```(*  Title:      HOL/Analysis/Radon_Nikodym.thy
Author:     Johannes Hölzl, TU München
*)

imports Bochner_Integration
begin

definition✐‹tag important› diff_measure :: "'a measure ⇒ 'a measure ⇒ 'a measure"
where
"diff_measure M N = measure_of (space M) (sets M) (λA. emeasure M A - emeasure N A)"

lemma
shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
by (auto simp: diff_measure_def)

lemma emeasure_diff_measure:
assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N"
assumes pos: "⋀A. A ∈ sets M ⟹ emeasure N A ≤ emeasure M A" and A: "A ∈ sets M"
shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?μ A")
unfolding diff_measure_def
proof (rule emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) ?μ"
using pos by (simp add: positive_def)
fix A :: "nat ⇒ _"  assume A: "range A ⊆ sets M" and "disjoint_family A"
then have suminf:
"(∑i. emeasure M (A i)) = emeasure M (⋃i. A i)"
"(∑i. emeasure N (A i)) = emeasure N (⋃i. A i)"
with A have "(∑i. emeasure M (A i) - emeasure N (A i)) =
(∑i. emeasure M (A i)) - (∑i. emeasure N (A i))"
using fin pos[of "A _"]
by (intro ennreal_suminf_minus)
(auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure)
then show "(∑i. emeasure M (A i) - emeasure N (A i)) =
emeasure M (⋃i. A i) - emeasure N (⋃i. A i) "
qed
qed fact

text ‹An equivalent characterization of sigma-finite spaces is the existence of integrable
positive functions (or, still equivalently, the existence of a probability measure which is in
the same measure class as the original measure).›

proposition (in sigma_finite_measure) obtain_positive_integrable_function:
obtains f::"'a ⇒ real" where
"f ∈ borel_measurable M"
"⋀x. f x > 0"
"⋀x. f x ≤ 1"
"integrable M f"
proof -
obtain A :: "nat ⇒ 'a set" where "range A ⊆ sets M" "(⋃i. A i) = space M" "⋀i. emeasure M (A i) ≠ ∞"
using sigma_finite by auto
then have [measurable]:"A n ∈ sets M" for n by auto
define g where "g = (λx. (∑n. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n))))"
have [measurable]: "g ∈ borel_measurable M" unfolding g_def by auto
have *: "summable (λn. (1/2)^(Suc n) * indicator (A n) x / (1+ measure M (A n)))" for x
apply (rule summable_comparison_test'[of "λn. (1/2)^(Suc n)" 0])
using power_half_series summable_def by (auto simp add: indicator_def divide_simps)
have "g x ≤ (∑n. (1/2)^(Suc n))" for x unfolding g_def
apply (rule suminf_le) using * power_half_series summable_def by (auto simp add: indicator_def divide_simps)
then have g_le_1: "g x ≤ 1" for x using power_half_series sums_unique by fastforce

have g_pos: "g x > 0" if "x ∈ space M" for x
unfolding g_def proof (subst suminf_pos_iff[OF *[of x]], auto)
obtain i where "x ∈ A i" using ‹(⋃i. A i) = space M› ‹x ∈ space M› by auto
then have "0 < (1 / 2) ^ Suc i * indicator (A i) x / (1 + Sigma_Algebra.measure M (A i))"
unfolding indicator_def apply (auto simp add: divide_simps) using measure_nonneg[of M "A i"]
by (auto, meson add_nonneg_nonneg linorder_not_le mult_nonneg_nonneg zero_le_numeral zero_le_one zero_le_power)
then show "∃i. 0 < (1 / 2) ^ i * indicator (A i) x / (2 + 2 * Sigma_Algebra.measure M (A i))"
by auto
qed

have "integrable M g"
unfolding g_def proof (rule integrable_suminf)
fix n
show "integrable M (λx. (1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n)))"
using ‹emeasure M (A n) ≠ ∞›
by (auto intro!: integrable_mult_right integrable_divide_zero integrable_real_indicator simp add: top.not_eq_extremum)
next
show "AE x in M. summable (λn. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))))"
using * by auto
show "summable (λn. (∫x. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))) ∂M))"
apply (rule summable_comparison_test'[of "λn. (1/2)^(Suc n)" 0], auto)
using power_half_series summable_def apply auto[1]
apply (auto simp add: field_split_simps) using measure_nonneg[of M] not_less by fastforce
qed

define f where "f = (λx. if x ∈ space M then g x else 1)"
have "f x > 0" for x unfolding f_def using g_pos by auto
moreover have "f x ≤ 1" for x unfolding f_def using g_le_1 by auto
moreover have [measurable]: "f ∈ borel_measurable M" unfolding f_def by auto
moreover have "integrable M f"
apply (subst integrable_cong[of _ _ _ g]) unfolding f_def using ‹integrable M g› by auto
ultimately show "(⋀f. f ∈ borel_measurable M ⟹ (⋀x. 0 < f x) ⟹ (⋀x. f x ≤ 1) ⟹ integrable M f ⟹ thesis) ⟹ thesis"
by (meson that)
qed

lemma (in sigma_finite_measure) Ex_finite_integrable_function:
"∃h∈borel_measurable M. integral⇧N M h ≠ ∞ ∧ (∀x∈space M. 0 < h x ∧ h x < ∞)"
proof -
obtain A :: "nat ⇒ 'a set" where
range[measurable]: "range A ⊆ sets M" and
space: "(⋃i. A i) = space M" and
measure: "⋀i. emeasure M (A i) ≠ ∞" and
disjoint: "disjoint_family A"
using sigma_finite_disjoint by blast
let ?B = "λi. 2^Suc i * emeasure M (A i)"
have [measurable]: "⋀i. A i ∈ sets M"
using range by fastforce+
have "∀i. ∃x. 0 < x ∧ x < inverse (?B i)"
proof
fix i show "∃x. 0 < x ∧ x < inverse (?B i)"
using measure[of i]
by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal)
qed
from choice[OF this] obtain n where n: "⋀i. 0 < n i"
"⋀i. n i < inverse (2^Suc i * emeasure M (A i))" by auto
{ fix i have "0 ≤ n i" using n(1)[of i] by auto } note pos = this
let ?h = "λx. ∑i. n i * indicator (A i) x"
show ?thesis
proof (safe intro!: bexI[of _ ?h] del: notI)
have "integral⇧N M ?h = (∑i. n i * emeasure M (A i))" using pos
also have "… ≤ (∑i. ennreal ((1/2)^Suc i))"
proof (intro suminf_le allI)
fix N
have "n N * emeasure M (A N) ≤ inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)"
using n[of N] by (intro mult_right_mono) auto
also have "… = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))"
using measure[of N]
by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult
power_eq_top_ennreal less_top[symmetric] mult_ac
del: power_Suc)
also have "… ≤ inverse (ennreal 2) ^ Suc N"
using measure[of N]
by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0")
(auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc)
also have "… = ennreal (inverse 2 ^ Suc N)"
by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal)
finally show "n N * emeasure M (A N) ≤ ennreal ((1/2)^Suc N)"
by simp
qed auto
also have "… < top"
unfolding less_top[symmetric]
by (rule ennreal_suminf_neq_top)
(auto simp: summable_geometric summable_Suc_iff simp del: power_Suc)
finally show "integral⇧N M ?h ≠ ∞"
by (auto simp: top_unique)
next
{ fix x assume "x ∈ space M"
then obtain i where "x ∈ A i" using space[symmetric] by auto
with disjoint n have "?h x = n i"
by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
then show "0 < ?h x" and "?h x < ∞" using n[of i] by (auto simp: less_top[symmetric]) }
note pos = this
qed measurable
qed

subsection "Absolutely continuous"

definition✐‹tag important› absolutely_continuous :: "'a measure ⇒ 'a measure ⇒ bool" where
"absolutely_continuous M N ⟷ null_sets M ⊆ null_sets N"

lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M"
unfolding absolutely_continuous_def by (auto simp: null_sets_count_space)

lemma absolutely_continuousI_density:
"f ∈ borel_measurable M ⟹ absolutely_continuous M (density M f)"
by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in)

lemma absolutely_continuousI_point_measure_finite:
"(⋀x. ⟦ x ∈ A ; f x ≤ 0 ⟧ ⟹ g x ≤ 0) ⟹ absolutely_continuous (point_measure A f) (point_measure A g)"
unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff)

lemma absolutely_continuousD:
"absolutely_continuous M N ⟹ A ∈ sets M ⟹ emeasure M A = 0 ⟹ emeasure N A = 0"
by (auto simp: absolutely_continuous_def null_sets_def)

lemma absolutely_continuous_AE:
assumes sets_eq: "sets M' = sets M"
and "absolutely_continuous M M'" "AE x in M. P x"
shows "AE x in M'. P x"
proof -
from ‹AE x in M. P x› obtain N where N: "N ∈ null_sets M" "{x∈space M. ¬ P x} ⊆ N"
unfolding eventually_ae_filter by auto
show "AE x in M'. P x"
proof (rule AE_I')
show "{x∈space M'. ¬ P x} ⊆ N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
from ‹absolutely_continuous M M'› show "N ∈ null_sets M'"
using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
qed
qed

subsection "Existence of the Radon-Nikodym derivative"

proposition
assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M"
assumes "absolutely_continuous M N"
shows "∃f ∈ borel_measurable M. density M f = N"
proof -
interpret N: finite_measure N by fact
define G where "G = {g ∈ borel_measurable M. ∀A∈sets M. (∫⇧+x. g x * indicator A x ∂M) ≤ N A}"
have [measurable_dest]: "f ∈ G ⟹ f ∈ borel_measurable M"
and G_D: "⋀A. f ∈ G ⟹ A ∈ sets M ⟹ (∫⇧+x. f x * indicator A x ∂M) ≤ N A" for f
by (auto simp: G_def)
note this[measurable_dest]
have "(λx. 0) ∈ G" unfolding G_def by auto
hence "G ≠ {}" by auto
{ fix f g assume f[measurable]: "f ∈ G" and g[measurable]: "g ∈ G"
have "(λx. max (g x) (f x)) ∈ G" (is "?max ∈ G") unfolding G_def
proof safe
let ?A = "{x ∈ space M. f x ≤ g x}"
have "?A ∈ sets M" using f g unfolding G_def by auto
fix A assume [measurable]: "A ∈ sets M"
have union: "((?A ∩ A) ∪ ((space M - ?A) ∩ A)) = A"
using sets.sets_into_space[OF ‹A ∈ sets M›] by auto
have "⋀x. x ∈ space M ⟹ max (g x) (f x) * indicator A x =
g x * indicator (?A ∩ A) x + f x * indicator ((space M - ?A) ∩ A) x"
by (auto simp: indicator_def max_def)
hence "(∫⇧+x. max (g x) (f x) * indicator A x ∂M) =
(∫⇧+x. g x * indicator (?A ∩ A) x ∂M) +
(∫⇧+x. f x * indicator ((space M - ?A) ∩ A) x ∂M)"
by (auto cong: nn_integral_cong intro!: nn_integral_add)
also have "… ≤ N (?A ∩ A) + N ((space M - ?A) ∩ A)"
using f g unfolding G_def by (auto intro!: add_mono)
also have "… = N A"
using union by (subst plus_emeasure) auto
finally show "(∫⇧+x. max (g x) (f x) * indicator A x ∂M) ≤ N A" .
qed auto }
note max_in_G = this
{ fix f assume  "incseq f" and f: "⋀i. f i ∈ G"
then have [measurable]: "⋀i. f i ∈ borel_measurable M" by (auto simp: G_def)
have "(λx. SUP i. f i x) ∈ G" unfolding G_def
proof safe
show "(λx. SUP i. f i x) ∈ borel_measurable M" by measurable
next
fix A assume "A ∈ sets M"
have "(∫⇧+x. (SUP i. f i x) * indicator A x ∂M) =
(∫⇧+x. (SUP i. f i x * indicator A x) ∂M)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "… = (SUP i. (∫⇧+x. f i x * indicator A x ∂M))"
using ‹incseq f› f ‹A ∈ sets M›
by (intro nn_integral_monotone_convergence_SUP)
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
finally show "(∫⇧+x. (SUP i. f i x) * indicator A x ∂M) ≤ N A"
using f ‹A ∈ sets M› by (auto intro!: SUP_least simp: G_D)
qed }
note SUP_in_G = this
let ?y = "SUP g ∈ G. integral⇧N M g"
have y_le: "?y ≤ N (space M)" unfolding G_def
proof (safe intro!: SUP_least)
fix g assume "∀A∈sets M. (∫⇧+x. g x * indicator A x ∂M) ≤ N A"
from this[THEN bspec, OF sets.top] show "integral⇧N M g ≤ N (space M)"
by (simp cong: nn_integral_cong)
qed
from ennreal_SUP_countable_SUP [OF ‹G ≠ {}›, of "integral⇧N M"]
obtain ys :: "nat ⇒ ennreal"
where ys: "range ys ⊆ integral⇧N M ` G ∧ Sup (integral⇧N M ` G) = Sup (range ys)"
by auto
then have "∀n. ∃g. g∈G ∧ integral⇧N M g = ys n"
proof safe
fix n assume "range ys ⊆ integral⇧N M ` G"
hence "ys n ∈ integral⇧N M ` G" by auto
thus "∃g. g∈G ∧ integral⇧N M g = ys n" by auto
qed
from choice[OF this] obtain gs where "⋀i. gs i ∈ G" "⋀n. integral⇧N M (gs n) = ys n" by auto
hence y_eq: "?y = (SUP i. integral⇧N M (gs i))" using ys by auto
let ?g = "λi x. Max ((λn. gs n x) ` {..i})"
define f where [abs_def]: "f x = (SUP i. ?g i x)" for x
let ?F = "λA x. f x * indicator A x"
have gs_not_empty: "⋀i x. (λn. gs n x) ` {..i} ≠ {}" by auto
{ fix i have "?g i ∈ G"
proof (induct i)
case 0 thus ?case by simp fact
next
case (Suc i)
with Suc gs_not_empty ‹gs (Suc i) ∈ G› show ?case
by (auto simp add: atMost_Suc intro!: max_in_G)
qed }
note g_in_G = this
have "incseq ?g" using gs_not_empty
by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)

from SUP_in_G[OF this g_in_G] have [measurable]: "f ∈ G" unfolding f_def .
then have [measurable]: "f ∈ borel_measurable M" unfolding G_def by auto

have "integral⇧N M f = (SUP i. integral⇧N M (?g i))" unfolding f_def
using g_in_G ‹incseq ?g› by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
also have "… = ?y"
proof (rule antisym)
show "(SUP i. integral⇧N M (?g i)) ≤ ?y"
using g_in_G by (auto intro: SUP_mono)
show "?y ≤ (SUP i. integral⇧N M (?g i))" unfolding y_eq
by (auto intro!: SUP_mono nn_integral_mono Max_ge)
qed
finally have int_f_eq_y: "integral⇧N M f = ?y" .

have upper_bound: "∀A∈sets M. N A ≤ density M f A"
proof (rule ccontr)
assume "¬ ?thesis"
then obtain A where A[measurable]: "A ∈ sets M" and f_less_N: "density M f A < N A"
by (auto simp: not_le)
then have pos_A: "0 < M A"
using ‹absolutely_continuous M N›[THEN absolutely_continuousD, OF A]
by (auto simp: zero_less_iff_neq_zero)

define b where "b = (N A - density M f A) / M A / 2"
with f_less_N pos_A have "0 < b" "b ≠ top"
by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff)

let ?f = "λx. f x + b"
have "nn_integral M f ≠ top"
using ‹f ∈ G›[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong)
with ‹b ≠ top› interpret Mf: finite_measure "density M ?f"
by (intro finite_measureI)
(auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff
cong: nn_integral_cong)

from unsigned_Hahn_decomposition[of "density M ?f" N A]
obtain Y where [measurable]: "Y ∈ sets M" and [simp]: "Y ⊆ A"
and Y1: "⋀C. C ∈ sets M ⟹ C ⊆ Y ⟹ density M ?f C ≤ N C"
and Y2: "⋀C. C ∈ sets M ⟹ C ⊆ A ⟹ C ∩ Y = {} ⟹ N C ≤ density M ?f C"
by auto

let ?f' = "λx. f x + b * indicator Y x"
have "M Y ≠ 0"
proof
assume "M Y = 0"
then have "N Y = 0"
using ‹absolutely_continuous M N›[THEN absolutely_continuousD, of Y] by auto
then have "N A = N (A - Y)"
by (subst emeasure_Diff) auto
also have "… ≤ density M ?f (A - Y)"
by (rule Y2) auto
also have "… ≤ density M ?f A - density M ?f Y"
by (subst emeasure_Diff) auto
also have "… ≤ density M ?f A - 0"
by (intro ennreal_minus_mono) auto
also have "density M ?f A = b * M A + density M f A"
also have "… < N A"
using f_less_N pos_A
by (cases "density M f A"; cases "M A"; cases "N A")
(auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric]
ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps
simp del: ennreal_numeral ennreal_plus)
finally show False
by simp
qed
then have "nn_integral M f < nn_integral M ?f'"
using ‹0 < b› ‹nn_integral M f ≠ top›
moreover
have "?f' ∈ G"
unfolding G_def
proof safe
fix X assume [measurable]: "X ∈ sets M"
have "(∫⇧+ x. ?f' x * indicator X x ∂M) = density M f (X - Y) + density M ?f (X ∩ Y)"
also have "… ≤ N (X - Y) + N (X ∩ Y)"
using G_D[OF ‹f ∈ G›] by (intro add_mono Y1) (auto simp: emeasure_density)
also have "… = N X"
by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure])
finally show "(∫⇧+ x. ?f' x * indicator X x ∂M) ≤ N X" .
qed simp
then have "nn_integral M ?f' ≤ ?y"
by (rule SUP_upper)
ultimately show False
qed
show ?thesis
proof (intro bexI[of _ f] measure_eqI conjI antisym)
fix A assume "A ∈ sets (density M f)" then show "emeasure (density M f) A ≤ emeasure N A"
by (auto simp: emeasure_density intro!: G_D[OF ‹f ∈ G›])
next
fix A assume A: "A ∈ sets (density M f)" then show "emeasure N A ≤ emeasure (density M f) A"
using upper_bound by auto
qed auto
qed

lemma (in finite_measure) split_space_into_finite_sets_and_rest:
assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M"
shows "∃B::nat⇒'a set. disjoint_family B ∧ range B ⊆ sets M ∧ (∀i. N (B i) ≠ ∞) ∧
(∀A∈sets M. A ∩ (⋃i. B i) = {} ⟶ (emeasure M A = 0 ∧ N A = 0) ∨ (emeasure M A > 0 ∧ N A = ∞))"
proof -
let ?Q = "{Q∈sets M. N Q ≠ ∞}"
let ?a = "SUP Q∈?Q. emeasure M Q"
have "{} ∈ ?Q" by auto
then have Q_not_empty: "?Q ≠ {}" by blast
have "?a ≤ emeasure M (space M)" using sets.sets_into_space
by (auto intro!: SUP_least emeasure_mono)
then have "?a ≠ ∞"
using finite_emeasure_space
by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff)
from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"]
obtain Q'' where "range Q'' ⊆ emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
by auto
then have "∀i. ∃Q'. Q'' i = emeasure M Q' ∧ Q' ∈ ?Q" by auto
from choice[OF this] obtain Q' where Q': "⋀i. Q'' i = emeasure M (Q' i)" "⋀i. Q' i ∈ ?Q"
by auto
then have a_Lim: "?a = (SUP i. emeasure M (Q' i))" using a by simp
let ?O = "λn. ⋃i≤n. Q' i"
have Union: "(SUP i. emeasure M (?O i)) = emeasure M (⋃i. ?O i)"
proof (rule SUP_emeasure_incseq[of ?O])
show "range ?O ⊆ sets M" using Q' by auto
show "incseq ?O" by (fastforce intro!: incseq_SucI)
qed
have Q'_sets[measurable]: "⋀i. Q' i ∈ sets M" using Q' by auto
have O_sets: "⋀i. ?O i ∈ sets M" using Q' by auto
then have O_in_G: "⋀i. ?O i ∈ ?Q"
proof (safe del: notI)
fix i have "Q' ` {..i} ⊆ sets M" using Q' by auto
then have "N (?O i) ≤ (∑i≤i. N (Q' i))"
also have "… < ∞" using Q' by (simp add: less_top)
finally show "N (?O i) ≠ ∞" by simp
qed auto
have O_mono: "⋀n. ?O n ⊆ ?O (Suc n)" by fastforce
have a_eq: "?a = emeasure M (⋃i. ?O i)" unfolding Union[symmetric]
proof (rule antisym)
show "?a ≤ (SUP i. emeasure M (?O i))" unfolding a_Lim
using Q' by (auto intro!: SUP_mono emeasure_mono)
show "(SUP i. emeasure M (?O i)) ≤ ?a"
proof (safe intro!: Sup_mono, unfold bex_simps)
fix i
have *: "(⋃(Q' ` {..i})) = ?O i" by auto
then show "∃x. (x ∈ sets M ∧ N x ≠ ∞) ∧
emeasure M (⋃(Q' ` {..i})) ≤ emeasure M x"
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
qed
qed
let ?O_0 = "(⋃i. ?O i)"
have "?O_0 ∈ sets M" using Q' by auto
have "disjointed Q' i ∈ sets M" for i
using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq)
note Q_sets = this
show ?thesis
proof (intro bexI exI conjI ballI impI allI)
show "disjoint_family (disjointed Q')"
by (rule disjoint_family_disjointed)
show "range (disjointed Q') ⊆ sets M"
using Q'_sets by (intro sets.range_disjointed_sets) auto
{ fix A assume A: "A ∈ sets M" "A ∩ (⋃i. disjointed Q' i) = {}"
then have A1: "A ∩ (⋃i. Q' i) = {}"
unfolding UN_disjointed_eq by auto
show "emeasure M A = 0 ∧ N A = 0 ∨ 0 < emeasure M A ∧ N A = ∞"
proof (rule disjCI, simp)
assume *: "emeasure M A = 0 ∨ N A ≠ top"
show "emeasure M A = 0 ∧ N A = 0"
proof (cases "emeasure M A = 0")
case True
with ac A have "N A = 0"
unfolding absolutely_continuous_def by auto
with True show ?thesis by simp
next
case False
with * have "N A ≠ ∞" by auto
with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 ∪ A)"
using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff)
also have "… = (SUP i. emeasure M (?O i ∪ A))"
proof (rule SUP_emeasure_incseq[of "λi. ?O i ∪ A", symmetric, simplified])
show "range (λi. ?O i ∪ A) ⊆ sets M"
using ‹N A ≠ ∞› O_sets A by auto
qed (fastforce intro!: incseq_SucI)
also have "… ≤ ?a"
proof (safe intro!: SUP_least)
fix i have "?O i ∪ A ∈ ?Q"
proof (safe del: notI)
show "?O i ∪ A ∈ sets M" using O_sets A by auto
from O_in_G[of i] have "N (?O i ∪ A) ≤ N (?O i) + N A"
using emeasure_subadditive[of "?O i" N A] A O_sets by auto
with O_in_G[of i] show "N (?O i ∪ A) ≠ ∞"
using ‹N A ≠ ∞› by (auto simp: top_unique)
qed
then show "emeasure M (?O i ∪ A) ≤ ?a" by (rule SUP_upper)
qed
finally have "emeasure M A = 0"
unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
with ‹emeasure M A ≠ 0› show ?thesis by auto
qed
qed }
{ fix i
have "N (disjointed Q' i) ≤ N (Q' i)"
by (auto intro!: emeasure_mono simp: disjointed_def)
then show "N (disjointed Q' i) ≠ ∞"
using Q'(2)[of i] by (auto simp: top_unique) }
qed
qed

assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
shows "∃f∈borel_measurable M. density M f = N"
proof -
from split_space_into_finite_sets_and_rest[OF assms]
obtain Q :: "nat ⇒ 'a set"
where Q: "disjoint_family Q" "range Q ⊆ sets M"
and in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ N A = 0 ∨ 0 < emeasure M A ∧ N A = ∞"
and Q_fin: "⋀i. N (Q i) ≠ ∞" by force
from Q have Q_sets: "⋀i. Q i ∈ sets M" by auto
let ?N = "λi. density N (indicator (Q i))" and ?M = "λi. density M (indicator (Q i))"
have "∀i. ∃f∈borel_measurable (?M i). density (?M i) f = ?N i"
fix i
from Q show "finite_measure (?M i)"
by (auto intro!: finite_measureI cong: nn_integral_cong
from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong)
with Q_fin show "finite_measure (?N i)"
by (auto intro!: finite_measureI)
show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
have [measurable]: "⋀A. A ∈ sets M ⟹ A ∈ sets N" by (simp add: sets_eq)
show "absolutely_continuous (?M i) (?N i)"
using ‹absolutely_continuous M N› ‹Q i ∈ sets M›
by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
intro!: absolutely_continuous_AE[OF sets_eq])
qed
from choice[OF this[unfolded Bex_def]]
obtain f where borel: "⋀i. f i ∈ borel_measurable M" "⋀i x. 0 ≤ f i x"
and f_density: "⋀i. density (?M i) (f i) = ?N i"
by force
{ fix A i assume A: "A ∈ sets M"
with Q borel have "(∫⇧+x. f i x * indicator (Q i ∩ A) x ∂M) = emeasure (density (?M i) (f i)) A"
by (auto simp add: emeasure_density nn_integral_density subset_eq
intro!: nn_integral_cong split: split_indicator)
also have "… = emeasure N (Q i ∩ A)"
using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
finally have "emeasure N (Q i ∩ A) = (∫⇧+x. f i x * indicator (Q i ∩ A) x ∂M)" .. }
note integral_eq = this
let ?f = "λx. (∑i. f i x * indicator (Q i) x) + ∞ * indicator (space M - (⋃i. Q i)) x"
show ?thesis
proof (safe intro!: bexI[of _ ?f])
show "?f ∈ borel_measurable M" using borel Q_sets
by (auto intro!: measurable_If)
show "density M ?f = N"
proof (rule measure_eqI)
fix A assume "A ∈ sets (density M ?f)"
then have "A ∈ sets M" by simp
have Qi: "⋀i. Q i ∈ sets M" using Q by auto
have [intro,simp]: "⋀i. (λx. f i x * indicator (Q i ∩ A) x) ∈ borel_measurable M"
"⋀i. AE x in M. 0 ≤ f i x * indicator (Q i ∩ A) x"
using borel Qi ‹A ∈ sets M› by auto
have "(∫⇧+x. ?f x * indicator A x ∂M) = (∫⇧+x. (∑i. f i x * indicator (Q i ∩ A) x) + ∞ * indicator ((space M - (⋃i. Q i)) ∩ A) x ∂M)"
using borel by (intro nn_integral_cong) (auto simp: indicator_def)
also have "… = (∫⇧+x. (∑i. f i x * indicator (Q i ∩ A) x) ∂M) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)"
using borel Qi ‹A ∈ sets M›
(auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
also have "… = (∑i. N (Q i ∩ A)) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)"
by (subst integral_eq[OF ‹A ∈ sets M›], subst nn_integral_suminf) auto
finally have "(∫⇧+x. ?f x * indicator A x ∂M) = (∑i. N (Q i ∩ A)) + ∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A)" .
moreover have "(∑i. N (Q i ∩ A)) = N ((⋃i. Q i) ∩ A)"
using Q Q_sets ‹A ∈ sets M›
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
moreover
have "(space M - (⋃x. Q x)) ∩ A ∩ (⋃x. Q x) = {}"
by auto
then have "∞ * emeasure M ((space M - (⋃i. Q i)) ∩ A) = N ((space M - (⋃i. Q i)) ∩ A)"
using in_Q0[of "(space M - (⋃i. Q i)) ∩ A"] ‹A ∈ sets M› Q by (auto simp: ennreal_top_mult)
moreover have "(space M - (⋃i. Q i)) ∩ A ∈ sets M" "((⋃i. Q i) ∩ A) ∈ sets M"
using Q_sets ‹A ∈ sets M› by auto
moreover have "((⋃i. Q i) ∩ A) ∪ ((space M - (⋃i. Q i)) ∩ A) = A" "((⋃i. Q i) ∩ A) ∩ ((space M - (⋃i. Q i)) ∩ A) = {}"
using ‹A ∈ sets M› sets.sets_into_space by auto
ultimately have "N A = (∫⇧+x. ?f x * indicator A x ∂M)"
using plus_emeasure[of "(⋃i. Q i) ∩ A" N "(space M - (⋃i. Q i)) ∩ A"] by (simp add: sets_eq)
with ‹A ∈ sets M› borel Q show "emeasure (density M ?f) A = N A"
by (auto simp: subset_eq emeasure_density)
qed
qed

assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
shows "∃f ∈ borel_measurable M. density M f = N"
proof -
from Ex_finite_integrable_function
obtain h where finite: "integral⇧N M h ≠ ∞" and
borel: "h ∈ borel_measurable M" and
nn: "⋀x. 0 ≤ h x" and
pos: "⋀x. x ∈ space M ⟹ 0 < h x" and
"⋀x. x ∈ space M ⟹ h x < ∞" by auto
let ?T = "λA. (∫⇧+x. h x * indicator A x ∂M)"
let ?MT = "density M h"
from borel finite nn interpret T: finite_measure ?MT
by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density)
have "absolutely_continuous ?MT N" "sets N = sets ?MT"
proof (unfold absolutely_continuous_def, safe)
fix A assume "A ∈ null_sets ?MT"
with borel have "A ∈ sets M" "AE x in M. x ∈ A ⟶ h x ≤ 0"
with pos sets.sets_into_space have "AE x in M. x ∉ A"
by (elim eventually_mono) (auto simp: not_le[symmetric])
then have "A ∈ null_sets M"
using ‹A ∈ sets M› by (simp add: AE_iff_null_sets)
with ac show "A ∈ null_sets N"
by (auto simp: absolutely_continuous_def)
obtain f where f_borel: "f ∈ borel_measurable M" "density ?MT f = N" by auto
with nn borel show ?thesis
by (auto intro!: bexI[of _ "λx. h x * f x"] simp: density_density_eq)
qed

subsection ‹Uniqueness of densities›

lemma finite_density_unique:
assumes borel: "f ∈ borel_measurable M" "g ∈ borel_measurable M"
assumes pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ g x"
and fin: "integral⇧N M f ≠ ∞"
shows "density M f = density M g ⟷ (AE x in M. f x = g x)"
proof (intro iffI ballI)
fix A assume eq: "AE x in M. f x = g x"
with borel show "density M f = density M g"
by (auto intro: density_cong)
next
let ?P = "λf A. ∫⇧+ x. f x * indicator A x ∂M"
assume "density M f = density M g"
with borel have eq: "∀A∈sets M. ?P f A = ?P g A"
from this[THEN bspec, OF sets.top] fin
have g_fin: "integral⇧N M g ≠ ∞" by (simp cong: nn_integral_cong)
{ fix f g assume borel: "f ∈ borel_measurable M" "g ∈ borel_measurable M"
and pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ g x"
and g_fin: "integral⇧N M g ≠ ∞" and eq: "∀A∈sets M. ?P f A = ?P g A"
let ?N = "{x∈space M. g x < f x}"
have N: "?N ∈ sets M" using borel by simp
have "?P g ?N ≤ integral⇧N M g" using pos
by (intro nn_integral_mono_AE) (auto split: split_indicator)
then have Pg_fin: "?P g ?N ≠ ∞" using g_fin by (auto simp: top_unique)
have "?P (λx. (f x - g x)) ?N = (∫⇧+x. f x * indicator ?N x - g x * indicator ?N x ∂M)"
by (auto intro!: nn_integral_cong simp: indicator_def)
also have "… = ?P f ?N - ?P g ?N"
proof (rule nn_integral_diff)
show "(λx. f x * indicator ?N x) ∈ borel_measurable M" "(λx. g x * indicator ?N x) ∈ borel_measurable M"
using borel N by auto
show "AE x in M. g x * indicator ?N x ≤ f x * indicator ?N x"
using pos by (auto split: split_indicator)
qed fact
also have "… = 0"
unfolding eq[THEN bspec, OF N] using Pg_fin by auto
finally have "AE x in M. f x ≤ g x"
using pos borel nn_integral_PInf_AE[OF borel(2) g_fin]
by (subst (asm) nn_integral_0_iff_AE)
(auto split: split_indicator simp: not_less ennreal_minus_eq_0) }
from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
show "AE x in M. f x = g x" by auto
qed

lemma (in finite_measure) density_unique_finite_measure:
assumes borel: "f ∈ borel_measurable M" "f' ∈ borel_measurable M"
assumes pos: "AE x in M. 0 ≤ f x" "AE x in M. 0 ≤ f' x"
assumes f: "⋀A. A ∈ sets M ⟹ (∫⇧+x. f x * indicator A x ∂M) = (∫⇧+x. f' x * indicator A x ∂M)"
(is "⋀A. A ∈ sets M ⟹ ?P f A = ?P f' A")
shows "AE x in M. f x = f' x"
proof -
let ?D = "λf. density M f"
let ?N = "λA. ?P f A" and ?N' = "λA. ?P f' A"
let ?f = "λA x. f x * indicator A x" and ?f' = "λA x. f' x * indicator A x"

have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
using borel by (auto intro!: absolutely_continuousI_density)
from split_space_into_finite_sets_and_rest[OF this]
obtain Q :: "nat ⇒ 'a set"
where Q: "disjoint_family Q" "range Q ⊆ sets M"
and in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ ?D f A = 0 ∨ 0 < emeasure M A ∧ ?D f A = ∞"
and Q_fin: "⋀i. ?D f (Q i) ≠ ∞" by force
with borel pos have in_Q0: "⋀A. A ∈ sets M ⟹ A ∩ (⋃i. Q i) = {} ⟹ emeasure M A = 0 ∧ ?N A = 0 ∨ 0 < emeasure M A ∧ ?N A = ∞"
and Q_fin: "⋀i. ?N (Q i) ≠ ∞" by (auto simp: emeasure_density subset_eq)

from Q have Q_sets[measurable]: "⋀i. Q i ∈ sets M" by auto
let ?D = "{x∈space M. f x ≠ f' x}"
have "?D ∈ sets M" using borel by auto
have *: "⋀i x A. ⋀y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i ∩ A) x"
unfolding indicator_def by auto
have "∀i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
by (intro finite_density_unique[THEN iffD1] allI)
(auto intro!: f measure_eqI simp: emeasure_density * subset_eq)
moreover have "AE x in M. ?f (space M - (⋃i. Q i)) x = ?f' (space M - (⋃i. Q i)) x"
proof (rule AE_I')
{ fix f :: "'a ⇒ ennreal" assume borel: "f ∈ borel_measurable M"
and eq: "⋀A. A ∈ sets M ⟹ ?N A = (∫⇧+x. f x * indicator A x ∂M)"
let ?A = "λi. (space M - (⋃i. Q i)) ∩ {x ∈ space M. f x < (i::nat)}"
have "(⋃i. ?A i) ∈ null_sets M"
proof (rule null_sets_UN)
fix i ::nat have "?A i ∈ sets M"
using borel by auto
have "?N (?A i) ≤ (∫⇧+x. (i::ennreal) * indicator (?A i) x ∂M)"
unfolding eq[OF ‹?A i ∈ sets M›]
by (auto intro!: nn_integral_mono simp: indicator_def)
also have "… = i * emeasure M (?A i)"
using ‹?A i ∈ sets M› by (auto intro!: nn_integral_cmult_indicator)
also have "… < ∞" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top)
finally have "?N (?A i) ≠ ∞" by simp
then show "?A i ∈ null_sets M" using in_Q0[OF ‹?A i ∈ sets M›] ‹?A i ∈ sets M› by auto
qed
also have "(⋃i. ?A i) = (space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}"
by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric])
finally have "(space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞} ∈ null_sets M" by simp }
from this[OF borel(1) refl] this[OF borel(2) f]
have "(space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞} ∈ null_sets M" "(space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞} ∈ null_sets M" by simp_all
then show "((space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}) ∪ ((space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞}) ∈ null_sets M" by (rule null_sets.Un)
show "{x ∈ space M. ?f (space M - (⋃i. Q i)) x ≠ ?f' (space M - (⋃i. Q i)) x} ⊆
((space M - (⋃i. Q i)) ∩ {x∈space M. f x ≠ ∞}) ∪ ((space M - (⋃i. Q i)) ∩ {x∈space M. f' x ≠ ∞})" by (auto simp: indicator_def)
qed
moreover have "AE x in M. (?f (space M - (⋃i. Q i)) x = ?f' (space M - (⋃i. Q i)) x) ⟶ (∀i. ?f (Q i) x = ?f' (Q i) x) ⟶
?f (space M) x = ?f' (space M) x"
by (auto simp: indicator_def)
ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
unfolding AE_all_countable[symmetric]
by eventually_elim (auto split: if_split_asm simp: indicator_def)
then show "AE x in M. f x = f' x" by auto
qed

proposition (in sigma_finite_measure) density_unique:
assumes f: "f ∈ borel_measurable M"
assumes f': "f' ∈ borel_measurable M"
assumes density_eq: "density M f = density M f'"
shows "AE x in M. f x = f' x"
proof -
obtain h where h_borel: "h ∈ borel_measurable M"
and fin: "integral⇧N M h ≠ ∞" and pos: "⋀x. x ∈ space M ⟹ 0 < h x ∧ h x < ∞" "⋀x. 0 ≤ h x"
using Ex_finite_integrable_function by auto
then have h_nn: "AE x in M. 0 ≤ h x" by auto
let ?H = "density M h"
interpret h: finite_measure ?H
using fin h_borel pos
by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin)
let ?fM = "density M f"
let ?f'M = "density M f'"
{ fix A assume "A ∈ sets M"
then have "{x ∈ space M. h x * indicator A x ≠ 0} = A"
using pos(1) sets.sets_into_space by (force simp: indicator_def)
then have "(∫⇧+x. h x * indicator A x ∂M) = 0 ⟷ A ∈ null_sets M"
using h_borel ‹A ∈ sets M› h_nn by (subst nn_integral_0_iff) auto }
note h_null_sets = this
{ fix A assume "A ∈ sets M"
have "(∫⇧+x. f x * (h x * indicator A x) ∂M) = (∫⇧+x. h x * indicator A x ∂?fM)"
using ‹A ∈ sets M› h_borel h_nn f f'
by (intro nn_integral_density[symmetric]) auto
also have "… = (∫⇧+x. h x * indicator A x ∂?f'M)"
also have "… = (∫⇧+x. f' x * (h x * indicator A x) ∂M)"
using ‹A ∈ sets M› h_borel h_nn f f'
by (intro nn_integral_density) auto
finally have "(∫⇧+x. h x * (f x * indicator A x) ∂M) = (∫⇧+x. h x * (f' x * indicator A x) ∂M)"
then have "(∫⇧+x. (f x * indicator A x) ∂?H) = (∫⇧+x. (f' x * indicator A x) ∂?H)"
using ‹A ∈ sets M› h_borel h_nn f f'
by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto
with AE_space[of M] pos show "AE x in M. f x = f' x"
unfolding AE_density[OF h_borel] by auto
qed

lemma (in sigma_finite_measure) density_unique_iff:
assumes f: "f ∈ borel_measurable M" and f': "f' ∈ borel_measurable M"
shows "density M f = density M f' ⟷ (AE x in M. f x = f' x)"
using density_unique[OF assms] density_cong[OF f f'] by auto

lemma sigma_finite_density_unique:
assumes borel: "f ∈ borel_measurable M" "g ∈ borel_measurable M"
and fin: "sigma_finite_measure (density M f)"
shows "density M f = density M g ⟷ (AE x in M. f x = g x)"
proof
assume "AE x in M. f x = g x" with borel show "density M f = density M g"
by (auto intro: density_cong)
next
assume eq: "density M f = density M g"
interpret f: sigma_finite_measure "density M f" by fact
from f.sigma_finite_incseq obtain A where cover: "range A ⊆ sets (density M f)"
"⋃ (range A) = space (density M f)"
"⋀i. emeasure (density M f) (A i) ≠ ∞"
"incseq A"
by auto
have "AE x in M. ∀i. x ∈ A i ⟶ f x = g x"
unfolding AE_all_countable
proof
fix i
have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))"
unfolding eq ..
moreover have "(∫⇧+x. f x * indicator (A i) x ∂M) ≠ ∞"
using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq)
ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x"
using borel cover(1)
by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq)
then show "AE x in M. x ∈ A i ⟶ f x = g x"
by auto
qed
with AE_space show "AE x in M. f x = g x"
apply eventually_elim
using cover(2)[symmetric]
apply auto
done
qed

lemma (in sigma_finite_measure) sigma_finite_iff_density_finite':
assumes f: "f ∈ borel_measurable M"
shows "sigma_finite_measure (density M f) ⟷ (AE x in M. f x ≠ ∞)"
(is "sigma_finite_measure ?N ⟷ _")
proof
assume "sigma_finite_measure ?N"
then interpret N: sigma_finite_measure ?N .
from N.Ex_finite_integrable_function obtain h where
h: "h ∈ borel_measurable M" "integral⇧N ?N h ≠ ∞" and
fin: "∀x∈space M. 0 < h x ∧ h x < ∞"
by auto
have "AE x in M. f x * h x ≠ ∞"
proof (rule AE_I')
have "integral⇧N ?N h = (∫⇧+x. f x * h x ∂M)"
using f h by (auto intro!: nn_integral_density)
then have "(∫⇧+x. f x * h x ∂M) ≠ ∞"
using h(2) by simp
then show "(λx. f x * h x) -` {∞} ∩ space M ∈ null_sets M"
using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage)
qed auto
then show "AE x in M. f x ≠ ∞"
using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top)
next
assume AE: "AE x in M. f x ≠ ∞"
from sigma_finite obtain Q :: "nat ⇒ 'a set"
where Q: "range Q ⊆ sets M" "⋃ (range Q) = space M" "⋀i. emeasure M (Q i) ≠ ∞"
by auto
define A where "A i =
f -` (case i of 0 ⇒ {∞} | Suc n ⇒ {.. ennreal(of_nat (Suc n))}) ∩ space M" for i
{ fix i j have "A i ∩ Q j ∈ sets M"
unfolding A_def using f Q
apply (rule_tac sets.Int)
by (cases i) (auto intro: measurable_sets[OF f(1)]) }
note A_in_sets = this

show "sigma_finite_measure ?N"
proof (standard, intro exI conjI ballI)
show "countable (range (λ(i, j). A i ∩ Q j))"
by auto
show "range (λ(i, j). A i ∩ Q j) ⊆ sets (density M f)"
using A_in_sets by auto
next
have "⋃(range (λ(i, j). A i ∩ Q j)) = (⋃i j. A i ∩ Q j)"
by auto
also have "… = (⋃i. A i) ∩ space M" using Q by auto
also have "(⋃i. A i) = space M"
proof safe
fix x assume x: "x ∈ space M"
show "x ∈ (⋃i. A i)"
proof (cases "f x" rule: ennreal_cases)
case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
next
case (real r)
with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n"
by auto
also have "n < (Suc n :: ennreal)"
by simp
finally show ?thesis
using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"])
qed
qed (auto simp: A_def)
finally show "⋃(range (λ(i, j). A i ∩ Q j)) = space ?N" by simp
next
fix X assume "X ∈ range (λ(i, j). A i ∩ Q j)"
then obtain i j where [simp]:"X = A i ∩ Q j" by auto
have "(∫⇧+x. f x * indicator (A i ∩ Q j) x ∂M) ≠ ∞"
proof (cases i)
case 0
have "AE x in M. f x * indicator (A i ∩ Q j) x = 0"
using AE by (auto simp: A_def ‹i = 0›)
from nn_integral_cong_AE[OF this] show ?thesis by simp
next
case (Suc n)
then have "(∫⇧+x. f x * indicator (A i ∩ Q j) x ∂M) ≤
(∫⇧+x. (Suc n :: ennreal) * indicator (Q j) x ∂M)"
by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat)
also have "… = Suc n * emeasure M (Q j)"
using Q by (auto intro!: nn_integral_cmult_indicator)
also have "… < ∞"
using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top)
finally show ?thesis by simp
qed
then show "emeasure ?N X ≠ ∞"
using A_in_sets Q f by (auto simp: emeasure_density)
qed
qed

lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
"f ∈ borel_measurable M ⟹ sigma_finite_measure (density M f) ⟷ (AE x in M. f x ≠ ∞)"
by (subst sigma_finite_iff_density_finite')
(auto simp: max_def intro!: measurable_If)

definition✐‹tag important› RN_deriv :: "'a measure ⇒ 'a measure ⇒ 'a ⇒ ennreal" where
"RN_deriv M N =
(if ∃f. f ∈ borel_measurable M ∧ density M f = N
then SOME f. f ∈ borel_measurable M ∧ density M f = N
else (λ_. 0))"

lemma RN_derivI:
assumes "f ∈ borel_measurable M" "density M f = N"
shows "density M (RN_deriv M N) = N"
proof -
have *: "∃f. f ∈ borel_measurable M ∧ density M f = N"
using assms by auto
then have "density M (SOME f. f ∈ borel_measurable M ∧ density M f = N) = N"
by (rule someI2_ex) auto
with * show ?thesis
by (auto simp: RN_deriv_def)
qed

lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N ∈ borel_measurable M"
proof -
{ assume ex: "∃f. f ∈ borel_measurable M ∧ density M f = N"
have 1: "(SOME f. f ∈ borel_measurable M ∧ density M f = N) ∈ borel_measurable M"
using ex by (rule someI2_ex) auto }
from this show ?thesis
by (auto simp: RN_deriv_def)
qed

lemma density_RN_deriv_density:
assumes f: "f ∈ borel_measurable M"
shows "density M (RN_deriv M (density M f)) = density M f"
by (rule RN_derivI[OF f]) simp

lemma (in sigma_finite_measure) density_RN_deriv:
"absolutely_continuous M N ⟹ sets N = sets M ⟹ density M (RN_deriv M N) = N"

lemma (in sigma_finite_measure) RN_deriv_nn_integral:
assumes N: "absolutely_continuous M N" "sets N = sets M"
and f: "f ∈ borel_measurable M"
shows "integral⇧N N f = (∫⇧+x. RN_deriv M N x * f x ∂M)"
proof -
have "integral⇧N N f = integral⇧N (density M (RN_deriv M N)) f"
using N by (simp add: density_RN_deriv)
also have "… = (∫⇧+x. RN_deriv M N x * f x ∂M)"
using f by (simp add: nn_integral_density)
finally show ?thesis by simp
qed

lemma (in sigma_finite_measure) RN_deriv_unique:
assumes f: "f ∈ borel_measurable M"
and eq: "density M f = N"
shows "AE x in M. f x = RN_deriv M N x"
unfolding eq[symmetric]
by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv
density_RN_deriv_density[symmetric])

lemma RN_deriv_unique_sigma_finite:
assumes f: "f ∈ borel_measurable M"
and eq: "density M f = N" and fin: "sigma_finite_measure N"
shows "AE x in M. f x = RN_deriv M N x"
using fin unfolding eq[symmetric]
by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv
density_RN_deriv_density[symmetric])

lemma (in sigma_finite_measure) RN_deriv_distr:
fixes T :: "'a ⇒ 'b"
assumes T: "T ∈ measurable M M'" and T': "T' ∈ measurable M' M"
and inv: "∀x∈space M. T' (T x) = x"
and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)"
and N: "sets N = sets M"
shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
proof (rule RN_deriv_unique)
have [simp]: "sets N = sets M" by fact
note sets_eq_imp_space_eq[OF N, simp]
have measurable_N[simp]: "⋀M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
{ fix A assume "A ∈ sets M"
with inv T T' sets.sets_into_space[OF this]
have "T -` T' -` A ∩ T -` space M' ∩ space M = A"
by (auto simp: measurable_def) }
note eq = this[simp]
{ fix A assume "A ∈ sets M"
with inv T T' sets.sets_into_space[OF this]
have "(T' ∘ T) -` A ∩ space M = A"
by (auto simp: measurable_def) }
note eq2 = this[simp]
let ?M' = "distr M M' T" and ?N' = "distr N M' T"
interpret M': sigma_finite_measure ?M'
proof
from sigma_finite_countable obtain F
where F: "countable F ∧ F ⊆ sets M ∧ ⋃ F = space M ∧ (∀a∈F. emeasure M a ≠ ∞)" ..
show "∃A. countable A ∧ A ⊆ sets (distr M M' T) ∧ ⋃A = space (distr M M' T) ∧ (∀a∈A. emeasure (distr M M' T) a ≠ ∞)"
proof (intro exI conjI ballI)
show *: "(λA. T' -` A ∩ space ?M') ` F ⊆ sets ?M'"
using F T' by (auto simp: measurable_def)
show "⋃((λA. T' -` A ∩ space ?M')`F) = space ?M'"
using F T'[THEN measurable_space] by (auto simp: set_eq_iff)
next
fix X assume "X ∈ (λA. T' -` A ∩ space ?M')`F"
then obtain A where [simp]: "X = T' -` A ∩ space ?M'" and "A ∈ F" by auto
have "X ∈ sets M'" using F T' ‹A∈F› by auto
moreover
have Fi: "A ∈ sets M" using F ‹A∈F› by auto
ultimately show "emeasure ?M' X ≠ ∞"
using F T T' ‹A∈F› by (simp add: emeasure_distr)
qed (use F in auto)
qed
have "(RN_deriv ?M' ?N') ∘ T ∈ borel_measurable M"
using T ac by measurable
then show "(λx. RN_deriv ?M' ?N' (T x)) ∈ borel_measurable M"

have "N = distr N M (T' ∘ T)"
by (subst measure_of_of_measure[of N, symmetric])
(auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed)
also have "… = distr (distr N M' T) M T'"
using T T' by (simp add: distr_distr)
also have "… = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
using ac by (simp add: M'.density_RN_deriv)
also have "… = density M (RN_deriv (distr M M' T) (distr N M' T) ∘ T)"
by (simp add: distr_density_distr[OF T T', OF inv])
finally show "density M (λx. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
qed

lemma (in sigma_finite_measure) RN_deriv_finite:
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
shows "AE x in M. RN_deriv M N x ≠ ∞"
proof -
interpret N: sigma_finite_measure N by fact
from N show ?thesis
using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac]
by simp
qed

lemma (in sigma_finite_measure)
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
and f: "f ∈ borel_measurable M"
shows RN_deriv_integrable: "integrable N f ⟷
integrable M (λx. enn2real (RN_deriv M N x) * f x)" (is ?integrable)
and RN_deriv_integral: "integral⇧L N f = (∫x. enn2real (RN_deriv M N x) * f x ∂M)" (is ?integral)
proof -
note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
interpret N: sigma_finite_measure N by fact

have eq: "density M (RN_deriv M N) = density M (λx. enn2real (RN_deriv M N x))"
proof (rule density_cong)
from RN_deriv_finite[OF assms(1,2,3)]
show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
by eventually_elim (auto simp: less_top)
qed (insert ac, auto)

show ?integrable
apply (subst density_RN_deriv[OF ac, symmetric])
unfolding eq
apply (intro integrable_real_density f AE_I2 enn2real_nonneg)
apply (insert ac, auto)
done

show ?integral
apply (subst density_RN_deriv[OF ac, symmetric])
unfolding eq
apply (intro integral_real_density f AE_I2 enn2real_nonneg)
apply (insert ac, auto)
done
qed

proposition (in sigma_finite_measure) real_RN_deriv:
assumes "finite_measure N"
assumes ac: "absolutely_continuous M N" "sets N = sets M"
obtains D where "D ∈ borel_measurable M"
and "AE x in M. RN_deriv M N x = ennreal (D x)"
and "AE x in N. 0 < D x"
and "⋀x. 0 ≤ D x"
proof
interpret N: finite_measure N by fact

note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac]

let ?RN = "λt. {x ∈ space M. RN_deriv M N x = t}"

show "(λx. enn2real (RN_deriv M N x)) ∈ borel_measurable M"
using RN by auto

have "N (?RN ∞) = (∫⇧+ x. RN_deriv M N x * indicator (?RN ∞) x ∂M)"
using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "… = (∫⇧+ x. ∞ * indicator (?RN ∞) x ∂M)"
by (intro nn_integral_cong) (auto simp: indicator_def)
also have "… = ∞ * emeasure M (?RN ∞)"
using RN by (intro nn_integral_cmult_indicator) auto
finally have eq: "N (?RN ∞) = ∞ * emeasure M (?RN ∞)" .
moreover
have "emeasure M (?RN ∞) = 0"
proof (rule ccontr)
assume "emeasure M {x ∈ space M. RN_deriv M N x = ∞} ≠ 0"
then have "0 < emeasure M {x ∈ space M. RN_deriv M N x = ∞}"
by (auto simp: zero_less_iff_neq_zero)
with eq have "N (?RN ∞) = ∞" by (simp add: ennreal_mult_eq_top_iff)
with N.emeasure_finite[of "?RN ∞"] RN show False by auto
qed
ultimately have "AE x in M. RN_deriv M N x < ∞"
using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric])
then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
by auto
then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
using ac absolutely_continuous_AE by auto

have "N (?RN 0) = (∫⇧+ x. RN_deriv M N x * indicator (?RN 0) x ∂M)"
by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
also have "… = (∫⇧+ x. 0 ∂M)"
by (intro nn_integral_cong) (auto simp: indicator_def)
finally have "AE x in N. RN_deriv M N x ≠ 0"
using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)"
by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero)
qed (rule enn2real_nonneg)

lemma (in sigma_finite_measure) RN_deriv_singleton:
assumes ac: "absolutely_continuous M N" "sets N = sets M"
and x: "{x} ∈ sets M"
shows "N {x} = RN_deriv M N x * emeasure M {x}"
proof -
from ‹{x} ∈ sets M›
have "density M (RN_deriv M N) {x} = (∫⇧+w. RN_deriv M N x * indicator {x} w ∂M)"
by (```