q_sets by auto
```
```   690     { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
```
```   691       show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
```
```   692       proof (rule disjCI, simp)
```
```   693         assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<infinity>"
```
```   694         show "\<mu> A = 0 \<and> \<nu> A = 0"
```
```   695         proof cases
```
```   696           assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
```
```   697             unfolding absolutely_continuous_def by auto
```
```   698           ultimately show ?thesis by simp
```
```   699         next
```
```   700           assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<infinity>" using positive_measure[OF A(1)] by auto
```
```   701           with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
```
```   702             using Q' by (auto intro!: measure_additive countable_UN)
```
```   703           also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
```
```   704           proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
```
```   705             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
```
```   706               using `\<nu> A \<noteq> \<infinity>` O_sets A by auto
```
```   707           qed (fastforce intro!: incseq_SucI)
```
```   708           also have "\<dots> \<le> ?a"
```
```   709           proof (safe intro!: SUP_least)
```
```   710             fix i have "?O i \<union> A \<in> ?Q"
```
```   711             proof (safe del: notI)
```
```   712               show "?O i \<union> A \<in> sets M" using O_sets A by auto
```
```   713               from O_in_G[of i]
```
```   714               moreover have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
```
```   715                 using v.measure_subadditive[of "?O i" A] A O_sets by auto
```
```   716               ultimately show "\<nu> (?O i \<union> A) \<noteq> \<infinity>"
```
```   717                 using `\<nu> A \<noteq> \<infinity>` by auto
```
```   718             qed
```
```   719             then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
```
```   720           qed
```
```   721           finally have "\<mu> A = 0"
```
```   722             unfolding a_eq using real_measure[OF `?O_0 \<in> sets M`] real_measure[OF A(1)] by auto
```
```   723           with `\<mu> A \<noteq> 0` show ?thesis by auto
```
```   724         qed
```
```   725       qed }
```
```   726     { fix i show "\<nu> (Q i) \<noteq> \<infinity>"
```
```   727       proof (cases i)
```
```   728         case 0 then show ?thesis
```
```   729           unfolding Q_def using Q'[of 0] by simp
```
```   730       next
```
```   731         case (Suc n)
```
```   732         then show ?thesis unfolding Q_def
```
```   733           using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
```
```   734           using v.measure_mono[OF O_mono, of n] v.positive_measure[of "?O n"] v.positive_measure[of "?O (Suc n)"]
```
```   735           using v.measure_Diff[of "?O n" "?O (Suc n)", OF _ _ _ O_mono]
```
```   736           by (cases rule: ereal2_cases[of "\<nu> (\<Union> x\<le>Suc n. Q' x)" "\<nu> (\<Union> i\<le>n. Q' i)"]) auto
```
```   737       qed }
```
```   738     show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
```
```   739     { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
```
```   740       proof (induct j)
```
```   741         case 0 then show ?case by (simp add: Q_def)
```
```   742       next
```
```   743         case (Suc j)
```
```   744         have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce
```
```   745         have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
```
```   746         then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
```
```   747           by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
```
```   748         then show ?case using Suc by (auto simp add: eq atMost_Suc)
```
```   749       qed }
```
```   750     then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
```
```   751     then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce
```
```   752   qed
```
```   753 qed
```
```   754
```
```   755 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
```
```   756   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
```
```   757   assumes "absolutely_continuous \<nu>"
```
```   758   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
```
```   759 proof -
```
```   760   interpret v: measure_space ?N
```
```   761     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
```
```   762     by fact auto
```
```   763   from split_space_into_finite_sets_and_rest[OF assms]
```
```   764   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
```
```   765     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
```
```   766     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
```
```   767     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
```
```   768     and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<infinity>" by force
```
```   769   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
```
```   770   have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
```
```   771     \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
```
```   772   proof
```
```   773     fix i
```
```   774     have indicator_eq: "\<And>f x A. (f x :: ereal) * indicator (Q i \<inter> A) x * indicator (Q i) x
```
```   775       = (f x * indicator (Q i) x) * indicator A x"
```
```   776       unfolding indicator_def by auto
```
```   777     have fm: "finite_measure (restricted_space (Q i))"
```
```   778       (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
```
```   779     then interpret R: finite_measure ?R .
```
```   780     have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
```
```   781     proof
```
```   782       show "measure_space ?Q"
```
```   783         using v.restricted_measure_space Q_sets[of i] by auto
```
```   784       show "measure ?Q (space ?Q) \<noteq> \<infinity>" using Q_fin by simp
```
```   785     qed
```
```   786     have "R.absolutely_continuous \<nu>"
```
```   787       using `absolutely_continuous \<nu>` `Q i \<in> sets M`
```
```   788       by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
```
```   789     from R.Radon_Nikodym_finite_measure[OF fmv this]
```
```   790     obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
```
```   791       and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x \<partial>M)"
```
```   792       unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
```
```   793         positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
```
```   794     then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
```
```   795       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
```
```   796       by (auto intro!: exI[of _ "\<lambda>x. max 0 (f x * indicator (Q i) x)"] positive_integral_cong_pos
```
```   797         split: split_indicator split_if_asm simp: max_def)
```
```   798   qed
```
```   799   from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```   800     and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
```
```   801       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
```
```   802     by auto
```
```   803   let "?f x" = "(\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
```
```   804   show ?thesis
```
```   805   proof (safe intro!: bexI[of _ ?f])
```
```   806     show "?f \<in> borel_measurable M" using Q0 borel Q_sets
```
```   807       by (auto intro!: measurable_If)
```
```   808     show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
```
```   809     fix A assume "A \<in> sets M"
```
```   810     have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
```
```   811     have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
```
```   812       "\<And>i. AE x. 0 \<le> f i x * indicator (Q i \<inter> A) x"
```
```   813       using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
```
```   814     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)"
```
```   815       using borel by (intro positive_integral_cong) (auto simp: indicator_def)
```
```   816     also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * \<mu> (Q0 \<inter> A)"
```
```   817       using borel Qi Q0(1) `A \<in> sets M`
```
```   818       by (subst positive_integral_add) (auto simp del: ereal_infty_mult
```
```   819           simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
```
```   820     also have "\<dots> = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)"
```
```   821       by (subst f[OF `A \<in> sets M`], subst positive_integral_suminf) auto
```
```   822     finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)" .
```
```   823     moreover have "(\<Sum>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
```
```   824       using Q Q_sets `A \<in> sets M`
```
```   825       by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
```
```   826          (auto simp: disjoint_family_on_def)
```
```   827     moreover have "\<infinity> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
```
```   828     proof -
```
```   829       have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
```
```   830       from in_Q0[OF this] show ?thesis by auto
```
```   831     qed
```
```   832     moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
```
```   833       using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
```
```   834     moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
```
```   835       using `A \<in> sets M` sets_into_space Q0 by auto
```
```   836     ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
```
```   837       using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
```
```   838       by simp
```
```   839   qed
```
```   840 qed
```
```   841
```
```   842 lemma (in sigma_finite_measure) Radon_Nikodym:
```
```   843   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
```
```   844   assumes ac: "absolutely_continuous \<nu>"
```
```   845   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
```
```   846 proof -
```
```   847   from Ex_finite_integrable_function
```
```   848   obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
```
```   849     borel: "h \<in> borel_measurable M" and
```
```   850     nn: "\<And>x. 0 \<le> h x" and
```
```   851     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
```
```   852     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
```
```   853   let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
```
```   854   let ?MT = "M\<lparr> measure := ?T \<rparr>"
```
```   855   interpret T: finite_measure ?MT
```
```   856     where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
```
```   857     using borel finite nn
```
```   858     by (auto intro!: measure_space_density finite_measureI cong: positive_integral_cong)
```
```   859   have "T.absolutely_continuous \<nu>"
```
```   860   proof (unfold T.absolutely_continuous_def, safe)
```
```   861     fix N assume "N \<in> sets M" "(\<integral>\<^isup>+x. h x * indicator N x \<partial>M) = 0"
```
```   862     with borel ac pos have "AE x. x \<notin> N"
```
```   863       by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_le)
```
```   864     then have "N \<in> null_sets" using `N \<in> sets M` sets_into_space
```
```   865       by (subst (asm) AE_iff_measurable[OF `N \<in> sets M`]) auto
```
```   866     then show "\<nu> N = 0" using ac by (auto simp: absolutely_continuous_def)
```
```   867   qed
```
```   868   from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
```
```   869   obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and
```
```   870     fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
```
```   871     by (auto simp: measurable_def)
```
```   872   show ?thesis
```
```   873   proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
```
```   874     show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
```
```   875       using borel f_borel by (auto intro: borel_measurable_ereal_times)
```
```   876     show "\<And>x. 0 \<le> h x * f x" using nn f_borel by auto
```
```   877     fix A assume "A \<in> sets M"
```
```   878     then show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
```
```   879       unfolding fT[OF `A \<in> sets M`] mult_assoc using nn borel f_borel
```
```   880       by (intro positive_integral_translated_density) auto
```
```   881   qed
```
```   882 qed
```
```   883
```
```   884 section "Uniqueness of densities"
```
```   885
```
```   886 lemma (in measure_space) finite_density_unique:
```
```   887   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   888   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
```
```   889   and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
```
```   890   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))
```
```   891     \<longleftrightarrow> (AE x. f x = g x)"
```
```   892     (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
```
```   893 proof (intro iffI ballI)
```
```   894   fix A assume eq: "AE x. f x = g x"
```
```   895   then show "?P f A = ?P g A"
```
```   896     by (auto intro: positive_integral_cong_AE)
```
```   897 next
```
```   898   assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   899   from this[THEN bspec, OF top] fin
```
```   900   have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
```
```   901   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   902       and pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
```
```   903       and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   904     let ?N = "{x\<in>space M. g x < f x}"
```
```   905     have N: "?N \<in> sets M" using borel by simp
```
```   906     have "?P g ?N \<le> integral\<^isup>P M g" using pos
```
```   907       by (intro positive_integral_mono_AE) (auto split: split_indicator)
```
```   908     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
```
```   909     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
```
```   910       by (auto intro!: positive_integral_cong simp: indicator_def)
```
```   911     also have "\<dots> = ?P f ?N - ?P g ?N"
```
```   912     proof (rule positive_integral_diff)
```
```   913       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
```
```   914         using borel N by auto
```
```   915       show "AE x. g x * indicator ?N x \<le> f x * indicator ?N x"
```
```   916            "AE x. 0 \<le> g x * indicator ?N x"
```
```   917         using pos by (auto split: split_indicator)
```
```   918     qed fact
```
```   919     also have "\<dots> = 0"
```
```   920       unfolding eq[THEN bspec, OF N] using positive_integral_positive Pg_fin by auto
```
```   921     finally have "AE x. f x \<le> g x"
```
```   922       using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
```
```   923       by (subst (asm) positive_integral_0_iff_AE)
```
```   924          (auto split: split_indicator simp: not_less ereal_minus_le_iff) }
```
```   925   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
```
```   926   show "AE x. f x = g x" by auto
```
```   927 qed
```
```   928
```
```   929 lemma (in finite_measure) density_unique_finite_measure:
```
```   930   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
```
```   931   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> f' x"
```
```   932   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)"
```
```   933     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
```
```   934   shows "AE x. f x = f' x"
```
```   935 proof -
```
```   936   let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A"
```
```   937   let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x"
```
```   938   interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
```
```   939     using borel(1) pos(1) by (rule measure_space_density) simp
```
```   940   have ac: "absolutely_continuous ?\<nu>"
```
```   941     using f by (rule density_is_absolutely_continuous)
```
```   942   from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
```
```   943   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
```
```   944     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
```
```   945     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
```
```   946     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<infinity>"
```
```   947     and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<infinity>" by force
```
```   948   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
```
```   949   let ?N = "{x\<in>space M. f x \<noteq> f' x}"
```
```   950   have "?N \<in> sets M" using borel by auto
```
```   951   have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
```
```   952     unfolding indicator_def by auto
```
```   953   have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
```
```   954     by (intro finite_density_unique[THEN iffD1] allI)
```
```   955        (auto intro!: borel_measurable_ereal_times f Int simp: *)
```
```   956   moreover have "AE x. ?f Q0 x = ?f' Q0 x"
```
```   957   proof (rule AE_I')
```
```   958     { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
```
```   959         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```   960       let "?A i" = "Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
```
```   961       have "(\<Union>i. ?A i) \<in> null_sets"
```
```   962       proof (rule null_sets_UN)
```
```   963         fix i ::nat have "?A i \<in> sets M"
```
```   964           using borel Q0(1) by auto
```
```   965         have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
```
```   966           unfolding eq[OF `?A i \<in> sets M`]
```
```   967           by (auto intro!: positive_integral_mono simp: indicator_def)
```
```   968         also have "\<dots> = i * \<mu> (?A i)"
```
```   969           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
```
```   970         also have "\<dots> < \<infinity>"
```
```   971           using `?A i \<in> sets M`[THEN finite_measure] by auto
```
```   972         finally have "?\<nu> (?A i) \<noteq> \<infinity>" by simp
```
```   973         then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
```
```   974       qed
```
```   975       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
```
```   976         by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
```
```   977       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" by simp }
```
```   978     from this[OF borel(1) refl] this[OF borel(2) f]
```
```   979     have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets" by simp_all
```
```   980     then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets" by (rule nullsets.Un)
```
```   981     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
```
```   982       (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)
```
```   983   qed
```
```   984   moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
```
```   985     ?f (space M) x = ?f' (space M) x"
```
```   986     by (auto simp: indicator_def Q0)
```
```   987   ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
```
```   988     by (auto simp: AE_all_countable[symmetric])
```
```   989   then show "AE x. f x = f' x" by auto
```
```   990 qed
```
```   991
```
```   992 lemma (in sigma_finite_measure) density_unique:
```
```   993   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```   994   assumes f': "f' \<in> borel_measurable M" "AE x. 0 \<le> f' x"
```
```   995   assumes eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
```
```   996     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
```
```   997   shows "AE x. f x = f' x"
```
```   998 proof -
```
```   999   obtain h where h_borel: "h \<in> borel_measurable M"
```
```  1000     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"
```
```  1001     using Ex_finite_integrable_function by auto
```
```  1002   then have h_nn: "AE x. 0 \<le> h x" by auto
```
```  1003   let ?H = "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
```
```  1004   have H: "measure_space ?H"
```
```  1005     using h_borel h_nn by (rule measure_space_density) simp
```
```  1006   then interpret h: measure_space ?H .
```
```  1007   interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
```
```  1008     by (intro H finite_measureI) (simp cong: positive_integral_cong add: fin)
```
```  1009   let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
```
```  1010   interpret f: measure_space ?fM
```
```  1011     using f by (rule measure_space_density) simp
```
```  1012   let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
```
```  1013   interpret f': measure_space ?f'M
```
```  1014     using f' by (rule measure_space_density) simp
```
```  1015   { fix A assume "A \<in> sets M"
```
```  1016     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
```
```  1017       using pos(1) sets_into_space by (force simp: indicator_def)
```
```  1018     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
```
```  1019       using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
```
```  1020   note h_null_sets = this
```
```  1021   { fix A assume "A \<in> sets M"
```
```  1022     have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
```
```  1023       using `A \<in> sets M` h_borel h_nn f f'
```
```  1024       by (intro positive_integral_translated_density[symmetric]) auto
```
```  1025     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
```
```  1026       by (rule f'.positive_integral_cong_measure) (simp_all add: eq)
```
```  1027     also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
```
```  1028       using `A \<in> sets M` h_borel h_nn f f'
```
```  1029       by (intro positive_integral_translated_density) auto
```
```  1030     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)"
```
```  1031       by (simp add: ac_simps)
```
```  1032     then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
```
```  1033       using `A \<in> sets M` h_borel h_nn f f'
```
```  1034       by (subst (asm) (1 2) positive_integral_translated_density[symmetric]) auto }
```
```  1035   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
```
```  1036     by (intro h.density_unique_finite_measure absolutely_continuous_AE[OF H] density_is_absolutely_continuous)
```
```  1037        simp_all
```
```  1038   then show "AE x. f x = f' x"
```
```  1039     unfolding h.almost_everywhere_def almost_everywhere_def
```
```  1040     by (auto simp add: h_null_sets)
```
```  1041 qed
```
```  1042
```
```  1043 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
```
```  1044   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
```
```  1045     and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```  1046     and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```  1047   shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<infinity>)"
```
```  1048 proof
```
```  1049   assume "sigma_finite_measure ?N"
```
```  1050   then interpret \<nu>: sigma_finite_measure ?N
```
```  1051     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
```
```  1052     and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
```
```  1053   from \<nu>.Ex_finite_integrable_function obtain h where
```
```  1054     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
```
```  1055     h_nn: "\<And>x. 0 \<le> h x" and
```
```  1056     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
```
```  1057   have "AE x. f x * h x \<noteq> \<infinity>"
```
```  1058   proof (rule AE_I')
```
```  1059     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
```
```  1060       by (subst \<nu>.positive_integral_cong_measure[symmetric,
```
```  1061           of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
```
```  1062          (auto intro!: positive_integral_translated_density simp: eq)
```
```  1063     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
```
```  1064       using h(2) by simp
```
```  1065     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets"
```
```  1066       using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
```
```  1067   qed auto
```
```  1068   then show "AE x. f x \<noteq> \<infinity>"
```
```  1069     using fin by (auto elim!: AE_Ball_mp)
```
```  1070 next
```
```  1071   assume AE: "AE x. f x \<noteq> \<infinity>"
```
```  1072   from sigma_finite guess Q .. note Q = this
```
```  1073   interpret \<nu>: measure_space ?N
```
```  1074     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
```
```  1075     and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
```
```  1076   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
```
```  1077   { fix i j have "A i \<inter> Q j \<in> sets M"
```
```  1078     unfolding A_def using f Q
```
```  1079     apply (rule_tac Int)
```
```  1080     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
```
```  1081   note A_in_sets = this
```
```  1082   let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
```
```  1083   show "sigma_finite_measure ?N"
```
```  1084   proof (default, intro exI conjI subsetI allI)
```
```  1085     fix x assume "x \<in> range ?A"
```
```  1086     then obtain n where n: "x = ?A n" by auto
```
```  1087     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
```
```  1088   next
```
```  1089     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
```
```  1090     proof safe
```
```  1091       fix x i j assume "x \<in> A i" "x \<in> Q j"
```
```  1092       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
```
```  1093         by (intro UN_I[of "prod_encode (i,j)"]) auto
```
```  1094     qed auto
```
```  1095     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
```
```  1096     also have "(\<Union>i. A i) = space M"
```
```  1097     proof safe
```
```  1098       fix x assume x: "x \<in> space M"
```
```  1099       show "x \<in> (\<Union>i. A i)"
```
```  1100       proof (cases "f x")
```
```  1101         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
```
```  1102       next
```
```  1103         case (real r)
```
```  1104         with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
```
```  1105         then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
```
```  1106       next
```
```  1107         case MInf with x show ?thesis
```
```  1108           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
```
```  1109       qed
```
```  1110     qed (auto simp: A_def)
```
```  1111     finally show "(\<Union>i. ?A i) = space ?N" by simp
```
```  1112   next
```
```  1113     fix n obtain i j where
```
```  1114       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
```
```  1115     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
```
```  1116     proof (cases i)
```
```  1117       case 0
```
```  1118       have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
```
```  1119         using AE by (auto simp: A_def `i = 0`)
```
```  1120       from positive_integral_cong_AE[OF this] show ?thesis by simp
```
```  1121     next
```
```  1122       case (Suc n)
```
```  1123       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
```
```  1124         (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
```
```  1125         by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
```
```  1126       also have "\<dots> = Suc n * \<mu> (Q j)"
```
```  1127         using Q by (auto intro!: positive_integral_cmult_indicator)
```
```  1128       also have "\<dots> < \<infinity>"
```
```  1129         using Q by (auto simp: real_eq_of_nat[symmetric])
```
```  1130       finally show ?thesis by simp
```
```  1131     qed
```
```  1132     then show "measure ?N (?A n) \<noteq> \<infinity>"
```
```  1133       using A_in_sets Q eq by auto
```
```  1134   qed
```
```  1135 qed
```
```  1136
```
```  1137 section "Radon-Nikodym derivative"
```
```  1138
```
```  1139 definition
```
```  1140   "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and>
```
```  1141     (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
```
```  1142
```
```  1143 lemma (in sigma_finite_measure) RN_deriv_cong:
```
```  1144   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
```
```  1145     and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
```
```  1146   shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
```
```  1147 proof -
```
```  1148   interpret \<mu>': sigma_finite_measure M'
```
```  1149     using cong by (rule sigma_finite_measure_cong)
```
```  1150   show ?thesis
```
```  1151     unfolding RN_deriv_def
```
```  1152     by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
```
```  1153 qed
```
```  1154
```
```  1155 lemma (in sigma_finite_measure) RN_deriv:
```
```  1156   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```  1157   assumes "absolutely_continuous \<nu>"
```
```  1158   shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
```
```  1159   and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
```
```  1160     (is "\<And>A. _ \<Longrightarrow> ?int A")
```
```  1161   and "0 \<le> RN_deriv M \<nu> x"
```
```  1162 proof -
```
```  1163   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
```
```  1164   then show ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
```
```  1165   from Ex show "0 \<le> RN_deriv M \<nu> x" unfolding RN_deriv_def
```
```  1166     by (rule someI2_ex) simp
```
```  1167   fix A assume "A \<in> sets M"
```
```  1168   from Ex show "?int A" unfolding RN_deriv_def
```
```  1169     by (rule someI2_ex) (simp add: `A \<in> sets M`)
```
```  1170 qed
```
```  1171
```
```  1172 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
```
```  1173   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
```
```  1174     and f: "f \<in> borel_measurable M"
```
```  1175   shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
```
```  1176 proof -
```
```  1177   interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```  1178   note RN = RN_deriv[OF \<nu>]
```
```  1179   have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>)"
```
```  1180     unfolding positive_integral_max_0 ..
```
```  1181   also have "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>) =
```
```  1182     (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>)"
```
```  1183     by (intro \<nu>.positive_integral_cong_measure[symmetric]) (simp_all add: RN(2))
```
```  1184   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * max 0 (f x) \<partial>M)"
```
```  1185     by (intro positive_integral_translated_density) (auto simp add: RN f)
```
```  1186   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
```
```  1187     using RN_deriv(3)[OF \<nu>]
```
```  1188     by (auto intro!: positive_integral_cong_pos split: split_if_asm
```
```  1189              simp: max_def ereal_mult_le_0_iff)
```
```  1190   finally show ?thesis .
```
```  1191 qed
```
```  1192
```
```  1193 lemma (in sigma_finite_measure) RN_deriv_unique:
```
```  1194   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
```
```  1195   and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
```
```  1196   and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
```
```  1197   shows "AE x. f x = RN_deriv M \<nu> x"
```
```  1198 proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
```
```  1199   show "AE x. 0 \<le> RN_deriv M \<nu> x" using RN_deriv[OF \<nu>] by auto
```
```  1200   fix A assume A: "A \<in> sets M"
```
```  1201   show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
```
```  1202     unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
```
```  1203 qed
```
```  1204
```
```  1205 lemma (in sigma_finite_measure) RN_deriv_vimage:
```
```  1206   assumes T: "T \<in> measure_preserving M M'"
```
```  1207     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
```
```  1208     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
```
```  1209   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
```
```  1210   shows "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
```
```  1211 proof (rule RN_deriv_unique)
```
```  1212   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
```
```  1213   show "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
```
```  1214     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
```
```  1215   interpret M': measure_space M'
```
```  1216   proof (rule measure_space_vimage)
```
```  1217     have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
```
```  1218     then show "sigma_algebra M'" by simp
```
```  1219   qed fact
```
```  1220   show "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
```
```  1221   proof safe
```
```  1222     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
```
```  1223     then have N': "T' -` N \<inter> space M' \<in> sets M'"
```
```  1224       using T' by (auto simp: measurable_def measure_preserving_def)
```
```  1225     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
```
```  1226       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
```
```  1227     then have "measure M' (T' -` N \<inter> space M') = 0"
```
```  1228       using measure_preservingD[OF T N'] N_0 by auto
```
```  1229     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
```
```  1230       unfolding M'.absolutely_continuous_def measurable_def by auto
```
```  1231   qed
```
```  1232   interpret M': sigma_finite_measure M'
```
```  1233   proof
```
```  1234     from sigma_finite guess F .. note F = this
```
```  1235     show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<infinity>)"
```
```  1236     proof (intro exI conjI allI)
```
```  1237       show *: "range (\<lambda>i. T' -` F i \<inter> space M') \<subseteq> sets M'"
```
```  1238         using F T' by (auto simp: measurable_def measure_preserving_def)
```
```  1239       show "(\<Union>i. T' -` F i \<inter> space M') = space M'"
```
```  1240         using F T' by (force simp: measurable_def measure_preserving_def)
```
```  1241       fix i
```
```  1242       have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
```
```  1243       note measure_preservingD[OF T this, symmetric]
```
```  1244       moreover
```
```  1245       have Fi: "F i \<in> sets M" using F by auto
```
```  1246       then have "T -` (T' -` F i \<inter> space M') \<inter> space M = F i"
```
```  1247         using T inv sets_into_space[OF Fi]
```
```  1248         by (auto simp: measurable_def measure_preserving_def)
```
```  1249       ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<infinity>"
```
```  1250         using F by simp
```
```  1251     qed
```
```  1252   qed
```
```  1253   have "(RN_deriv M' \<nu>') \<circ> T \<in> borel_measurable M"
```
```  1254     by (intro measurable_comp[where b=M'] M'.RN_deriv(1) measure_preservingD2[OF T]) fact+
```
```  1255   then show "(\<lambda>x. RN_deriv M' \<nu>' (T x)) \<in> borel_measurable M"
```
```  1256     by (simp add: comp_def)
```
```  1257   show "AE x. 0 \<le> RN_deriv M' \<nu>' (T x)" using M'.RN_deriv(3)[OF \<nu>'] by auto
```
```  1258   fix A let ?A = "T' -` A \<inter> space M'"
```
```  1259   assume A: "A \<in> sets M"
```
```  1260   then have A': "?A \<in> sets M'" using T' unfolding measurable_def measure_preserving_def
```
```  1261     by auto
```
```  1262   from A have "\<nu> A = \<nu>' ?A" using T'[THEN measure_preservingD] by simp
```
```  1263   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' x * indicator ?A x \<partial>M'"
```
```  1264     using A' by (rule M'.RN_deriv(2)[OF \<nu>'])
```
```  1265   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator ?A (T x) \<partial>M"
```
```  1266   proof (rule positive_integral_vimage)
```
```  1267     show "sigma_algebra M'" by default
```
```  1268     show "(\<lambda>x. RN_deriv M' \<nu>' x * indicator (T' -` A \<inter> space M') x) \<in> borel_measurable M'"
```
```  1269       by (auto intro!: A' M'.RN_deriv(1)[OF \<nu>'])
```
```  1270   qed fact
```
```  1271   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M"
```
```  1272     using T inv by (auto intro!: positive_integral_cong simp: measure_preserving_def measurable_def indicator_def)
```
```  1273   finally show "\<nu> A = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M" .
```
```  1274 qed
```
```  1275
```
```  1276 lemma (in sigma_finite_measure) RN_deriv_finite:
```
```  1277   assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
```
```  1278   shows "AE x. RN_deriv M \<nu> x \<noteq> \<infinity>"
```
```  1279 proof -
```
```  1280   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```  1281   have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```  1282   from sfm show ?thesis
```
```  1283     using sigma_finite_iff_density_finite[OF \<nu> RN_deriv(1)[OF \<nu> ac]] RN_deriv(2,3)[OF \<nu> ac] by simp
```
```  1284 qed
```
```  1285
```
```  1286 lemma (in sigma_finite_measure)
```
```  1287   assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
```
```  1288     and f: "f \<in> borel_measurable M"
```
```  1289   shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
```
```  1290       integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
```
```  1291     and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
```
```  1292       (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
```
```  1293 proof -
```
```  1294   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
```
```  1295   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
```
```  1296   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
```
```  1297   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
```
```  1298   have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f by simp
```
```  1299   { fix f :: "'a \<Rightarrow> real"
```
```  1300     { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<infinity>"
```
```  1301       have "ereal (real (RN_deriv M \<nu> x)) * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
```
```  1302         by (simp add: mult_le_0_iff)
```
```  1303       then have "RN_deriv M \<nu> x * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
```
```  1304         using RN_deriv(3)[OF ms \<nu>(2)] * by (auto simp add: ereal_real split: split_if_asm) }
```
```  1305     then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M \<nu> x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (f x) \<partial>M)"
```
```  1306               "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M \<nu> x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (- f x) \<partial>M)"
```
```  1307       using RN_deriv_finite[OF \<nu>] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
```
```  1308       by (auto intro!: positive_integral_cong_AE) }
```
```  1309   note * = this
```
```  1310   show ?integral ?integrable
```
```  1311     unfolding lebesgue_integral_def integrable_def *
```
```  1312     using f RN_deriv(1)[OF ms \<nu>(2)]
```
```  1313     by (auto simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
```
```  1314 qed
```
```  1315
```
```  1316 lemma (in sigma_finite_measure) real_RN_deriv:
```
```  1317   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?\<nu>")
```
```  1318   assumes ac: "absolutely_continuous \<nu>"
```
```  1319   obtains D where "D \<in> borel_measurable M"
```
```  1320     and "AE x. RN_deriv M \<nu> x = ereal (D x)"
```
```  1321     and "AE x in M\<lparr>measure := \<nu>\<rparr>. 0 < D x"
```
```  1322     and "\<And>x. 0 \<le> D x"
```
```  1323 proof
```
```  1324   interpret \<nu>: finite_measure ?\<nu> by fact
```
```  1325   have ms: "measure_space ?\<nu>" by default
```
```  1326   note RN = RN_deriv[OF ms ac]
```
```  1327
```
```  1328   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M \<nu> x = t}"
```
```  1329
```
```  1330   show "(\<lambda>x. real (RN_deriv M \<nu> x)) \<in> borel_measurable M"
```
```  1331     using RN by auto
```
```  1332
```
```  1333   have "\<nu> (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1334     using RN by auto
```
```  1335   also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1336     by (intro positive_integral_cong) (auto simp: indicator_def)
```
```  1337   also have "\<dots> = \<infinity> * \<mu> (?RN \<infinity>)"
```
```  1338     using RN by (intro positive_integral_cmult_indicator) auto
```
```  1339   finally have eq: "\<nu> (?RN \<infinity>) = \<infinity> * \<mu> (?RN \<infinity>)" .
```
```  1340   moreover
```
```  1341   have "\<mu> (?RN \<infinity>) = 0"
```
```  1342   proof (rule ccontr)
```
```  1343     assume "\<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>} \<noteq> 0"
```
```  1344     moreover from RN have "0 \<le> \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
```
```  1345     ultimately have "0 < \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
```
```  1346     with eq have "\<nu> (?RN \<infinity>) = \<infinity>" by simp
```
```  1347     with \<nu>.finite_measure[of "?RN \<infinity>"] RN show False by auto
```
```  1348   qed
```
```  1349   ultimately have "AE x. RN_deriv M \<nu> x < \<infinity>"
```
```  1350     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
```
```  1351   then show "AE x. RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
```
```  1352     using RN(3) by (auto simp: ereal_real)
```
```  1353   then have eq: "AE x in (M\<lparr>measure := \<nu>\<rparr>) . RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
```
```  1354     using ac absolutely_continuous_AE[OF ms] by auto
```
```  1355
```
```  1356   show "\<And>x. 0 \<le> real (RN_deriv M \<nu> x)"
```
```  1357     using RN by (auto intro: real_of_ereal_pos)
```
```  1358
```
```  1359   have "\<nu> (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN 0) x \<partial>M)"
```
```  1360     using RN by auto
```
```  1361   also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
```
```  1362     by (intro positive_integral_cong) (auto simp: indicator_def)
```
```  1363   finally have "AE x in (M\<lparr>measure := \<nu>\<rparr>). RN_deriv M \<nu> x \<noteq> 0"
```
```  1364     using RN by (intro \<nu>.AE_iff_measurable[THEN iffD2]) auto
```
```  1365   with RN(3) eq show "AE x in (M\<lparr>measure := \<nu>\<rparr>). 0 < real (RN_deriv M \<nu> x)"
```
```  1366     by (auto simp: zero_less_real_of_ereal le_less)
```
```  1367 qed
```
```  1368
```
```  1369 lemma (in sigma_finite_measure) RN_deriv_singleton:
```
```  1370   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```  1371   and ac: "absolutely_continuous \<nu>"
```
```  1372   and "{x} \<in> sets M"
```
```  1373   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
```
```  1374 proof -
```
```  1375   note deriv = RN_deriv[OF assms(1, 2)]
```
```  1376   from deriv(2)[OF `{x} \<in> sets M`]
```
```  1377   have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
```
```  1378     by (auto simp: indicator_def intro!: positive_integral_cong)
```
```  1379   thus ?thesis using positive_integral_cmult_indicator[OF _ `{x} \<in> sets M`] deriv(3)
```
```  1380     by auto
```
```  1381 qed
```
```  1382
```
```  1383 theorem (in finite_measure_space) RN_deriv_finite_measure:
```
```  1384   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
```
```  1385   and ac: "absolutely_continuous \<nu>"
```
```  1386   and "x \<in> space M"
```
```  1387   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
```
```  1388 proof -
```
```  1389   have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
```
```  1390   from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
```
```  1391 qed
```
```  1392
```
```  1393 end
```