# Theory Nonnegative_Lebesgue_Integration

theory Nonnegative_Lebesgue_Integration
imports Measure_Space Borel_Space
```(*  Title:      HOL/Analysis/Nonnegative_Lebesgue_Integration.thy
Author:     Johannes Hölzl, TU München
Author:     Armin Heller, TU München
*)

section ‹Lebesgue Integration for Nonnegative Functions›

theory Nonnegative_Lebesgue_Integration
imports Measure_Space Borel_Space
begin

subsection✐‹tag unimportant› ‹Approximating functions›

lemma AE_upper_bound_inf_ennreal:
fixes F G::"'a ⇒ ennreal"
assumes "⋀e. (e::real) > 0 ⟹ AE x in M. F x ≤ G x + e"
shows "AE x in M. F x ≤ G x"
proof -
have "AE x in M. ∀n::nat. F x ≤ G x + ennreal (1 / Suc n)"
using assms by (auto simp: AE_all_countable)
then show ?thesis
proof (eventually_elim)
fix x assume x: "∀n::nat. F x ≤ G x + ennreal (1 / Suc n)"
show "F x ≤ G x"
proof (rule ennreal_le_epsilon)
fix e :: real assume "0 < e"
then obtain n where n: "1 / Suc n < e"
by (blast elim: nat_approx_posE)
have "F x ≤ G x + 1 / Suc n"
using x by simp
also have "… ≤ G x + e"
using n by (intro add_mono ennreal_leI) auto
finally show "F x ≤ G x + ennreal e" .
qed
qed
qed

lemma AE_upper_bound_inf:
fixes F G::"'a ⇒ real"
assumes "⋀e. e > 0 ⟹ AE x in M. F x ≤ G x + e"
shows "AE x in M. F x ≤ G x"
proof -
have "AE x in M. F x ≤ G x + 1/real (n+1)" for n::nat
by (rule assms, auto)
then have "AE x in M. ∀n::nat ∈ UNIV. F x ≤ G x + 1/real (n+1)"
by (rule AE_ball_countable', auto)
moreover
{
fix x assume i: "∀n::nat ∈ UNIV. F x ≤ G x + 1/real (n+1)"
have "(λn. G x + 1/real (n+1)) ⇢ G x + 0"
by (rule tendsto_add, simp, rule LIMSEQ_ignore_initial_segment[OF lim_1_over_n, of 1])
then have "F x ≤ G x" using i LIMSEQ_le_const by fastforce
}
ultimately show ?thesis by auto
qed

lemma not_AE_zero_ennreal_E:
fixes f::"'a ⇒ ennreal"
assumes "¬ (AE x in M. f x = 0)" and [measurable]: "f ∈ borel_measurable M"
shows "∃A∈sets M. ∃e::real>0. emeasure M A > 0 ∧ (∀x ∈ A. f x ≥ e)"
proof -
{ assume "¬ (∃e::real>0. {x ∈ space M. f x ≥ e} ∉ null_sets M)"
then have "0 < e ⟹ AE x in M. f x ≤ e" for e :: real
by (auto simp: not_le less_imp_le dest!: AE_not_in)
then have "AE x in M. f x ≤ 0"
by (intro AE_upper_bound_inf_ennreal[where G="λ_. 0"]) simp
then have False
using assms by auto }
then obtain e::real where e: "e > 0" "{x ∈ space M. f x ≥ e} ∉ null_sets M" by auto
define A where "A = {x ∈ space M. f x ≥ e}"
have 1 [measurable]: "A ∈ sets M" unfolding A_def by auto
have 2: "emeasure M A > 0"
using e(2) A_def ‹A ∈ sets M› by auto
have 3: "⋀x. x ∈ A ⟹ f x ≥ e" unfolding A_def by auto
show ?thesis using e(1) 1 2 3 by blast
qed

lemma not_AE_zero_E:
fixes f::"'a ⇒ real"
assumes "AE x in M. f x ≥ 0"
"¬(AE x in M. f x = 0)"
and [measurable]: "f ∈ borel_measurable M"
shows "∃A e. A ∈ sets M ∧ e>0 ∧ emeasure M A > 0 ∧ (∀x ∈ A. f x ≥ e)"
proof -
have "∃e. e > 0 ∧ {x ∈ space M. f x ≥ e} ∉ null_sets M"
proof (rule ccontr)
assume *: "¬(∃e. e > 0 ∧ {x ∈ space M. f x ≥ e} ∉ null_sets M)"
{
fix e::real assume "e > 0"
then have "{x ∈ space M. f x ≥ e} ∈ null_sets M" using * by blast
then have "AE x in M. x ∉ {x ∈ space M. f x ≥ e}" using AE_not_in by blast
then have "AE x in M. f x ≤ e" by auto
}
then have "AE x in M. f x ≤ 0" by (rule AE_upper_bound_inf, auto)
then have "AE x in M. f x = 0" using assms(1) by auto
then show False using assms(2) by auto
qed
then obtain e where e: "e>0" "{x ∈ space M. f x ≥ e} ∉ null_sets M" by auto
define A where "A = {x ∈ space M. f x ≥ e}"
have 1 [measurable]: "A ∈ sets M" unfolding A_def by auto
have 2: "emeasure M A > 0"
using e(2) A_def ‹A ∈ sets M› by auto
have 3: "⋀x. x ∈ A ⟹ f x ≥ e" unfolding A_def by auto
show ?thesis
using e(1) 1 2 3 by blast
qed

subsection "Simple function"

text ‹

Our simple functions are not restricted to nonnegative real numbers. Instead
they are just functions with a finite range and are measurable when singleton
sets are measurable.

›

definition✐‹tag important› "simple_function M g ⟷
finite (g ` space M) ∧
(∀x ∈ g ` space M. g -` {x} ∩ space M ∈ sets M)"

lemma simple_functionD:
assumes "simple_function M g"
shows "finite (g ` space M)" and "g -` X ∩ space M ∈ sets M"
proof -
show "finite (g ` space M)"
using assms unfolding simple_function_def by auto
have "g -` X ∩ space M = g -` (X ∩ g`space M) ∩ space M" by auto
also have "… = (⋃x∈X ∩ g`space M. g-`{x} ∩ space M)" by auto
finally show "g -` X ∩ space M ∈ sets M" using assms
by (auto simp del: UN_simps simp: simple_function_def)
qed

lemma measurable_simple_function[measurable_dest]:
"simple_function M f ⟹ f ∈ measurable M (count_space UNIV)"
unfolding simple_function_def measurable_def
proof safe
fix A assume "finite (f ` space M)" "∀x∈f ` space M. f -` {x} ∩ space M ∈ sets M"
then have "(⋃x∈f ` space M. if x ∈ A then f -` {x} ∩ space M else {}) ∈ sets M"
by (intro sets.finite_UN) auto
also have "(⋃x∈f ` space M. if x ∈ A then f -` {x} ∩ space M else {}) = f -` A ∩ space M"
by (auto split: if_split_asm)
finally show "f -` A ∩ space M ∈ sets M" .
qed simp

lemma borel_measurable_simple_function:
"simple_function M f ⟹ f ∈ borel_measurable M"
by (auto dest!: measurable_simple_function simp: measurable_def)

lemma simple_function_measurable2[intro]:
assumes "simple_function M f" "simple_function M g"
shows "f -` A ∩ g -` B ∩ space M ∈ sets M"
proof -
have "f -` A ∩ g -` B ∩ space M = (f -` A ∩ space M) ∩ (g -` B ∩ space M)"
by auto
then show ?thesis using assms[THEN simple_functionD(2)] by auto
qed

lemma simple_function_indicator_representation:
fixes f ::"'a ⇒ ennreal"
assumes f: "simple_function M f" and x: "x ∈ space M"
shows "f x = (∑y ∈ f ` space M. y * indicator (f -` {y} ∩ space M) x)"
(is "?l = ?r")
proof -
have "?r = (∑y ∈ f ` space M.
(if y = f x then y * indicator (f -` {y} ∩ space M) x else 0))"
by (auto intro!: sum.cong)
also have "... =  f x *  indicator (f -` {f x} ∩ space M) x"
using assms by (auto dest: simple_functionD simp: sum.delta)
also have "... = f x" using x by (auto simp: indicator_def)
finally show ?thesis by auto
qed

lemma simple_function_notspace:
"simple_function M (λx. h x * indicator (- space M) x::ennreal)" (is "simple_function M ?h")
proof -
have "?h ` space M ⊆ {0}" unfolding indicator_def by auto
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
have "?h -` {0} ∩ space M = space M" by auto
thus ?thesis unfolding simple_function_def by (auto simp add: image_constant_conv)
qed

lemma simple_function_cong:
assumes "⋀t. t ∈ space M ⟹ f t = g t"
shows "simple_function M f ⟷ simple_function M g"
proof -
have "⋀x. f -` {x} ∩ space M = g -` {x} ∩ space M"
using assms by auto
with assms show ?thesis
by (simp add: simple_function_def cong: image_cong)
qed

lemma simple_function_cong_algebra:
assumes "sets N = sets M" "space N = space M"
shows "simple_function M f ⟷ simple_function N f"
unfolding simple_function_def assms ..

lemma simple_function_borel_measurable:
fixes f :: "'a ⇒ 'x::{t2_space}"
assumes "f ∈ borel_measurable M" and "finite (f ` space M)"
shows "simple_function M f"
using assms unfolding simple_function_def
by (auto intro: borel_measurable_vimage)

lemma simple_function_iff_borel_measurable:
fixes f :: "'a ⇒ 'x::{t2_space}"
shows "simple_function M f ⟷ finite (f ` space M) ∧ f ∈ borel_measurable M"
by (metis borel_measurable_simple_function simple_functionD(1) simple_function_borel_measurable)

lemma simple_function_eq_measurable:
"simple_function M f ⟷ finite (f`space M) ∧ f ∈ measurable M (count_space UNIV)"
using measurable_simple_function[of M f] by (fastforce simp: simple_function_def)

lemma simple_function_const[intro, simp]:
"simple_function M (λx. c)"
by (auto intro: finite_subset simp: simple_function_def)
lemma simple_function_compose[intro, simp]:
assumes "simple_function M f"
shows "simple_function M (g ∘ f)"
unfolding simple_function_def
proof safe
show "finite ((g ∘ f) ` space M)"
using assms unfolding simple_function_def image_comp [symmetric] by auto
next
fix x assume "x ∈ space M"
let ?G = "g -` {g (f x)} ∩ (f`space M)"
have *: "(g ∘ f) -` {(g ∘ f) x} ∩ space M =
(⋃x∈?G. f -` {x} ∩ space M)" by auto
show "(g ∘ f) -` {(g ∘ f) x} ∩ space M ∈ sets M"
using assms unfolding simple_function_def *
by (rule_tac sets.finite_UN) auto
qed

lemma simple_function_indicator[intro, simp]:
assumes "A ∈ sets M"
shows "simple_function M (indicator A)"
proof -
have "indicator A ` space M ⊆ {0, 1}" (is "?S ⊆ _")
by (auto simp: indicator_def)
hence "finite ?S" by (rule finite_subset) simp
moreover have "- A ∩ space M = space M - A" by auto
ultimately show ?thesis unfolding simple_function_def
using assms by (auto simp: indicator_def [abs_def])
qed

lemma simple_function_Pair[intro, simp]:
assumes "simple_function M f"
assumes "simple_function M g"
shows "simple_function M (λx. (f x, g x))" (is "simple_function M ?p")
unfolding simple_function_def
proof safe
show "finite (?p ` space M)"
using assms unfolding simple_function_def
by (rule_tac finite_subset[of _ "f`space M × g`space M"]) auto
next
fix x assume "x ∈ space M"
have "(λx. (f x, g x)) -` {(f x, g x)} ∩ space M =
(f -` {f x} ∩ space M) ∩ (g -` {g x} ∩ space M)"
by auto
with ‹x ∈ space M› show "(λx. (f x, g x)) -` {(f x, g x)} ∩ space M ∈ sets M"
using assms unfolding simple_function_def by auto
qed

lemma simple_function_compose1:
assumes "simple_function M f"
shows "simple_function M (λx. g (f x))"
using simple_function_compose[OF assms, of g]

lemma simple_function_compose2:
assumes "simple_function M f" and "simple_function M g"
shows "simple_function M (λx. h (f x) (g x))"
proof -
have "simple_function M ((λ(x, y). h x y) ∘ (λx. (f x, g x)))"
using assms by auto
thus ?thesis by (simp_all add: comp_def)
qed

lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="(+)"]
and simple_function_diff[intro, simp] = simple_function_compose2[where h="(-)"]
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
and simple_function_mult[intro, simp] = simple_function_compose2[where h="(*)"]
and simple_function_div[intro, simp] = simple_function_compose2[where h="(/)"]
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
and simple_function_max[intro, simp] = simple_function_compose2[where h=max]

lemma simple_function_sum[intro, simp]:
assumes "⋀i. i ∈ P ⟹ simple_function M (f i)"
shows "simple_function M (λx. ∑i∈P. f i x)"
proof cases
assume "finite P" from this assms show ?thesis by induct auto
qed auto

lemma simple_function_ennreal[intro, simp]:
fixes f g :: "'a ⇒ real" assumes sf: "simple_function M f"
shows "simple_function M (λx. ennreal (f x))"
by (rule simple_function_compose1[OF sf])

lemma simple_function_real_of_nat[intro, simp]:
fixes f g :: "'a ⇒ nat" assumes sf: "simple_function M f"
shows "simple_function M (λx. real (f x))"
by (rule simple_function_compose1[OF sf])

lemma✐‹tag important› borel_measurable_implies_simple_function_sequence:
fixes u :: "'a ⇒ ennreal"
assumes u[measurable]: "u ∈ borel_measurable M"
shows "∃f. incseq f ∧ (∀i. (∀x. f i x < top) ∧ simple_function M (f i)) ∧ u = (SUP i. f i)"
proof -
define f where [abs_def]:
"f i x = real_of_int (floor (enn2real (min i (u x)) * 2^i)) / 2^i" for i x

have [simp]: "0 ≤ f i x" for i x
by (auto simp: f_def intro!: divide_nonneg_nonneg mult_nonneg_nonneg enn2real_nonneg)

have *: "2^n * real_of_int x = real_of_int (2^n * x)" for n x
by simp

have "real_of_int ⌊real i * 2 ^ i⌋ = real_of_int ⌊i * 2 ^ i⌋" for i
by (intro arg_cong[where f=real_of_int]) simp
then have [simp]: "real_of_int ⌊real i * 2 ^ i⌋ = i * 2 ^ i" for i
unfolding floor_of_nat by simp

have "incseq f"
proof (intro monoI le_funI)
fix m n :: nat and x assume "m ≤ n"
moreover
{ fix d :: nat
have "⌊2^d::real⌋ * ⌊2^m * enn2real (min (of_nat m) (u x))⌋ ≤
⌊2^d * (2^m * enn2real (min (of_nat m) (u x)))⌋"
by (rule le_mult_floor) (auto simp: enn2real_nonneg)
also have "… ≤ ⌊2^d * (2^m * enn2real (min (of_nat d + of_nat m) (u x)))⌋"
by (intro floor_mono mult_mono enn2real_mono min.mono)
(auto simp: enn2real_nonneg min_less_iff_disj of_nat_less_top)
finally have "f m x ≤ f (m + d) x"
unfolding f_def
by (auto simp: field_simps power_add * simp del: of_int_mult) }
ultimately show "f m x ≤ f n x"
qed
then have inc_f: "incseq (λi. ennreal (f i x))" for x
by (auto simp: incseq_def le_fun_def)
then have "incseq (λi x. ennreal (f i x))"
by (auto simp: incseq_def le_fun_def)
moreover
have "simple_function M (f i)" for i
proof (rule simple_function_borel_measurable)
have "⌊enn2real (min (of_nat i) (u x)) * 2 ^ i⌋ ≤ ⌊int i * 2 ^ i⌋" for x
by (cases "u x" rule: ennreal_cases)
(auto split: split_min intro!: floor_mono)
then have "f i ` space M ⊆ (λn. real_of_int n / 2^i) ` {0 .. of_nat i * 2^i}"
unfolding floor_of_int by (auto simp: f_def enn2real_nonneg intro!: imageI)
then show "finite (f i ` space M)"
by (rule finite_subset) auto
show "f i ∈ borel_measurable M"
unfolding f_def enn2real_def by measurable
qed
moreover
{ fix x
have "(SUP i. ennreal (f i x)) = u x"
proof (cases "u x" rule: ennreal_cases)
case top then show ?thesis
by (simp add: f_def inf_min[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric]
ennreal_SUP_of_nat_eq_top)
next
case (real r)
obtain n where "r ≤ of_nat n" using real_arch_simple by auto
then have min_eq_r: "∀⇩F x in sequentially. min (real x) r = r"
by (auto simp: eventually_sequentially intro!: exI[of _ n] split: split_min)

have "(λi. real_of_int ⌊min (real i) r * 2^i⌋ / 2^i) ⇢ r"
proof (rule tendsto_sandwich)
show "(λn. r - (1/2)^n) ⇢ r"
by (auto intro!: tendsto_eq_intros LIMSEQ_power_zero)
show "∀⇩F n in sequentially. real_of_int ⌊min (real n) r * 2 ^ n⌋ / 2 ^ n ≤ r"
using min_eq_r by eventually_elim (auto simp: field_simps)
have *: "r * (2 ^ n * 2 ^ n) ≤ 2^n + 2^n * real_of_int ⌊r * 2 ^ n⌋" for n
using real_of_int_floor_ge_diff_one[of "r * 2^n", THEN mult_left_mono, of "2^n"]
by (auto simp: field_simps)
show "∀⇩F n in sequentially. r - (1/2)^n ≤ real_of_int ⌊min (real n) r * 2 ^ n⌋ / 2 ^ n"
using min_eq_r by eventually_elim (insert *, auto simp: field_simps)
qed auto
then have "(λi. ennreal (f i x)) ⇢ ennreal r"
by (simp add: real f_def ennreal_of_nat_eq_real_of_nat min_ennreal)
from LIMSEQ_unique[OF LIMSEQ_SUP[OF inc_f] this]
show ?thesis
qed }
ultimately show ?thesis
by (intro exI [of _ "λi x. ennreal (f i x)"]) (auto simp add: image_comp)
qed

lemma borel_measurable_implies_simple_function_sequence':
fixes u :: "'a ⇒ ennreal"
assumes u: "u ∈ borel_measurable M"
obtains f where
"⋀i. simple_function M (f i)" "incseq f" "⋀i x. f i x < top" "⋀x. (SUP i. f i x) = u x"
using borel_measurable_implies_simple_function_sequence [OF u]
by (metis SUP_apply)

lemma✐‹tag important› simple_function_induct
[consumes 1, case_names cong set mult add, induct set: simple_function]:
fixes u :: "'a ⇒ ennreal"
assumes u: "simple_function M u"
assumes cong: "⋀f g. simple_function M f ⟹ simple_function M g ⟹ (AE x in M. f x = g x) ⟹ P f ⟹ P g"
assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)"
assumes mult: "⋀u c. P u ⟹ P (λx. c * u x)"
assumes add: "⋀u v. P u ⟹ P v ⟹ P (λx. v x + u x)"
shows "P u"
proof (rule cong)
from AE_space show "AE x in M. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x"
proof eventually_elim
fix x assume x: "x ∈ space M"
from simple_function_indicator_representation[OF u x]
show "(∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x" ..
qed
next
from u have "finite (u ` space M)"
unfolding simple_function_def by auto
then show "P (λx. ∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x)"
proof induct
case empty show ?case
using set[of "{}"] by (simp add: indicator_def[abs_def])
qed (auto intro!: add mult set simple_functionD u)
next
show "simple_function M (λx. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x))"
apply (subst simple_function_cong)
apply (rule simple_function_indicator_representation[symmetric])
apply (auto intro: u)
done
qed fact

lemma simple_function_induct_nn[consumes 1, case_names cong set mult add]:
fixes u :: "'a ⇒ ennreal"
assumes u: "simple_function M u"
assumes cong: "⋀f g. simple_function M f ⟹ simple_function M g ⟹ (⋀x. x ∈ space M ⟹ f x = g x) ⟹ P f ⟹ P g"
assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)"
assumes mult: "⋀u c. simple_function M u ⟹ P u ⟹ P (λx. c * u x)"
assumes add: "⋀u v. simple_function M u ⟹ P u ⟹ simple_function M v ⟹ (⋀x. x ∈ space M ⟹ u x = 0 ∨ v x = 0) ⟹ P v ⟹ P (λx. v x + u x)"
shows "P u"
proof -
show ?thesis
proof (rule cong)
fix x assume x: "x ∈ space M"
from simple_function_indicator_representation[OF u x]
show "(∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x" ..
next
show "simple_function M (λx. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x))"
apply (subst simple_function_cong)
apply (rule simple_function_indicator_representation[symmetric])
apply (auto intro: u)
done
next
from u have "finite (u ` space M)"
unfolding simple_function_def by auto
then show "P (λx. ∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x)"
proof induct
case empty show ?case
using set[of "{}"] by (simp add: indicator_def[abs_def])
next
case (insert x S)
{ fix z have "(∑y∈S. y * indicator (u -` {y} ∩ space M) z) = 0 ∨
x * indicator (u -` {x} ∩ space M) z = 0"
using insert by (subst sum_eq_0_iff) (auto simp: indicator_def) }
note disj = this
from insert show ?case
by (auto intro!: add mult set simple_functionD u simple_function_sum disj)
qed
qed fact
qed

lemma✐‹tag important› borel_measurable_induct
[consumes 1, case_names cong set mult add seq, induct set: borel_measurable]:
fixes u :: "'a ⇒ ennreal"
assumes u: "u ∈ borel_measurable M"
assumes cong: "⋀f g. f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ f x = g x) ⟹ P g ⟹ P f"
assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)"
assumes mult': "⋀u c. c < top ⟹ u ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ u x < top) ⟹ P u ⟹ P (λx. c * u x)"
assumes add: "⋀u v. u ∈ borel_measurable M⟹ (⋀x. x ∈ space M ⟹ u x < top) ⟹ P u ⟹ v ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ v x < top) ⟹ (⋀x. x ∈ space M ⟹ u x = 0 ∨ v x = 0) ⟹ P v ⟹ P (λx. v x + u x)"
assumes seq: "⋀U. (⋀i. U i ∈ borel_measurable M) ⟹ (⋀i x. x ∈ space M ⟹ U i x < top) ⟹ (⋀i. P (U i)) ⟹ incseq U ⟹ u = (SUP i. U i) ⟹ P (SUP i. U i)"
shows "P u"
using u
proof (induct rule: borel_measurable_implies_simple_function_sequence')
fix U assume U: "⋀i. simple_function M (U i)" "incseq U" "⋀i x. U i x < top" and sup: "⋀x. (SUP i. U i x) = u x"
have u_eq: "u = (SUP i. U i)"
using u by (auto simp add: image_comp sup)

have not_inf: "⋀x i. x ∈ space M ⟹ U i x < top"
using U by (auto simp: image_iff eq_commute)

from U have "⋀i. U i ∈ borel_measurable M"

show "P u"
unfolding u_eq
proof (rule seq)
fix i show "P (U i)"
using ‹simple_function M (U i)› not_inf[of _ i]
proof (induct rule: simple_function_induct_nn)
case (mult u c)
show ?case
proof cases
assume "c = 0 ∨ space M = {} ∨ (∀x∈space M. u x = 0)"
with mult(1) show ?thesis
by (intro cong[of "λx. c * u x" "indicator {}"] set)
(auto dest!: borel_measurable_simple_function)
next
assume "¬ (c = 0 ∨ space M = {} ∨ (∀x∈space M. u x = 0))"
then obtain x where "space M ≠ {}" and x: "x ∈ space M" "u x ≠ 0" "c ≠ 0"
by auto
with mult(3)[of x] have "c < top"
by (auto simp: ennreal_mult_less_top)
then have u_fin: "x' ∈ space M ⟹ u x' < top" for x'
using mult(3)[of x'] ‹c ≠ 0› by (auto simp: ennreal_mult_less_top)
then have "P u"
by (rule mult)
with u_fin ‹c < top› mult(1) show ?thesis
by (intro mult') (auto dest!: borel_measurable_simple_function)
qed
qed (auto intro: cong intro!: set add dest!: borel_measurable_simple_function)
qed fact+
qed

lemma simple_function_If_set:
assumes sf: "simple_function M f" "simple_function M g" and A: "A ∩ space M ∈ sets M"
shows "simple_function M (λx. if x ∈ A then f x else g x)" (is "simple_function M ?IF")
proof -
define F where "F x = f -` {x} ∩ space M" for x
define G where "G x = g -` {x} ∩ space M" for x
show ?thesis unfolding simple_function_def
proof safe
have "?IF ` space M ⊆ f ` space M ∪ g ` space M" by auto
from finite_subset[OF this] assms
show "finite (?IF ` space M)" unfolding simple_function_def by auto
next
fix x assume "x ∈ space M"
then have *: "?IF -` {?IF x} ∩ space M = (if x ∈ A
then ((F (f x) ∩ (A ∩ space M)) ∪ (G (f x) - (G (f x) ∩ (A ∩ space M))))
else ((F (g x) ∩ (A ∩ space M)) ∪ (G (g x) - (G (g x) ∩ (A ∩ space M)))))"
using sets.sets_into_space[OF A] by (auto split: if_split_asm simp: G_def F_def)
have [intro]: "⋀x. F x ∈ sets M" "⋀x. G x ∈ sets M"
unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
show "?IF -` {?IF x} ∩ space M ∈ sets M" unfolding * using A by auto
qed
qed

lemma simple_function_If:
assumes sf: "simple_function M f" "simple_function M g" and P: "{x∈space M. P x} ∈ sets M"
shows "simple_function M (λx. if P x then f x else g x)"
proof -
have "{x∈space M. P x} = {x. P x} ∩ space M" by auto
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
qed

lemma simple_function_subalgebra:
assumes "simple_function N f"
and N_subalgebra: "sets N ⊆ sets M" "space N = space M"
shows "simple_function M f"
using assms unfolding simple_function_def by auto

lemma simple_function_comp:
assumes T: "T ∈ measurable M M'"
and f: "simple_function M' f"
shows "simple_function M (λx. f (T x))"
proof (intro simple_function_def[THEN iffD2] conjI ballI)
have "(λx. f (T x)) ` space M ⊆ f ` space M'"
using T unfolding measurable_def by auto
then show "finite ((λx. f (T x)) ` space M)"
using f unfolding simple_function_def by (auto intro: finite_subset)
fix i assume i: "i ∈ (λx. f (T x)) ` space M"
then have "i ∈ f ` space M'"
using T unfolding measurable_def by auto
then have "f -` {i} ∩ space M' ∈ sets M'"
using f unfolding simple_function_def by auto
then have "T -` (f -` {i} ∩ space M') ∩ space M ∈ sets M"
using T unfolding measurable_def by auto
also have "T -` (f -` {i} ∩ space M') ∩ space M = (λx. f (T x)) -` {i} ∩ space M"
using T unfolding measurable_def by auto
finally show "(λx. f (T x)) -` {i} ∩ space M ∈ sets M" .
qed

subsection "Simple integral"

definition✐‹tag important› simple_integral :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ ennreal" ("integral⇧S") where
"integral⇧S M f = (∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M))"

syntax
"_simple_integral" :: "pttrn ⇒ ennreal ⇒ 'a measure ⇒ ennreal" ("∫⇧S _. _ ∂_" [60,61] 110)

translations
"∫⇧S x. f ∂M" == "CONST simple_integral M (%x. f)"

lemma simple_integral_cong:
assumes "⋀t. t ∈ space M ⟹ f t = g t"
shows "integral⇧S M f = integral⇧S M g"
proof -
have "f ` space M = g ` space M"
"⋀x. f -` {x} ∩ space M = g -` {x} ∩ space M"
using assms by (auto intro!: image_eqI)
thus ?thesis unfolding simple_integral_def by simp
qed

lemma simple_integral_const[simp]:
"(∫⇧Sx. c ∂M) = c * (emeasure M) (space M)"
proof (cases "space M = {}")
case True thus ?thesis unfolding simple_integral_def by simp
next
case False hence "(λx. c) ` space M = {c}" by auto
thus ?thesis unfolding simple_integral_def by simp
qed

lemma simple_function_partition:
assumes f: "simple_function M f" and g: "simple_function M g"
assumes sub: "⋀x y. x ∈ space M ⟹ y ∈ space M ⟹ g x = g y ⟹ f x = f y"
assumes v: "⋀x. x ∈ space M ⟹ f x = v (g x)"
shows "integral⇧S M f = (∑y∈g ` space M. v y * emeasure M {x∈space M. g x = y})"
(is "_ = ?r")
proof -
from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
by (auto simp: simple_function_def)
from f g have [measurable]: "f ∈ measurable M (count_space UNIV)" "g ∈ measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function)

{ fix y assume "y ∈ space M"
then have "f ` space M ∩ {i. ∃x∈space M. i = f x ∧ g y = g x} = {v (g y)}"
by (auto cong: sub simp: v[symmetric]) }
note eq = this

have "integral⇧S M f =
(∑y∈f`space M. y * (∑z∈g`space M.
if ∃x∈space M. y = f x ∧ z = g x then emeasure M {x∈space M. g x = z} else 0))"
unfolding simple_integral_def
proof (safe intro!: sum.cong ennreal_mult_left_cong)
fix y assume y: "y ∈ space M" "f y ≠ 0"
have [simp]: "g ` space M ∩ {z. ∃x∈space M. f y = f x ∧ z = g x} =
{z. ∃x∈space M. f y = f x ∧ z = g x}"
by auto
have eq:"(⋃i∈{z. ∃x∈space M. f y = f x ∧ z = g x}. {x ∈ space M. g x = i}) =
f -` {f y} ∩ space M"
by (auto simp: eq_commute cong: sub rev_conj_cong)
have "finite (g`space M)" by simp
then have "finite {z. ∃x∈space M. f y = f x ∧ z = g x}"
by (rule rev_finite_subset) auto
then show "emeasure M (f -` {f y} ∩ space M) =
(∑z∈g ` space M. if ∃x∈space M. f y = f x ∧ z = g x then emeasure M {x ∈ space M. g x = z} else 0)"
apply (subst sum_emeasure)
apply (auto simp: disjoint_family_on_def eq)
done
qed
also have "… = (∑y∈f`space M. (∑z∈g`space M.
if ∃x∈space M. y = f x ∧ z = g x then y * emeasure M {x∈space M. g x = z} else 0))"
by (auto intro!: sum.cong simp: sum_distrib_left)
also have "… = ?r"
by (subst sum.swap)
(auto intro!: sum.cong simp: sum.If_cases scaleR_sum_right[symmetric] eq)
finally show "integral⇧S M f = ?r" .
qed

assumes f: "simple_function M f" and "⋀x. 0 ≤ f x" and g: "simple_function M g" and "⋀x. 0 ≤ g x"
shows "(∫⇧Sx. f x + g x ∂M) = integral⇧S M f + integral⇧S M g"
proof -
have "(∫⇧Sx. f x + g x ∂M) =
(∑y∈(λx. (f x, g x))`space M. (fst y + snd y) * emeasure M {x∈space M. (f x, g x) = y})"
by (intro simple_function_partition) (auto intro: f g)
also have "… = (∑y∈(λx. (f x, g x))`space M. fst y * emeasure M {x∈space M. (f x, g x) = y}) +
(∑y∈(λx. (f x, g x))`space M. snd y * emeasure M {x∈space M. (f x, g x) = y})"
using assms(2,4) by (auto intro!: sum.cong distrib_right simp: sum.distrib[symmetric])
also have "(∑y∈(λx. (f x, g x))`space M. fst y * emeasure M {x∈space M. (f x, g x) = y}) = (∫⇧Sx. f x ∂M)"
by (intro simple_function_partition[symmetric]) (auto intro: f g)
also have "(∑y∈(λx. (f x, g x))`space M. snd y * emeasure M {x∈space M. (f x, g x) = y}) = (∫⇧Sx. g x ∂M)"
by (intro simple_function_partition[symmetric]) (auto intro: f g)
finally show ?thesis .
qed

lemma simple_integral_sum[simp]:
assumes "⋀i x. i ∈ P ⟹ 0 ≤ f i x"
assumes "⋀i. i ∈ P ⟹ simple_function M (f i)"
shows "(∫⇧Sx. (∑i∈P. f i x) ∂M) = (∑i∈P. integral⇧S M (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
by induct (auto simp: simple_function_sum simple_integral_add sum_nonneg)
qed auto

lemma simple_integral_mult[simp]:
assumes f: "simple_function M f"
shows "(∫⇧Sx. c * f x ∂M) = c * integral⇧S M f"
proof -
have "(∫⇧Sx. c * f x ∂M) = (∑y∈f ` space M. (c * y) * emeasure M {x∈space M. f x = y})"
using f by (intro simple_function_partition) auto
also have "… = c * integral⇧S M f"
using f unfolding simple_integral_def
by (subst sum_distrib_left) (auto simp: mult.assoc Int_def conj_commute)
finally show ?thesis .
qed

lemma simple_integral_mono_AE:
assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g"
and mono: "AE x in M. f x ≤ g x"
shows "integral⇧S M f ≤ integral⇧S M g"
proof -
let ?μ = "λP. emeasure M {x∈space M. P x}"
have "integral⇧S M f = (∑y∈(λx. (f x, g x))`space M. fst y * ?μ (λx. (f x, g x) = y))"
using f g by (intro simple_function_partition) auto
also have "… ≤ (∑y∈(λx. (f x, g x))`space M. snd y * ?μ (λx. (f x, g x) = y))"
proof (clarsimp intro!: sum_mono)
fix x assume "x ∈ space M"
let ?M = "?μ (λy. f y = f x ∧ g y = g x)"
show "f x * ?M ≤ g x * ?M"
proof cases
assume "?M ≠ 0"
then have "0 < ?M"
also have "… ≤ ?μ (λy. f x ≤ g x)"
using mono by (intro emeasure_mono_AE) auto
finally have "¬ ¬ f x ≤ g x"
by (intro notI) auto
then show ?thesis
by (intro mult_right_mono) auto
qed simp
qed
also have "… = integral⇧S M g"
using f g by (intro simple_function_partition[symmetric]) auto
finally show ?thesis .
qed

lemma simple_integral_mono:
assumes "simple_function M f" and "simple_function M g"
and mono: "⋀ x. x ∈ space M ⟹ f x ≤ g x"
shows "integral⇧S M f ≤ integral⇧S M g"
using assms by (intro simple_integral_mono_AE) auto

lemma simple_integral_cong_AE:
assumes "simple_function M f" and "simple_function M g"
and "AE x in M. f x = g x"
shows "integral⇧S M f = integral⇧S M g"
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)

lemma simple_integral_cong':
assumes sf: "simple_function M f" "simple_function M g"
and mea: "(emeasure M) {x∈space M. f x ≠ g x} = 0"
shows "integral⇧S M f = integral⇧S M g"
proof (intro simple_integral_cong_AE sf AE_I)
show "(emeasure M) {x∈space M. f x ≠ g x} = 0" by fact
show "{x ∈ space M. f x ≠ g x} ∈ sets M"
using sf[THEN borel_measurable_simple_function] by auto
qed simp

lemma simple_integral_indicator:
assumes A: "A ∈ sets M"
assumes f: "simple_function M f"
shows "(∫⇧Sx. f x * indicator A x ∂M) =
(∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M ∩ A))"
proof -
have eq: "(λx. (f x, indicator A x)) ` space M ∩ {x. snd x = 1} = (λx. (f x, 1::ennreal))`A"
using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: if_split_asm)
have eq2: "⋀x. f x ∉ f ` A ⟹ f -` {f x} ∩ space M ∩ A = {}"
by (auto simp: image_iff)

have "(∫⇧Sx. f x * indicator A x ∂M) =
(∑y∈(λx. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x∈space M. (f x, indicator A x) = y})"
using assms by (intro simple_function_partition) auto
also have "… = (∑y∈(λx. (f x, indicator A x::ennreal))`space M.
if snd y = 1 then fst y * emeasure M (f -` {fst y} ∩ space M ∩ A) else 0)"
by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="(*)"] arg_cong2[where f=emeasure] sum.cong)
also have "… = (∑y∈(λx. (f x, 1::ennreal))`A. fst y * emeasure M (f -` {fst y} ∩ space M ∩ A))"
using assms by (subst sum.If_cases) (auto intro!: simple_functionD(1) simp: eq)
also have "… = (∑y∈fst`(λx. (f x, 1::ennreal))`A. y * emeasure M (f -` {y} ∩ space M ∩ A))"
by (subst sum.reindex [of fst]) (auto simp: inj_on_def)
also have "… = (∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M ∩ A))"
using A[THEN sets.sets_into_space]
by (intro sum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
finally show ?thesis .
qed

lemma simple_integral_indicator_only[simp]:
assumes "A ∈ sets M"
shows "integral⇧S M (indicator A) = emeasure M A"
using simple_integral_indicator[OF assms, of "λx. 1"] sets.sets_into_space[OF assms]
by (simp_all add: image_constant_conv Int_absorb1 split: if_split_asm)

lemma simple_integral_null_set:
assumes "simple_function M u" "⋀x. 0 ≤ u x" and "N ∈ null_sets M"
shows "(∫⇧Sx. u x * indicator N x ∂M) = 0"
proof -
have "AE x in M. indicator N x = (0 :: ennreal)"
using ‹N ∈ null_sets M› by (auto simp: indicator_def intro!: AE_I[of _ _ N])
then have "(∫⇧Sx. u x * indicator N x ∂M) = (∫⇧Sx. 0 ∂M)"
using assms apply (intro simple_integral_cong_AE) by auto
then show ?thesis by simp
qed

lemma simple_integral_cong_AE_mult_indicator:
assumes sf: "simple_function M f" and eq: "AE x in M. x ∈ S" and "S ∈ sets M"
shows "integral⇧S M f = (∫⇧Sx. f x * indicator S x ∂M)"
using assms by (intro simple_integral_cong_AE) auto

lemma simple_integral_cmult_indicator:
assumes A: "A ∈ sets M"
shows "(∫⇧Sx. c * indicator A x ∂M) = c * emeasure M A"
using simple_integral_mult[OF simple_function_indicator[OF A]]
unfolding simple_integral_indicator_only[OF A] by simp

lemma simple_integral_nonneg:
assumes f: "simple_function M f" and ae: "AE x in M. 0 ≤ f x"
shows "0 ≤ integral⇧S M f"
proof -
have "integral⇧S M (λx. 0) ≤ integral⇧S M f"
using simple_integral_mono_AE[OF _ f ae] by auto
then show ?thesis by simp
qed

subsection ‹Integral on nonnegative functions›

definition✐‹tag important› nn_integral :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ ennreal" ("integral⇧N") where
"integral⇧N M f = (SUP g ∈ {g. simple_function M g ∧ g ≤ f}. integral⇧S M g)"

syntax
"_nn_integral" :: "pttrn ⇒ ennreal ⇒ 'a measure ⇒ ennreal" ("∫⇧+((2 _./ _)/ ∂_)" [60,61] 110)

translations
"∫⇧+x. f ∂M" == "CONST nn_integral M (λx. f)"

lemma nn_integral_def_finite:
"integral⇧N M f = (SUP g ∈ {g. simple_function M g ∧ g ≤ f ∧ (∀x. g x < top)}. integral⇧S M g)"
(is "_ = Sup (?A ` ?f)")
unfolding nn_integral_def
proof (safe intro!: antisym SUP_least)
fix g assume g[measurable]: "simple_function M g" "g ≤ f"

show "integral⇧S M g ≤ Sup (?A ` ?f)"
proof cases
assume ae: "AE x in M. g x ≠ top"
let ?G = "{x ∈ space M. g x ≠ top}"
have "integral⇧S M g = integral⇧S M (λx. g x * indicator ?G x)"
proof (rule simple_integral_cong_AE)
show "AE x in M. g x = g x * indicator ?G x"
using ae AE_space by eventually_elim auto
qed (insert g, auto)
also have "… ≤ Sup (?A ` ?f)"
using g by (intro SUP_upper) (auto simp: le_fun_def less_top split: split_indicator)
finally show ?thesis .
next
assume nAE: "¬ (AE x in M. g x ≠ top)"
then have "emeasure M {x∈space M. g x = top} ≠ 0" (is "emeasure M ?G ≠ 0")
by (subst (asm) AE_iff_measurable[OF _ refl]) auto
then have "top = (SUP n. (∫⇧Sx. of_nat n * indicator ?G x ∂M))"
by (simp add: ennreal_SUP_of_nat_eq_top ennreal_top_eq_mult_iff SUP_mult_right_ennreal[symmetric])
also have "… ≤ Sup (?A ` ?f)"
using g
by (safe intro!: SUP_least SUP_upper)
(auto simp: le_fun_def of_nat_less_top top_unique[symmetric] split: split_indicator
intro: order_trans[of _ "g x" "f x" for x, OF order_trans[of _ top]])
finally show ?thesis
by (simp add: top_unique del: SUP_eq_top_iff Sup_eq_top_iff)
qed
qed (auto intro: SUP_upper)

lemma nn_integral_mono_AE:
assumes ae: "AE x in M. u x ≤ v x" shows "integral⇧N M u ≤ integral⇧N M v"
unfolding nn_integral_def
proof (safe intro!: SUP_mono)
fix n assume n: "simple_function M n" "n ≤ u"
from ae[THEN AE_E] guess N . note N = this
then have ae_N: "AE x in M. x ∉ N" by (auto intro: AE_not_in)
let ?n = "λx. n x * indicator (space M - N) x"
have "AE x in M. n x ≤ ?n x" "simple_function M ?n"
using n N ae_N by auto
moreover
{ fix x have "?n x ≤ v x"
proof cases
assume x: "x ∈ space M - N"
with N have "u x ≤ v x" by auto
with n(2)[THEN le_funD, of x] x show ?thesis
by (auto simp: max_def split: if_split_asm)
qed simp }
then have "?n ≤ v" by (auto simp: le_funI)
moreover have "integral⇧S M n ≤ integral⇧S M ?n"
using ae_N N n by (auto intro!: simple_integral_mono_AE)
ultimately show "∃m∈{g. simple_function M g ∧ g ≤ v}. integral⇧S M n ≤ integral⇧S M m"
by force
qed

lemma nn_integral_mono:
"(⋀x. x ∈ space M ⟹ u x ≤ v x) ⟹ integral⇧N M u ≤ integral⇧N M v"
by (auto intro: nn_integral_mono_AE)

lemma mono_nn_integral: "mono F ⟹ mono (λx. integral⇧N M (F x))"
by (auto simp add: mono_def le_fun_def intro!: nn_integral_mono)

lemma nn_integral_cong_AE:
"AE x in M. u x = v x ⟹ integral⇧N M u = integral⇧N M v"
by (auto simp: eq_iff intro!: nn_integral_mono_AE)

lemma nn_integral_cong:
"(⋀x. x ∈ space M ⟹ u x = v x) ⟹ integral⇧N M u = integral⇧N M v"
by (auto intro: nn_integral_cong_AE)

lemma nn_integral_cong_simp:
"(⋀x. x ∈ space M =simp=> u x = v x) ⟹ integral⇧N M u = integral⇧N M v"
by (auto intro: nn_integral_cong simp: simp_implies_def)

lemma incseq_nn_integral:
assumes "incseq f" shows "incseq (λi. integral⇧N M (f i))"
proof -
have "⋀i x. f i x ≤ f (Suc i) x"
using assms by (auto dest!: incseq_SucD simp: le_fun_def)
then show ?thesis
by (auto intro!: incseq_SucI nn_integral_mono)
qed

lemma nn_integral_eq_simple_integral:
assumes f: "simple_function M f" shows "integral⇧N M f = integral⇧S M f"
proof -
let ?f = "λx. f x * indicator (space M) x"
have f': "simple_function M ?f" using f by auto
have "integral⇧N M ?f ≤ integral⇧S M ?f" using f'
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def nn_integral_def)
moreover have "integral⇧S M ?f ≤ integral⇧N M ?f"
unfolding nn_integral_def
using f' by (auto intro!: SUP_upper)
ultimately show ?thesis
by (simp cong: nn_integral_cong simple_integral_cong)
qed

text ‹Beppo-Levi monotone convergence theorem›
lemma nn_integral_monotone_convergence_SUP:
assumes f: "incseq f" and [measurable]: "⋀i. f i ∈ borel_measurable M"
shows "(∫⇧+ x. (SUP i. f i x) ∂M) = (SUP i. integral⇧N M (f i))"
proof (rule antisym)
show "(∫⇧+ x. (SUP i. f i x) ∂M) ≤ (SUP i. (∫⇧+ x. f i x ∂M))"
unfolding nn_integral_def_finite[of _ "λx. SUP i. f i x"]
proof (safe intro!: SUP_least)
fix u assume sf_u[simp]: "simple_function M u" and
u: "u ≤ (λx. SUP i. f i x)" and u_range: "∀x. u x < top"
note sf_u[THEN borel_measurable_simple_function, measurable]
show "integral⇧S M u ≤ (SUP j. ∫⇧+x. f j x ∂M)"
proof (rule ennreal_approx_unit)
fix a :: ennreal assume "a < 1"
let ?au = "λx. a * u x"

let ?B = "λc i. {x∈space M. ?au x = c ∧ c ≤ f i x}"
have "integral⇧S M ?au = (∑c∈?au`space M. c * (SUP i. emeasure M (?B c i)))"
unfolding simple_integral_def
proof (intro sum.cong ennreal_mult_left_cong refl)
fix c assume "c ∈ ?au ` space M" "c ≠ 0"
{ fix x' assume x': "x' ∈ space M" "?au x' = c"
with ‹c ≠ 0› u_range have "?au x' < 1 * u x'"
by (intro ennreal_mult_strict_right_mono ‹a < 1›) (auto simp: less_le)
also have "… ≤ (SUP i. f i x')"
using u by (auto simp: le_fun_def)
finally have "∃i. ?au x' ≤ f i x'"
by (auto simp: less_SUP_iff intro: less_imp_le) }
then have *: "?au -` {c} ∩ space M = (⋃i. ?B c i)"
by auto
show "emeasure M (?au -` {c} ∩ space M) = (SUP i. emeasure M (?B c i))"
unfolding * using f
by (intro SUP_emeasure_incseq[symmetric])
(auto simp: incseq_def le_fun_def intro: order_trans)
qed
also have "… = (SUP i. ∑c∈?au`space M. c * emeasure M (?B c i))"
unfolding SUP_mult_left_ennreal using f
by (intro ennreal_SUP_sum[symmetric])
(auto intro!: mult_mono emeasure_mono simp: incseq_def le_fun_def intro: order_trans)
also have "… ≤ (SUP i. integral⇧N M (f i))"
proof (intro SUP_subset_mono order_refl)
fix i
have "(∑c∈?au`space M. c * emeasure M (?B c i)) =
(∫⇧Sx. (a * u x) * indicator {x∈space M. a * u x ≤ f i x} x ∂M)"
by (subst simple_integral_indicator)
(auto intro!: sum.cong ennreal_mult_left_cong arg_cong2[where f=emeasure])
also have "… = (∫⇧+x. (a * u x) * indicator {x∈space M. a * u x ≤ f i x} x ∂M)"
by (rule nn_integral_eq_simple_integral[symmetric]) simp
also have "… ≤ (∫⇧+x. f i x ∂M)"
by (intro nn_integral_mono) (auto split: split_indicator)
finally show "(∑c∈?au`space M. c * emeasure M (?B c i)) ≤ (∫⇧+x. f i x ∂M)" .
qed
finally show "a * integral⇧S M u ≤ (SUP i. integral⇧N M (f i))"
by simp
qed
qed
qed (auto intro!: SUP_least SUP_upper nn_integral_mono)

lemma sup_continuous_nn_integral[order_continuous_intros]:
assumes f: "⋀y. sup_continuous (f y)"
assumes [measurable]: "⋀x. (λy. f y x) ∈ borel_measurable M"
shows "sup_continuous (λx. (∫⇧+y. f y x ∂M))"
unfolding sup_continuous_def
proof safe
fix C :: "nat ⇒ 'b" assume C: "incseq C"
with sup_continuous_mono[OF f] show "(∫⇧+ y. f y (Sup (C ` UNIV)) ∂M) = (SUP i. ∫⇧+ y. f y (C i) ∂M)"
unfolding sup_continuousD[OF f C]
by (subst nn_integral_monotone_convergence_SUP) (auto simp: mono_def le_fun_def)
qed

theorem nn_integral_monotone_convergence_SUP_AE:
assumes f: "⋀i. AE x in M. f i x ≤ f (Suc i) x" "⋀i. f i ∈ borel_measurable M"
shows "(∫⇧+ x. (SUP i. f i x) ∂M) = (SUP i. integral⇧N M (f i))"
proof -
from f have "AE x in M. ∀i. f i x ≤ f (Suc i) x"
from this[THEN AE_E] guess N . note N = this
let ?f = "λi x. if x ∈ space M - N then f i x else 0"
have f_eq: "AE x in M. ∀i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
then have "(∫⇧+ x. (SUP i. f i x) ∂M) = (∫⇧+ x. (SUP i. ?f i x) ∂M)"
by (auto intro!: nn_integral_cong_AE)
also have "… = (SUP i. (∫⇧+ x. ?f i x ∂M))"
proof (rule nn_integral_monotone_convergence_SUP)
show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
{ fix i show "(λx. if x ∈ space M - N then f i x else 0) ∈ borel_measurable M"
using f N(3) by (intro measurable_If_set) auto }
qed
also have "… = (SUP i. (∫⇧+ x. f i x ∂M))"
using f_eq by (force intro!: arg_cong[where f = "λf. Sup (range f)"] nn_integral_cong_AE ext)
finally show ?thesis .
qed

lemma nn_integral_monotone_convergence_simple:
"incseq f ⟹ (⋀i. simple_function M (f i)) ⟹ (SUP i. ∫⇧S x. f i x ∂M) = (∫⇧+x. (SUP i. f i x) ∂M)"
using nn_integral_monotone_convergence_SUP[of f M]

lemma SUP_simple_integral_sequences:
assumes f: "incseq f" "⋀i. simple_function M (f i)"
and g: "incseq g" "⋀i. simple_function M (g i)"
and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
shows "(SUP i. integral⇧S M (f i)) = (SUP i. integral⇧S M (g i))"
(is "Sup (?F ` _) = Sup (?G ` _)")
proof -
have "(SUP i. integral⇧S M (f i)) = (∫⇧+x. (SUP i. f i x) ∂M)"
using f by (rule nn_integral_monotone_convergence_simple)
also have "… = (∫⇧+x. (SUP i. g i x) ∂M)"
unfolding eq[THEN nn_integral_cong_AE] ..
also have "… = (SUP i. ?G i)"
using g by (rule nn_integral_monotone_convergence_simple[symmetric])
finally show ?thesis by simp
qed

lemma nn_integral_const[simp]: "(∫⇧+ x. c ∂M) = c * emeasure M (space M)"
by (subst nn_integral_eq_simple_integral) auto

lemma nn_integral_linear:
assumes f: "f ∈ borel_measurable M" and g: "g ∈ borel_measurable M"
shows "(∫⇧+ x. a * f x + g x ∂M) = a * integral⇧N M f + integral⇧N M g"
(is "integral⇧N M ?L = _")
proof -
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
note u = nn_integral_monotone_convergence_simple[OF this(2,1)] this
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
note v = nn_integral_monotone_convergence_simple[OF this(2,1)] this
let ?L' = "λi x. a * u i x + v i x"

have "?L ∈ borel_measurable M" using assms by auto
from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
note l = nn_integral_monotone_convergence_simple[OF this(2,1)] this

have inc: "incseq (λi. a * integral⇧S M (u i))" "incseq (λi. integral⇧S M (v i))"
using u v by (auto simp: incseq_Suc_iff le_fun_def intro!: add_mono mult_left_mono simple_integral_mono)

have l': "(SUP i. integral⇧S M (l i)) = (SUP i. integral⇧S M (?L' i))"
proof (rule SUP_simple_integral_sequences[OF l(3,2)])
show "incseq ?L'" "⋀i. simple_function M (?L' i)"
using u v unfolding incseq_Suc_iff le_fun_def
{ fix x
have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
using u(3) v(3) u(4)[of _ x] v(4)[of _ x] unfolding SUP_mult_left_ennreal
then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
unfolding l(5) using u(5) v(5) by (intro AE_I2) auto
qed
also have "… = (SUP i. a * integral⇧S M (u i) + integral⇧S M (v i))"
using u(2) v(2) by auto
finally show ?thesis
unfolding l(5)[symmetric] l(1)[symmetric]
qed

lemma nn_integral_cmult: "f ∈ borel_measurable M ⟹ (∫⇧+ x. c * f x ∂M) = c * integral⇧N M f"
using nn_integral_linear[of f M "λx. 0" c] by simp

lemma nn_integral_multc: "f ∈ borel_measurable M ⟹ (∫⇧+ x. f x * c ∂M) = integral⇧N M f * c"
unfolding mult.commute[of _ c] nn_integral_cmult by simp

lemma nn_integral_divide: "f ∈ borel_measurable M ⟹ (∫⇧+ x. f x / c ∂M) = (∫⇧+ x. f x ∂M) / c"
unfolding divide_ennreal_def by (rule nn_integral_multc)

lemma nn_integral_indicator[simp]: "A ∈ sets M ⟹ (∫⇧+ x. indicator A x∂M) = (emeasure M) A"
by (subst nn_integral_eq_simple_integral) (auto simp: simple_integral_indicator)

lemma nn_integral_cmult_indicator: "A ∈ sets M ⟹ (∫⇧+ x. c * indicator A x ∂M) = c * emeasure M A"
by (subst nn_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)

lemma nn_integral_indicator':
assumes [measurable]: "A ∩ space M ∈ sets M"
shows "(∫⇧+ x. indicator A x ∂M) = emeasure M (A ∩ space M)"
proof -
have "(∫⇧+ x. indicator A x ∂M) = (∫⇧+ x. indicator (A ∩ space M) x ∂M)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "… = emeasure M (A ∩ space M)"
by simp
finally show ?thesis .
qed

lemma nn_integral_indicator_singleton[simp]:
assumes [measurable]: "{y} ∈ sets M" shows "(∫⇧+x. f x * indicator {y} x ∂M) = f y * emeasure M {y}"
proof -
have "(∫⇧+x. f x * indicator {y} x ∂M) = (∫⇧+x. f y * indicator {y} x ∂M)"
by (auto intro!: nn_integral_cong split: split_indicator)
then show ?thesis
qed

lemma nn_integral_set_ennreal:
"(∫⇧+x. ennreal (f x) * indicator A x ∂M) = (∫⇧+x. ennreal (f x * indicator A x) ∂M)"
by (rule nn_integral_cong) (simp split: split_indicator)

lemma nn_integral_indicator_singleton'[simp]:
assumes [measurable]: "{y} ∈ sets M"
shows "(∫⇧+x. ennreal (f x * indicator {y} x) ∂M) = f y * emeasure M {y}"
by (subst nn_integral_set_ennreal[symmetric]) (simp add: nn_integral_indicator_singleton)

"f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (∫⇧+ x. f x + g x ∂M) = integral⇧N M f + integral⇧N M g"
using nn_integral_linear[of f M g 1] by simp

lemma nn_integral_sum:
"(⋀i. i ∈ P ⟹ f i ∈ borel_measurable M) ⟹ (∫⇧+ x. (∑i∈P. f i x) ∂M) = (∑i∈P. integral⇧N M (f i))"
by (induction P rule: infinite_finite_induct) (auto simp: nn_integral_add)

theorem nn_integral_suminf:
assumes f: "⋀i. f i ∈ borel_measurable M"
shows "(∫⇧+ x. (∑i. f i x) ∂M) = (∑i. integral⇧N M (f i))"
proof -
have all_pos: "AE x in M. ∀i. 0 ≤ f i x"
using assms by (auto simp: AE_all_countable)
have "(∑i. integral⇧N M (f i)) = (SUP n. ∑i<n. integral⇧N M (f i))"
by (rule suminf_eq_SUP)
also have "… = (SUP n. ∫⇧+x. (∑i<n. f i x) ∂M)"
unfolding nn_integral_sum[OF f] ..
also have "… = ∫⇧+x. (SUP n. ∑i<n. f i x) ∂M" using f all_pos
by (intro nn_integral_monotone_convergence_SUP_AE[symmetric])
(elim AE_mp, auto simp: sum_nonneg simp del: sum.lessThan_Suc intro!: AE_I2 sum_mono2)
also have "… = ∫⇧+x. (∑i. f i x) ∂M" using all_pos
by (intro nn_integral_cong_AE) (auto simp: suminf_eq_SUP)
finally show ?thesis by simp
qed

lemma nn_integral_bound_simple_function:
assumes bnd: "⋀x. x ∈ space M ⟹ f x < ∞"
assumes f[measurable]: "simple_function M f"
assumes supp: "emeasure M {x∈space M. f x ≠ 0} < ∞"
shows "nn_integral M f < ∞"
proof cases
assume "space M = {}"
then have "nn_integral M f = (∫⇧+x. 0 ∂M)"
by (intro nn_integral_cong) auto
then show ?thesis by simp
next
assume "space M ≠ {}"
with simple_functionD(1)[OF f] bnd have bnd: "0 ≤ Max (f`space M) ∧ Max (f`space M) < ∞"
by (subst Max_less_iff) (auto simp: Max_ge_iff)

have "nn_integral M f ≤ (∫⇧+x. Max (f`space M) * indicator {x∈space M. f x ≠ 0} x ∂M)"
proof (rule nn_integral_mono)
fix x assume "x ∈ space M"
with f show "f x ≤ Max (f ` space M) * indicator {x ∈ space M. f x ≠ 0} x"
by (auto split: split_indicator intro!: Max_ge simple_functionD)
qed
also have "… < ∞"
using bnd supp by (subst nn_integral_cmult) (auto simp: ennreal_mult_less_top)
finally show ?thesis .
qed

theorem nn_integral_Markov_inequality:
assumes u: "u ∈ borel_measurable M" and "A ∈ sets M"
shows "(emeasure M) ({x∈space M. 1 ≤ c * u x} ∩ A) ≤ c * (∫⇧+ x. u x * indicator A x ∂M)"
(is "(emeasure M) ?A ≤ _ * ?PI")
proof -
have "?A ∈ sets M"
using ‹A ∈ sets M› u by auto
hence "(emeasure M) ?A = (∫⇧+ x. indicator ?A x ∂M)"
using nn_integral_indicator by simp
also have "… ≤ (∫⇧+ x. c * (u x * indicator A x) ∂M)"
using u by (auto intro!: nn_integral_mono_AE simp: indicator_def)
also have "… = c * (∫⇧+ x. u x * indicator A x ∂M)"
using assms by (auto intro!: nn_integral_cmult)
finally show ?thesis .
qed

lemma nn_integral_noteq_infinite:
assumes g: "g ∈ borel_measurable M" and "integral⇧N M g ≠ ∞"
shows "AE x in M. g x ≠ ∞"
proof (rule ccontr)
assume c: "¬ (AE x in M. g x ≠ ∞)"
have "(emeasure M) {x∈space M. g x = ∞} ≠ 0"
using c g by (auto simp add: AE_iff_null)
then have "0 < (emeasure M) {x∈space M. g x = ∞}"
by (auto simp: zero_less_iff_neq_zero)
then have "∞ = ∞ * (emeasure M) {x∈space M. g x = ∞}"
by (auto simp: ennreal_top_eq_mult_iff)
also have "… ≤ (∫⇧+x. ∞ * indicator {x∈space M. g x = ∞} x ∂M)"
using g by (subst nn_integral_cmult_indicator) auto
also have "… ≤ integral⇧N M g"
using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def)
finally show False
using ‹integral⇧N M g ≠ ∞› by (auto simp: top_unique)
qed

lemma nn_integral_PInf:
assumes f: "f ∈ borel_measurable M" and not_Inf: "integral⇧N M f ≠ ∞"
shows "emeasure M (f -` {∞} ∩ space M) = 0"
proof -
have "∞ * emeasure M (f -` {∞} ∩ space M) = (∫⇧+ x. ∞ * indicator (f -` {∞} ∩ space M) x ∂M)"
using f by (subst nn_integral_cmult_indicator) (auto simp: measurable_sets)
also have "… ≤ integral⇧N M f"
by (auto intro!: nn_integral_mono simp: indicator_def)
finally have "∞ * (emeasure M) (f -` {∞} ∩ space M) ≤ integral⇧N M f"
by simp
then show ?thesis
using assms by (auto simp: ennreal_top_mult top_unique split: if_split_asm)
qed

lemma simple_integral_PInf:
"simple_function M f ⟹ integral⇧S M f ≠ ∞ ⟹ emeasure M (f -` {∞} ∩ space M) = 0"
by (rule nn_integral_PInf) (auto simp: nn_integral_eq_simple_integral borel_measurable_simple_function)

lemma nn_integral_PInf_AE:
assumes "f ∈ borel_measurable M" "integral⇧N M f ≠ ∞" shows "AE x in M. f x ≠ ∞"
proof (rule AE_I)
show "(emeasure M) (f -` {∞} ∩ space M) = 0"
by (rule nn_integral_PInf[OF assms])
show "f -` {∞} ∩ space M ∈ sets M"
using assms by (auto intro: borel_measurable_vimage)
qed auto

lemma nn_integral_diff:
assumes f: "f ∈ borel_measurable M"
and g: "g ∈ borel_measurable M"
and fin: "integral⇧N M g ≠ ∞"
and mono: "AE x in M. g x ≤ f x"
shows "(∫⇧+ x. f x - g x ∂M) = integral⇧N M f - integral⇧N M g"
proof -
have diff: "(λx. f x - g x) ∈ borel_measurable M"
using assms by auto
have "AE x in M. f x = f x - g x + g x"
using diff_add_cancel_ennreal mono nn_integral_noteq_infinite[OF g fin] assms by auto
then have **: "integral⇧N M f = (∫⇧+x. f x - g x ∂M) + integral⇧N M g"
by (rule nn_integral_cong_AE)
show ?thesis unfolding **
using fin
by (cases rule: ennreal2_cases[of "∫⇧+ x. f x - g x ∂M" "integral⇧N M g"]) auto
qed

lemma nn_integral_mult_bounded_inf:
assumes f: "f ∈ borel_measurable M" "(∫⇧+x. f x ∂M) < ∞" and c: "c ≠ ∞" and ae: "AE x in M. g x ≤ c * f x"
shows "(∫⇧+x. g x ∂M) < ∞"
proof -
have "(∫⇧+x. g x ∂M) ≤ (∫⇧+x. c * f x ∂M)"
by (intro nn_integral_mono_AE ae)
also have "(∫⇧+x. c * f x ∂M) < ∞"
using c f by (subst nn_integral_cmult) (auto simp: ennreal_mult_less_top top_unique not_less)
finally show ?thesis .
qed

text ‹Fatou's lemma: convergence theorem on limes inferior›

lemma nn_integral_monotone_convergence_INF_AE':
assumes f: "⋀i. AE x in M. f (Suc i) x ≤ f i x" and [measurable]: "⋀i. f i ∈ borel_measurable M"
and *: "(∫⇧+ x. f 0 x ∂M) < ∞"
shows "(∫⇧+ x. (INF i. f i x) ∂M) = (INF i. integral⇧N M (f i))"
proof (rule ennreal_minus_cancel)
have "integral⇧N M (f 0) - (∫⇧+ x. (INF i. f i x) ∂M) = (∫⇧+x. f 0 x - (INF i. f i x) ∂M)"
proof (rule nn_integral_diff[symmetric])
have "(∫⇧+ x. (INF i. f i x) ∂M) ≤ (∫⇧+ x. f 0 x ∂M)"
by (intro nn_integral_mono INF_lower) simp
with * show "(∫⇧+ x. (INF i. f i x) ∂M) ≠ ∞"
by simp
qed (auto intro: INF_lower)
also have "… = (∫⇧+x. (SUP i. f 0 x - f i x) ∂M)"
also have "… = (SUP i. (∫⇧+x. f 0 x - f i x ∂M))"
proof (intro nn_integral_monotone_convergence_SUP_AE)
show "AE x in M. f 0 x - f i x ≤ f 0 x - f (Suc i) x" for i
using f[of i] by eventually_elim (auto simp: ennreal_mono_minus)
qed simp
also have "… = (SUP i. nn_integral M (f 0) - (∫⇧+x. f i x ∂M))"
proof (subst nn_integral_diff[symmetric])
fix i
have dec: "AE x in M. ∀i. f (Suc i) x ≤ f i x"
unfolding AE_all_countable using f by auto
then show "AE x in M. f i x ≤ f 0 x"
using dec by eventually_elim (auto intro: lift_Suc_antimono_le[of "λi. f i x" 0 i for x])
then have "(∫⇧+ x. f i x ∂M) ≤ (∫⇧+ x. f 0 x ∂M)"
by (rule nn_integral_mono_AE)
with * show "(∫⇧+ x. f i x ∂M) ≠ ∞"
by simp
qed (insert f, auto simp: decseq_def le_fun_def)
finally show "integral⇧N M (f 0) - (∫⇧+ x. (INF i. f i x) ∂M) =
integral⇧N M (f 0) - (INF i. ∫⇧+ x. f i x ∂M)"
qed (insert *, auto intro!: nn_integral_mono intro: INF_lower)

theorem nn_integral_monotone_convergence_INF_AE:
fixes f :: "nat ⇒ 'a ⇒ ennreal"
assumes f: "⋀i. AE x in M. f (Suc i) x ≤ f i x"
and [measurable]: "⋀i. f i ∈ borel_measurable M"
and fin: "(∫⇧+ x. f i x ∂M) < ∞"
shows "(∫⇧+ x. (INF i. f i x) ∂M) = (INF i. integral⇧N M (f i))"
proof -
{ fix f :: "nat ⇒ ennreal" and j assume "decseq f"
then have "(INF i. f i) = (INF i. f (i + j))"
apply (intro INF_eq)
apply (rule_tac x="i" in bexI)
apply (auto simp: decseq_def le_fun_def)
done }
note INF_shift = this
have mono: "AE x in M. ∀i. f (Suc i) x ≤ f i x"
using f by (auto simp: AE_all_countable)
then have "AE x in M. (INF i. f i x) = (INF n. f (n + i) x)"
by eventually_elim (auto intro!: decseq_SucI INF_shift)
then have "(∫⇧+ x. (INF i. f i x) ∂M) = (∫⇧+ x. (INF n. f (n + i) x) ∂M)"
by (rule nn_integral_cong_AE)
also have "… = (INF n. (∫⇧+ x. f (n + i) x ∂M))"
by (rule nn_integral_monotone_convergence_INF_AE') (insert assms, auto)
also have "… = (INF n. (∫⇧+ x. f n x ∂M))"
by (intro INF_shift[symmetric] decseq_SucI nn_integral_mono_AE f)
finally show ?thesis .
qed

lemma nn_integral_monotone_convergence_INF_decseq:
assumes f: "decseq f" and *: "⋀i. f i ∈ borel_measurable M" "(∫⇧+ x. f i x ∂M) < ∞"
shows "(∫⇧+ x. (INF i. f i x) ∂M) = (INF i. integral⇧N M (f i))"
using nn_integral_monotone_convergence_INF_AE[of f M i, OF _ *] f by (auto simp: decseq_Suc_iff le_fun_def)

theorem nn_integral_liminf:
fixes u :: "nat ⇒ 'a ⇒ ennreal"
assumes u: "⋀i. u i ∈ borel_measurable M"
shows "(∫⇧+ x. liminf (λn. u n x) ∂M) ≤ liminf (λn. integral⇧N M (u n))"
proof -
have "(∫⇧+ x. liminf (λn. u n x) ∂M) = (SUP n. ∫⇧+ x. (INF i∈{n..}. u i x) ∂M)"
unfolding liminf_SUP_INF using u
by (intro nn_integral_monotone_convergence_SUP_AE)
(auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
also have "… ≤ liminf (λn. integral⇧N M (u n))"
by (auto simp: liminf_SUP_INF intro!: SUP_mono INF_greatest nn_integral_mono INF_lower)
finally show ?thesis .
qed

theorem nn_integral_limsup:
fixes u :: "nat ⇒ 'a ⇒ ennreal"
assumes [measurable]: "⋀i. u i ∈ borel_measurable M" "w ∈ borel_measurable M"
assumes bounds: "⋀i. AE x in M. u i x ≤ w x" and w: "(∫⇧+x. w x ∂M) < ∞"
shows "limsup (λn. integral⇧N M (u n)) ≤ (∫⇧+ x. limsup (λn. u n x) ∂M)"
proof -
have bnd: "AE x in M. ∀i. u i x ≤ w x"
using bounds by (auto simp: AE_all_countable)
then have "(∫⇧+ x. (SUP n. u n x) ∂M) ≤ (∫⇧+ x. w x ∂M)"
by (auto intro!: nn_integral_mono_AE elim: eventually_mono intro: SUP_least)
then have "(∫⇧+ x. limsup (λn. u n x) ∂M) = (INF n. ∫⇧+ x. (SUP i∈{n..}. u i x) ∂M)"
unfolding limsup_INF_SUP using bnd w
by (intro nn_integral_monotone_convergence_INF_AE')
(auto intro!: AE_I2 intro: SUP_least SUP_subset_mono)
also have "… ≥ limsup (λn. integral⇧N M (u n))"
by (auto simp: limsup_INF_SUP intro!: INF_mono SUP_least exI nn_integral_mono SUP_upper)
finally (xtrans) show ?thesis .
qed

lemma nn_integral_LIMSEQ:
assumes f: "incseq f" "⋀i. f i ∈ borel_measurable M"
and u: "⋀x. (λi. f i x) ⇢ u x"
shows "(λn. integral⇧N M (f n)) ⇢ integral⇧N M u"
proof -
have "(λn. integral⇧N M (f n)) ⇢ (SUP n. integral⇧N M (f n))"
using f by (intro LIMSEQ_SUP[of "λn. integral⇧N M (f n)"] incseq_nn_integral)
also have "(SUP n. integral⇧N M (f n)) = integral⇧N M (λx. SUP n. f n x)"
using f by (intro nn_integral_monotone_convergence_SUP[symmetric])
also have "integral⇧N M (λx. SUP n. f n x) = integral⇧N M (λx. u x)"
using f by (subst LIMSEQ_SUP[THEN LIMSEQ_unique, OF _ u]) (auto simp: incseq_def le_fun_def)
finally show ?thesis .
qed

theorem nn_integral_dominated_convergence:
assumes [measurable]:
"⋀i. u i ∈ borel_measurable M" "u' ∈ borel_measurable M" "w ∈ borel_measurable M"
and bound: "⋀j. AE x in M. u j x ≤ w x"
and w: "(∫⇧+x. w x ∂M) < ∞"
and u': "AE x in M. (λi. u i x) ⇢ u' x"
shows "(λi. (∫⇧+x. u i x ∂M)) ⇢ (∫⇧+x. u' x ∂M)"
proof -
have "limsup (λn. integral⇧N M (u n)) ≤ (∫⇧+ x. limsup (λn. u n x) ∂M)"
by (intro nn_integral_limsup[OF _ _ bound w]) auto
moreover have "(∫⇧+ x. limsup (λn. u n x) ∂M) = (∫⇧+ x. u' x ∂M)"
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
moreover have "(∫⇧+ x. liminf (λn. u n x) ∂M) = (∫⇧+ x. u' x ∂M)"
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
moreover have "(∫⇧+x. liminf (λn. u n x) ∂M) ≤ liminf (λn. integral⇧N M (u n))"
by (intro nn_integral_liminf) auto
moreover have "liminf (λn. integral⇧N M (u n)) ≤ limsup (λn. integral⇧N M (u n))"
by (intro Liminf_le_Limsup sequentially_bot)
ultimately show ?thesis
by (intro Liminf_eq_Limsup) auto
qed

lemma inf_continuous_nn_integral[order_continuous_intros]:
assumes f: "⋀y. inf_continuous (f y)"
assumes [measurable]: "⋀x. (λy. f y x) ∈ borel_measurable M"
assumes bnd: "⋀x. (∫⇧+ y. f y x ∂M) ≠ ∞"
shows "inf_continuous (λx. (∫⇧+y. f y x ∂M))"
unfolding inf_continuous_def
proof safe
fix C :: "nat ⇒ 'b" assume C: "decseq C"
then show "(∫⇧+ y. f y (Inf (C ` UNIV)) ∂M) = (INF i. ∫⇧+ y. f y (C i) ∂M)"
using inf_continuous_mono[OF f] bnd
by (auto simp add: inf_continuousD[OF f C] fun_eq_iff antimono_def mono_def le_fun_def less_top
intro!: nn_integral_monotone_convergence_INF_decseq)
qed

lemma nn_integral_null_set:
assumes "N ∈ null_sets M" shows "(∫⇧+ x. u x * indicator N x ∂M) = 0"
proof -
have "(∫⇧+ x. u x * indicator N x ∂M) = (∫⇧+ x. 0 ∂M)"
proof (intro nn_integral_cong_AE AE_I)
show "{x ∈ space M. u x * indicator N x ≠ 0} ⊆ N"
by (auto simp: indicator_def)
show "(emeasure M) N = 0" "N ∈ sets M"
using assms by auto
qed
then show ?thesis by simp
qed

lemma nn_integral_0_iff:
assumes u: "u ∈ borel_measurable M"
shows "integral⇧N M u = 0 ⟷ emeasure M {x∈space M. u x ≠ 0} = 0"
(is "_ ⟷ (emeasure M) ?A = 0")
proof -
have u_eq: "(∫⇧+ x. u x * indicator ?A x ∂M) = integral⇧N M u"
by (auto intro!: nn_integral_cong simp: indicator_def)
show ?thesis
proof
assume "(emeasure M) ?A = 0"
with nn_integral_null_set[of ?A M u] u
show "integral⇧N M u = 0" by (simp add: u_eq null_sets_def)
next
assume *: "integral⇧N M u = 0"
let ?M = "λn. {x ∈ space M. 1 ≤ real (n::nat) * u x}"
have "0 = (SUP n. (emeasure M) (?M n ∩ ?A))"
proof -
{ fix n :: nat
from nn_integral_Markov_inequality[OF u, of ?A "of_nat n"] u
have "(emeasure M) (?M n ∩ ?A) ≤ 0"
by (simp add: ennreal_of_nat_eq_real_of_nat u_eq *)
moreover have "0 ≤ (emeasure M) (?M n ∩ ?A)" using u by auto
ultimately have "(emeasure M) (?M n ∩ ?A) = 0" by auto }
thus ?thesis by simp
qed
also have "… = (emeasure M) (⋃n. ?M n ∩ ?A)"
proof (safe intro!: SUP_emeasure_incseq)
fix n show "?M n ∩ ?A ∈ sets M"
using u by (auto intro!: sets.Int)
next
show "incseq (λn. {x ∈ space M. 1 ≤ real n * u x} ∩ {x ∈ space M. u x ≠ 0})"
proof (safe intro!: incseq_SucI)
fix n :: nat and x
assume *: "1 ≤ real n * u x"
also have "real n * u x ≤ real (Suc n) * u x"
by (auto intro!: mult_right_mono)
finally show "1 ≤ real (Suc n) * u x" by auto
qed
qed
also have "… = (emeasure M) {x∈space M. 0 < u x}"
proof (safe intro!: arg_cong[where f="(emeasure M)"])
fix x assume "0 < u x" and [simp, intro]: "x ∈ space M"
show "x ∈ (⋃n. ?M n ∩ ?A)"
proof (cases "u x" rule: ennreal_cases)
case (real r) with ‹0 < u x› have "0 < r" by auto
obtain j :: nat where "1 / r ≤ real j" using real_arch_simple ..
hence "1 / r * r ≤ real j * r" unfolding mult_le_cancel_right using ‹0 < r› by auto
hence "1 ≤ real j * r" using real ‹0 < r› by auto
thus ?thesis using ‹0 < r› real
by (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_1[symmetric] ennreal_mult[symmetric]
simp del: ennreal_1)
qed (insert ‹0 < u x›, auto simp: ennreal_mult_top)
qed (auto simp: zero_less_iff_neq_zero)
finally show "emeasure M ?A = 0"
qed
qed

lemma nn_integral_0_iff_AE:
assumes u: "u ∈ borel_measurable M"
shows "integral⇧N M u = 0 ⟷ (AE x in M. u x = 0)"
proof -
have sets: "{x∈space M. u x ≠ 0} ∈ sets M"
using u by auto
show "integral⇧N M u = 0 ⟷ (AE x in M. u x = 0)"
using nn_integral_0_iff[of u] AE_iff_null[OF sets] u by auto
qed

lemma AE_iff_nn_integral:
"{x∈space M. P x} ∈ sets M ⟹ (AE x in M. P x) ⟷ integral⇧N M (indicator {x. ¬ P x}) = 0"
by (subst nn_integral_0_iff_AE) (auto simp: indicator_def[abs_def])

lemma nn_integral_less:
assumes [measurable]: "f ∈ borel_measurable M" "g ∈ borel_measurable M"
assumes f: "(∫⇧+x. f x ∂M) ≠ ∞"
assumes ord: "AE x in M. f x ≤ g x" "¬ (AE x in M. g x ≤ f x)"
shows "(∫⇧+x. f x ∂M) < (∫⇧+x. g x ∂M)"
proof -
have "0 < (∫⇧+x. g x - f x ∂M)"
proof (intro order_le_neq_trans notI)
assume "0 = (∫⇧+x. g x - f x ∂M)"
then have "AE x in M. g x - f x = 0"
using nn_integral_0_iff_AE[of "λx. g x - f x" M] by simp
with ord(1) have "AE x in M. g x ≤ f x"
by eventually_elim (auto simp: ennreal_minus_eq_0)
with ord show False
by simp
qed simp
also have "… = (∫⇧+x. g x ∂M) - (∫⇧+x. f x ∂M)"
using f by (subst nn_integral_diff) (auto simp: ord)
finally show ?thesis
using f by (auto dest!: ennreal_minus_pos_iff[rotated] simp: less_top)
qed

lemma nn_integral_subalgebra:
assumes f: "f ∈ borel_measurable N"
and N: "sets N ⊆ sets M" "space N = space M" "⋀A. A ∈ sets N ⟹ emeasure N A = emeasure M A"
shows "integral⇧N N f = integral⇧N M f"
proof -
have [simp]: "⋀f :: 'a ⇒ ennreal. f ∈ borel_measurable N ⟹ f ∈ borel_measurable M"
using N by (auto simp: measurable_def)
have [simp]: "⋀P. (AE x in N. P x) ⟹ (AE x in M. P x)"
using N by (auto simp add: eventually_ae_filter null_sets_def subset_eq)
have [simp]: "⋀A. A ∈ sets N ⟹ A ∈ sets M"
using N by auto
from f show ?thesis
apply induct
apply (auto intro!: nn_integral_cong cong: nn_integral_cong simp: N(2)[symmetric])
done
qed

lemma nn_integral_nat_function:
fixes f :: "'a ⇒ nat"
assumes "f ∈ measurable M (count_space UNIV)"
shows "(∫⇧+x. of_nat (f x) ∂M) = (∑t. emeasure M {x∈space M. t < f x})"
proof -
define F where "F i = {x∈space M. i < f x}" for i
with assms have [measurable]: "⋀i. F i ∈ sets M"
by auto

{ fix x assume "x ∈ space M"
have "(λi. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
using sums_If_finite[of "λi. i < f x" "λ_. 1::real"] by simp
then have "(λi. ennreal (if i < f x then 1 else 0)) sums of_nat(f x)"
unfolding ennreal_of_nat_eq_real_of_nat
by (subst sums_ennreal) auto
moreover have "⋀i. ennreal (if i < f x then 1 else 0) = indicator (F i) x"
using ‹x ∈ space M› by (simp add: one_ennreal_def F_def)
ultimately have "of_nat (f x) = (∑i. indicator (F i) x :: ennreal)"
then have "(∫⇧+x. of_nat (f x) ∂M) = (∫⇧+x. (∑i. indicator (F i) x) ∂M)"
by (simp cong: nn_integral_cong)
also have "… = (∑i. emeasure M (F i))"
finally show ?thesis
qed

theorem nn_integral_lfp:
assumes sets[simp]: "⋀s. sets (M s) = sets N"
assumes f: "sup_continuous f"
assumes g: "sup_continuous g"
assumes meas: "⋀F. F ∈ borel_measurable N ⟹ f F ∈ borel_measurable N"
assumes step: "⋀F s. F ∈ borel_measurable N ⟹ integral⇧N (M s) (f F) = g (λs. integral⇧N (M s) F) s"
shows "(∫⇧+ω. lfp f ω ∂M s) = lfp g s"
proof (subst lfp_transfer_bounded[where α="λF s. ∫⇧+x. F x ∂M s" and g=g and f=f and P="λf. f ∈ borel_measurable N", symmetric])
fix C :: "nat ⇒ 'b ⇒ ennreal" assume "incseq C" "⋀i. C i ∈ borel_measurable N"
then show "(λs. ∫⇧+x. (SUP i. C i) x ∂M s) = (SUP i. (λs. ∫⇧+x. C i x ∂M s))"
unfolding SUP_apply[abs_def]
by (subst nn_integral_monotone_convergence_SUP)
(auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets)
qed (auto simp add: step le_fun_def SUP_apply[abs_def] bot_fun_def bot_ennreal intro!: meas f g)

theorem nn_integral_gfp:
assumes sets[simp]: "⋀s. sets (M s) = sets N"
assumes f: "inf_continuous f" and g: "inf_continuous g"
assumes meas: "⋀F. F ∈ borel_measurable N ⟹ f F ∈ borel_measurable N"
assumes bound: "⋀F s. F ∈ borel_measurable N ⟹ (∫⇧+x. f F x ∂M s) < ∞"
assumes non_zero: "⋀s. emeasure (M s) (space (M s)) ≠ 0"
assumes step: "⋀F s. F ∈ borel_measurable N ⟹ integral⇧N (M s) (f F) = g (λs. integral⇧N (M s) F) s"
shows "(∫⇧+ω. gfp f ω ∂M s) = gfp g s"
proof (subst gfp_transfer_bounded[where α="λF s. ∫⇧+x. F x ∂M s" and g=g and f=f
and P="λF. F ∈ borel_measurable N ∧ (∀s. (∫⇧+x. F x ∂M s) < ∞)", symmetric])
fix C :: "nat ⇒ 'b ⇒ ennreal" assume "decseq C" "⋀i. C i ∈ borel_measurable N ∧ (∀s. integral⇧N (M s) (C i) < ∞)"
then show "(λs. ∫⇧+x. (INF i. C i) x ∂M s) = (INF i. (λs. ∫⇧+x. C i x ∂M s))"
unfolding INF_apply[abs_def]
by (subst nn_integral_monotone_convergence_INF_decseq)
(auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets)
next
show "⋀x. g x ≤ (λs. integral⇧N (M s) (f top))"
by (subst step)
(auto simp add: top_fun_def less_le non_zero le_fun_def ennreal_top_mult
cong del: if_weak_cong intro!: monoD[OF inf_continuous_mono[OF g], THEN le_funD])
next
fix C assume "⋀i::nat. C i ∈ borel_measurable N ∧ (∀s. integral⇧N (M s) (C i) < ∞)" "decseq C"
with bound show "Inf (C ` UNIV) ∈ borel_measurable N ∧ (∀s. integral⇧N (M s) (Inf (C ` UNIV)) < ∞)"
unfolding INF_apply[abs_def]
by (subst nn_integral_monotone_convergence_INF_decseq)
(auto simp: INF_less_iff cong: measurable_cong_sets intro!: borel_measurable_INF)
next
show "⋀x. x ∈ borel_measurable N ∧ (∀s. integral⇧N (M s) x < ∞) ⟹
(λs. integral⇧N (M s) (f x)) = g (λs. integral⇧N (M s) x)"
by (subst step) auto
qed (insert bound, auto simp add: le_fun_def INF_apply[abs_def] top_fun_def intro!: meas f g)

(* TODO: rename? *)
subsection ‹Integral under concrete measures›

lemma nn_integral_mono_measure:
assumes "sets M = sets N" "M ≤ N" shows "nn_integral M f ≤ nn_integral N f"
unfolding nn_integral_def
proof (intro SUP_subset_mono)
note ‹sets M = sets N›[simp]  ‹sets M = sets N›[THEN sets_eq_imp_space_eq, simp]
show "{g. simple_function M g ∧ g ≤ f} ⊆ {g. simple_function N g ∧ g ≤ f}"
show "integral⇧S M x ≤ integral⇧S N x" for x
using le_measureD3[OF ‹M ≤ N›]
by (auto simp add: simple_integral_def intro!: sum_mono mult_mono)
qed

lemma nn_integral_empty:
assumes "space M = {}"
shows "nn_integral M f = 0"
proof -
have "(∫⇧+ x. f x ∂M) = (∫⇧+ x. 0 ∂M)"
thus ?thesis by simp
qed

lemma nn_integral_bot[simp]: "nn_integral bot f = 0"

subsubsection✐‹tag unimportant› ‹Distributions›

lemma nn_integral_distr:
assumes T: "T ∈ measurable M M'" and f: "f ∈ borel_measurable (distr M M' T)"
shows "integral⇧N (distr M M' T) f = (∫⇧+ x. f (T x) ∂M)"
using f
proof induct
case (cong f g)
with T show ?case
apply (subst nn_integral_cong[of _ f g])
apply simp
apply (subst nn_integral_cong[of _ "λx. f (T x)" "λx. g (T x)"])
apply simp
done
next
case (set A)
then have eq: "⋀x. x ∈ space M ⟹ indicator A (T x) = indicator (T -` A ∩ space M) x"
by (auto simp: indicator_def)
from set T show ?case
by (subst nn_integral_cong[OF eq])
(auto simp add: emeasure_distr intro!: nn_integral_indicator[symmetric] measurable_sets)
nn_integral_monotone_convergence_SUP le_fun_def incseq_def image_comp)

subsubsection✐‹tag unimportant› ‹Counting space›

lemma simple_function_count_space[simp]:
"simple_function (count_space A) f ⟷ finite (f ` A)"
unfolding simple_function_def by simp

lemma nn_integral_count_space:
assumes A: "finite {a∈A. 0 < f a}"
shows "integral⇧N (count_space A) f = (∑a|a∈A ∧ 0 < f a. f a)"
proof -
have *: "(∫⇧+x. max 0 (f x) ∂count_space A) =
(∫⇧+ x. (∑a|a∈A ∧ 0 < f a. f a * indicator {a} x) ∂count_space A)"
by (auto intro!: nn_integral_cong
simp add: indicator_def if_distrib sum.If_cases[OF A] max_def le_less)
also have "… = (∑a|a∈A ∧ 0 < f a. ∫⇧+ x. f a * indicator {a} x ∂count_space A)"
by (subst nn_integral_sum) (simp_all add: AE_count_space  less_imp_le)
also have "… = (∑a|a∈A ∧ 0 < f a. f a)"
by (auto intro!: sum.cong simp: one_ennreal_def[symmetric] max_def)
finally show ?thesis by (simp add: max.absorb2)
qed

lemma nn_integral_count_space_finite:
"finite A ⟹ (∫⇧+x. f x ∂count_space A) = (∑a∈A. f a)"
by (auto intro!: sum.mono_neutral_left simp: nn_integral_count_space less_le)

lemma nn_integral_count_space':
assumes "finite A" "⋀x. x ∈ B ⟹ x ∉ A ⟹ f x = 0" "A ⊆ B"
shows "(∫⇧+x. f x ∂count_space B) = (∑x∈A. f x)"
proof -
have "(∫⇧+x. f x ∂count_space B) = (∑a | a ∈ B ∧ 0 < f a. f a)"
using assms(2,3)
by (intro nn_integral_count_space finite_subset[OF _ ‹finite A›]) (auto simp: less_le)
also have "… = (∑a∈A. f a)"
using assms by (intro sum.mono_neutral_cong_left) (auto simp: less_le)
finally show ?thesis .
qed

lemma nn_integral_bij_count_space:
assumes g: "bij_betw g A B"
shows "(∫⇧+x. f (g x) ∂count_space A) = (∫⇧+x. f x ∂count_space B)"
using g[THEN bij_betw_imp_funcset]
by (subst distr_bij_count_space[OF g, symmetric])
(auto intro!: nn_integral_distr[symmetric])

lemma nn_integral_indicator_finite:
fixes f :: "'a ⇒ ennreal"
assumes f: "finite A" and [measurable]: "⋀a. a ∈ A ⟹ {a} ∈ sets M"
shows "(∫⇧+x. f x * indicator A x ∂M) = (∑x∈A. f x * emeasure M {x})"
proof -
from f have "(∫⇧+x. f x * indicator A x ∂M) = (∫⇧+x. (∑a∈A. f a * indicator {a} x) ∂M)"
by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="λa. x * a" for x] sum.If_cases)
also have "… = (∑a∈A. f a * emeasure M {a})"
by (subst nn_integral_sum) auto
finally show ?thesis .
qed

lemma nn_integral_count_space_nat:
fixes f :: "nat ⇒ ennreal"
shows "(∫⇧+i. f i ∂count_space UNIV) = (∑i. f i)"
proof -
have "(∫⇧+i. f i ∂count_space UNIV) =
(∫⇧+i. (∑j. f j * indicator {j} i) ∂count_space UNIV)"
proof (intro nn_integral_cong)
fix i
have "f i = (∑j∈{i}. f j * indicator {j} i)"
by simp
also have "… = (∑j. f j * indicator {j} i)"
by (rule suminf_finite[symmetric]) auto
finally show "f i = (∑j. f j * indicator {j} i)" .
qed
also have "… = (∑j. (∫⇧+i. f j * indicator {j} i ∂count_space UNIV))"
by (rule nn_integral_suminf) auto
finally show ?thesis
by simp
qed

lemma nn_integral_enat_function:
assumes f: "f ∈ measurable M (count_space UNIV)"
shows "(∫⇧+ x. ennreal_of_enat (f x) ∂M) = (∑t. emeasure M {x ∈ space M. t < f x})"
proof -
define F where "F i = {x∈space M. i < f x}" for i :: nat
with assms have [measurable]: "⋀i. F i ∈ sets M"
by auto

{ fix x assume "x ∈ space M"
have "(λi::nat. if i < f x then 1 else 0) sums ennreal_of_enat (f x)"
using sums_If_finite[of "λr. r < f x" "λ_. 1 :: ennreal"]
by (cases "f x") (simp_all add: sums_def of_nat_tendsto_top_ennreal)
also have "(λi. (if i < f x then 1 else 0)) = (λi. indicator (F i) x)"
using ‹x ∈ space M› by (simp add: one_ennreal_def F_def fun_eq_iff)
finally have "ennreal_of_enat (f x) = (∑i. indicator (F i) x)"
then have "(∫⇧+x. ennreal_of_enat (f x) ∂M) = (∫⇧+x. (∑i. indicator (F i) x) ∂M)"
by (simp cong: nn_integral_cong)
also have "… = (∑i. emeasure M (F i))"
finally show ?thesis
qed

lemma nn_integral_count_space_nn_integral:
fixes f :: "'i ⇒ 'a ⇒ ennreal"
assumes "countable I" and [measurable]: "⋀i. i ∈ I ⟹ f i ∈ borel_measurable M"
shows "(∫⇧+x. ∫⇧+i. f i x ∂count_space I ∂M) = (∫⇧+i. ∫⇧+x. f i x ∂M ∂count_space I)"
proof cases
assume "finite I" then show ?thesis
next
assume "infinite I"
then have [simp]: "I ≠ {}"
by auto
note * = bij_betw_from_nat_into[OF ‹countable I› ‹infinite I›]
have **: "⋀f. (⋀i. 0 ≤ f i) ⟹ (∫⇧+i. f i ∂count_space I) = (∑n. f (from_nat_into I n))"
by (simp add: nn_integral_bij_count_space[symmetric, OF *] nn_integral_count_space_nat)
show ?thesis
by (simp add: ** nn_integral_suminf from_nat_into)
qed

lemma of_bool_Bex_eq_nn_integral:
assumes unique: "⋀x y. x ∈ X ⟹ y ∈ X ⟹ P x ⟹ P y ⟹ x = y"
shows "of_bool (∃y∈X. P y) = (∫⇧+y. of_bool (P y) ∂count_space X)"
proof cases
assume "∃y∈X. P y"
then obtain y where "P y" "y ∈ X" by auto
then show ?thesis
by (subst nn_integral_count_space'[where A="{y}"]) (auto dest: unique)
qed (auto cong: nn_integral_cong_simp)

lemma emeasure_UN_countable:
assumes sets[measurable]: "⋀i. i ∈ I ⟹ X i ∈ sets M" and I[simp]: "countable I"
assumes disj: "disjoint_family_on X I"
shows "emeasure M (⋃(X ` I)) = (∫⇧+i. emeasure M (X i) ∂count_space I)"
proof -
have eq: "⋀x. indicator (⋃(X ` I)) x = ∫⇧+ i. indicator (X i) x ∂count_space I"
proof cases
fix x assume x: "x ∈ ⋃(X ` I)"
then obtain j where j: "x ∈ X j" "j ∈ I"
by auto
with disj have "⋀i. i ∈ I ⟹ indicator (X i) x = (indicator {j} i::ennreal)"
by (auto simp: disjoint_family_on_def split: split_indicator)
with x j show "?thesis x"
by (simp cong: nn_integral_cong_simp)
qed (auto simp: nn_integral_0_iff_AE)

note sets.countable_UN'[unfolded subset_eq, measurable]
have "emeasure M (⋃(X ` I)) = (∫⇧+x. indicator (⋃(X ` I)) x ∂M)"
by simp
also have "… = (∫⇧+i. ∫⇧+x. indicator (X i) x ∂M ∂count_space I)"
finally show ?thesis
by (simp cong: nn_integral_cong_simp)
qed

lemma emeasure_countable_singleton:
assumes sets: "⋀x. x ∈ X ⟹ {x} ∈ sets M" and X: "countable X"
shows "emeasure M X = (∫⇧+x. emeasure M {x} ∂count_space X)"
proof -
have "emeasure M (⋃i∈X. {i}) = (∫⇧+x. emeasure M {x} ∂count_space X)"
using assms by (intro emeasure_UN_countable) (auto simp: disjoint_family_on_def)
also have "(⋃i∈X. {i}) = X" by auto
finally show ?thesis .
qed

lemma measure_eqI_countable:
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and A: "countable A"
assumes eq: "⋀a. a ∈ A ⟹ emeasure M {a} = emeasure N {a}"
shows "M = N"
proof (rule measure_eqI)
fix X assume "X ∈ sets M"
then have X: "X ⊆ A" by auto
moreover from A X have "countable X" by (auto dest: countable_subset)
ultimately have
"emeasure M X = (∫⇧+a. emeasure M {a} ∂count_space X)"
"emeasure N X = (∫⇧+a. emeasure N {a} ∂count_space X)"
by (auto intro!: emeasure_countable_singleton)
moreover have "(∫⇧+a. emeasure M {a} ∂count_space X) = (∫⇧+a. emeasure N {a} ∂count_space X)"
using X by (intro nn_integral_cong eq) auto
ultimately show "emeasure M X = emeasure N X"
by simp
qed simp

lemma measure_eqI_countable_AE:
assumes [simp]: "sets M = UNIV" "sets N = UNIV"
assumes ae: "AE x in M. x ∈ Ω" "AE x in N. x ∈ Ω" and [simp]: "countable Ω"
assumes eq: "⋀x. x ∈ Ω ⟹ emeasure M {x} = emeasure N {x}"
shows "M = N"
proof (rule measure_eqI)
fix A
have "emeasure N A = emeasure N {x∈Ω. x ∈ A}"
using ae by (intro emeasure_eq_AE) auto
also have "… = (∫⇧+x. emeasure N {x} ∂count_space {x∈Ω. x ∈ A})"
by (intro emeasure_countable_singleton) auto
also have "… = (∫⇧+x. emeasure M {x} ∂count_space {x∈Ω. x ∈ A})"
by (intro nn_integral_cong eq[symmetric]) auto
also have "… = emeasure M {x∈Ω. x ∈ A}"
by (intro emeasure_countable_singleton[symmetric]) auto
also have "… = emeasure M A"
using ae by (intro emeasure_eq_AE) auto
finally show "emeasure M A = emeasure N A" ..
qed simp

lemma nn_integral_monotone_convergence_SUP_nat:
fixes f :: "'a ⇒ nat ⇒ ennreal"
assumes chain: "Complete_Partial_Order.chain (≤) (f ` Y)"
and nonempty: "Y ≠ {}"
shows "(∫⇧+ x. (SUP i∈Y. f i x) ∂count_space UNIV) = (SUP i∈Y. (∫⇧+ x. f i x ∂count_space UNIV))"
(is "?lhs = ?rhs" is "integral⇧N ?M _ = _")
proof (rule order_class.order.antisym)
show "?rhs ≤ ?lhs"
by (auto intro!: SUP_least SUP_upper nn_integral_mono)
next
have "∃g. incseq g ∧ range g ⊆ (λi. f i x) ` Y ∧ (SUP i∈Y. f i x) = (SUP i. g i)" for x
by (rule ennreal_Sup_countable_SUP) (simp add: nonempty)
then obtain g where incseq: "⋀x. incseq (g x)"
and range: "⋀x. range (g x) ⊆ (λi. f i x) ` Y"
and sup: "⋀x. (SUP i∈Y. f i x) = (SUP i. g x i)" by moura
from incseq have incseq': "incseq (λi x. g x i)"
by(blast intro: incseq_SucI le_funI dest: incseq_SucD)

have "?lhs = ∫⇧+ x. (SUP i. g x i) ∂?M" by(simp add: sup)
also have "… = (SUP i. ∫⇧+ x. g x i ∂?M)" using incseq'
by(rule nn_integral_monotone_convergence_SUP) simp
also have "… ≤ (SUP i∈Y. ∫⇧+ x. f i x ∂?M)"
proof(rule SUP_least)
fix n
have "⋀x. ∃i. g x n = f i x ∧ i ∈ Y" using range by blast
then obtain I where I: "⋀x. g x n = f (I x) x" "⋀x. I x ∈ Y" by moura

have "(∫⇧+ x. g x n ∂count_space UNIV) = (∑x. g x n)"
by(rule nn_integral_count_space_nat)
also have "… = (SUP m. ∑x<m. g x n)"
by(rule suminf_eq_SUP)
also have "… ≤ (SUP i∈Y. ∫⇧+ x. f i x ∂?M)"
proof(rule SUP_mono)
fix m
show "∃m'∈Y. (∑x<m. g x n) ≤ (∫⇧+ x. f m' x ∂?M)"
proof(cases "m > 0")
case False
thus ?thesis using nonempty by auto
next
case True
let ?Y = "I ` {..<m}"
have "f ` ?Y ⊆ f ` Y" using I by auto
with chain have chain': "Complete_Partial_Order.chain (≤) (f ` ?Y)" by(rule chain_subset)
hence "Sup (f ` ?Y) ∈ f ` ?Y"
by(rule ccpo_class.in_chain_finite)(auto simp add: True lessThan_empty_iff)
then obtain m' where "m' < m" and m': "(SUP i∈?Y. f i) = f (I m')" by auto
have "I m' ∈ Y" using I by blast
have "(∑x<m. g x n) ≤ (∑x<m. f (I m') x)"
proof(rule sum_mono)
fix x
assume "x ∈ {..<m}"
hence "x < m" by simp
have "g x n = f (I x) x" by(simp add: I)
also have "… ≤ (SUP i∈?Y. f i) x" unfolding Sup_fun_def image_image
using ‹x ∈ {..<m}› by (rule Sup_upper [OF imageI])
also have "… = f (I m') x" unfolding m' by simp
finally show "g x n ≤ f (I m') x" .
qed
also have "… ≤ (SUP m. (∑x<m. f (I m') x))"
by(rule SUP_upper) simp
also have "… = (∑x. f (I m') x)"
by(rule suminf_eq_SUP[symmetric])
also have "… = (∫⇧+ x. f (I m') x ∂?M)"
by(rule nn_integral_count_space_nat[symmetric])
finally show ?thesis using ‹I m' ∈ Y› by blast
qed
qed
finally show "(∫⇧+ x. g x n ∂count_space UNIV) ≤ …" .
qed
finally show "?lhs ≤ ?rhs" .
qed

lemma power_series_tendsto_at_left:
assumes nonneg: "⋀i. 0 ≤ f i" and summable: "⋀z. 0 ≤ z ⟹ z < 1 ⟹ summable (λn. f n * z^n)"
shows "((λz. ennreal (∑n. f n * z^n)) ⤏ (∑n. ennreal (f n))) (at_left (1::real))"
proof (intro tendsto_at_left_sequentially)
show "0 < (1::real)" by simp
fix S :: "nat ⇒ real" assume S: "⋀n. S n < 1" "⋀n. 0 < S n" "S ⇢ 1" "incseq S"
then have S_nonneg: "⋀i. 0 ≤ S i" by (auto intro: less_imp_le)

have "(λi. (∫⇧+n. f n * S i^n ∂count_space UNIV)) ⇢ (∫⇧+n. ennreal (f n) ∂count_space UNIV)"
proof (rule nn_integral_LIMSEQ)
show "incseq (λi n. ennreal (f n * S i^n))"
using S by (auto intro!: mult_mono power_mono nonneg ennreal_leI
simp: incseq_def le_fun_def less_imp_le)
fix n have "(λi. ennreal (f n * S i^n)) ⇢ ennreal (f n * 1^n)"
by (intro tendsto_intros tendsto_ennrealI S)
then show "(λi. ennreal (f n * S i^n)) ⇢ ennreal (f n)"
by simp
qed (auto simp: S_nonneg intro!: mult_nonneg_nonneg nonneg)
also have "(λi. (∫⇧+n. f n * S i^n ∂count_space UNIV)) = (λi. ∑n. f n * S i^n)"
by (subst nn_integral_count_space_nat)
(intro ext suminf_ennreal2 mult_nonneg_nonneg nonneg S_nonneg
zero_le_power summable S)+
also have "(∫⇧+n. ennreal (f n) ∂count_space UNIV) = (∑n. ennreal (f n))"
finally show "(λn. ennreal (∑na. f na * S n ^ na)) ⇢ (∑n. ennreal (f n))" .
qed

subsubsection ‹Measures with Restricted Space›

lemma simple_function_restrict_space_ennreal:
fixes f :: "'a ⇒ ennreal"
assumes "Ω ∩ space M ∈ sets M"
shows "simple_function (restrict_space M Ω) f ⟷ simple_function M (λx. f x * indicator Ω x)"
proof -
{ assume "finite (f ` space (restrict_space M Ω))"
then have "finite (f ` space (restrict_space M Ω) ∪ {0})" by simp
then have "finite ((λx. f x * indicator Ω x) ` space M)"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
moreover
{ assume "finite ((λx. f x * indicator Ω x) ` space M)"
then have "finite (f ` space (restrict_space M Ω))"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
ultimately show ?thesis
unfolding
simple_function_iff_borel_measurable borel_measurable_restrict_space_iff_ennreal[OF assms]
by auto
qed

lemma simple_function_restrict_space:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes "Ω ∩ space M ∈ sets M"
shows "simple_function (restrict_space M Ω) f ⟷ simple_function M (λx. indicator Ω x *⇩R f x)"
proof -
{ assume "finite (f ` space (restrict_space M Ω))"
then have "finite (f ` space (restrict_space M Ω) ∪ {0})" by simp
then have "finite ((λx. indicator Ω x *⇩R f x) ` space M)"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
moreover
{ assume "finite ((λx. indicator Ω x *⇩R f x) ` space M)"
then have "finite (f ` space (restrict_space M Ω))"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
ultimately show ?thesis
unfolding simple_function_iff_borel_measurable
borel_measurable_restrict_space_iff[OF assms]
by auto
qed

lemma simple_integral_restrict_space:
assumes Ω: "Ω ∩ space M ∈ sets M" "simple_function (restrict_space M Ω) f"
shows "simple_integral (restrict_space M Ω) f = simple_integral M (λx. f x * indicator Ω x)"
using simple_function_restrict_space_ennreal[THEN iffD1, OF Ω, THEN simple_functionD(1)]
by (auto simp add: space_restrict_space emeasure_restrict_space[OF Ω(1)] le_infI2 simple_integral_def
split: split_indicator split_indicator_asm
intro!: sum.mono_neutral_cong_left ennreal_mult_left_cong arg_cong2[where f=emeasure])

lemma nn_integral_restrict_space:
assumes Ω[simp]: "Ω ∩ space M ∈ sets M"
shows "nn_integral (restrict_space M Ω) f = nn_integral M (λx. f x * indicator Ω x)"
proof -
let ?R = "restrict_space M Ω" and ?X = "λM f. {s. simple_function M s ∧ s ≤ f ∧ (∀x. s x < top)}"
have "integral⇧S ?R ` ?X ?R f = integral⇧S M ` ?X M (λx. f x * indicator Ω x)"
proof (safe intro!: image_eqI)
fix s assume s: "simple_function ?R s" "s ≤ f" "∀x. s x < top"
from s show "integral⇧S (restrict_space M Ω) s = integral⇧S M (λx. s x * indicator Ω x)"
by (intro simple_integral_restrict_space) auto
from s show "simple_function M (λx. s x * indicator Ω x)"
from s show "(λx. s x * indicator Ω x) ≤ (λx. f x * indicator Ω x)"
"⋀x. s x * indicator Ω x < top"
by (auto split: split_indicator simp: le_fun_def image_subset_iff)
next
fix s assume s: "simple_function M s" "s ≤ (λx. f x * indicator Ω x)" "∀x. s x < top"
then have "simple_function M (λx. s x * indicator (Ω ∩ space M) x)" (is ?s')
by (intro simple_function_mult simple_function_indicator) auto
also have "?s' ⟷ simple_function M (λx. s x * indicator Ω x)"
by (rule simple_function_cong) (auto split: split_indicator)
finally show sf: "simple_function (restrict_space M Ω) s"

from s have s_eq: "s = (λx. s x * indicator Ω x)"
by (auto simp add: fun_eq_iff le_fun_def image_subset_iff
split: split_indicator split_indicator_asm
intro: antisym)

show "integral⇧S M s = integral⇧S (restrict_space M Ω) s"
by (subst s_eq) (rule simple_integral_restrict_space[symmetric, OF Ω sf])
show "⋀x. s x < top"
using s by (auto simp: image_subset_iff)
from s show "s ≤ f"
by (subst s_eq) (auto simp: image_subset_iff le_fun_def split: split_indicator split_indicator_asm)
qed
then show ?thesis
unfolding nn_integral_def_finite by (simp cong del: SUP_cong_simp)
qed

lemma nn_integral_count_space_indicator:
assumes "NO_MATCH (UNIV::'a set) (X::'a set)"
shows "(∫⇧+x. f x ∂count_space X) = (∫⇧+x. f x * indicator X x ∂count_space UNIV)"

lemma nn_integral_count_space_eq:
"(⋀x. x ∈ A - B ⟹ f x = 0) ⟹ (⋀x. x ∈ B - A ⟹ f x = 0) ⟹
(∫⇧+x. f x ∂count_space A) = (∫⇧+x. f x ∂count_space B)"
by (auto simp: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)

lemma nn_integral_ge_point:
assumes "x ∈ A"
shows "p x ≤ ∫⇧+ x. p x ∂count_space A"
proof -
from assms have "p x ≤ ∫⇧+ x. p x ∂count_space {x}"
also have "… = ∫⇧+ x'. p x' * indicator {x} x' ∂count_space A"
using assms by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
also have "… ≤ ∫⇧+ x. p x ∂count_space A"
finally show ?thesis .
qed

subsubsection ‹Measure spaces with an associated density›

definition✐‹tag important› density :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ 'a measure" where
"density M f = measure_of (space M) (sets M) (λA. ∫⇧+ x. f x * indicator A x ∂M)"

lemma
shows sets_density[simp, measurable_cong]: "sets (density M f) = sets M"
and space_density[simp]: "space (density M f) = space M"
by (auto simp: density_def)

(* FIXME: add conversion to simplify space, sets and measurable *)
lemma space_density_imp[measurable_dest]:
"⋀x M f. x ∈ space (density M f) ⟹ x ∈ space M" by auto

lemma
shows measurable_density_eq1[simp]: "g ∈ measurable (density Mg f) Mg' ⟷ g ∈ measurable Mg Mg'"
and measurable_density_eq2[simp]: "h ∈ measurable Mh (density Mh' f) ⟷ h ∈ measurable Mh Mh'"
and simple_function_density_eq[simp]: "simple_function (density Mu f) u ⟷ simple_function Mu u"
unfolding measurable_def simple_function_def by simp_all

lemma density_cong: "f ∈ borel_measurable M ⟹ f' ∈ borel_measurable M ⟹
(AE x in M. f x = f' x) ⟹ density M f = density M f'"
unfolding density_def by (auto intro!: measure_of_eq nn_integral_cong_AE sets.space_closed)

lemma emeasure_density:
assumes f[measurable]: "f ∈ borel_measurable M" and A[measurable]: "A ∈ sets M"
shows "emeasure (density M f) A = (∫⇧+ x. f x * indicator A x ∂M)"
(is "_ = ?μ A")
unfolding density_def
proof (rule emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) ?μ"
using f by (auto simp: positive_def)
fix A :: "nat ⇒ 'a set" assume "range A ⊆ sets M"
then have "⋀i. A i ∈ sets M" by auto
then have *: "⋀i. (λx. f x * indicator (A i) x) ∈ borel_measurable M"
by auto
assume disj: "disjoint_family A"
then have "(∑n. ?μ (A n)) = (∫⇧+ x. (∑n. f x * indicator (A n) x) ∂M)"
using f * by (subst nn_integral_suminf) auto
also have "(∫⇧+ x. (∑n. f x * indicator (A n) x) ∂M) = (∫⇧+ x. f x * (∑n. indicator (A n) x) ∂M)"
using f by (auto intro!: ennreal_suminf_cmult nn_integral_cong_AE)
also have "… = (∫⇧+ x. f x * indicator (⋃n. A n) x ∂M)"
unfolding suminf_indicator[OF disj] ..
finally show "(∑i. ∫⇧+ x. f x * indicator (A i) x ∂M) = ∫⇧+ x. f x * indicator (⋃i. A i) x ∂M" .
qed
qed fact

lemma null_sets_density_iff:
assumes f: "f ∈ borel_measurable M"
shows "A ∈ null_sets (density M f) ⟷ A ∈ sets M ∧ (AE x in M. x ∈ A ⟶ f x = 0)"
proof -
{ assume "A ∈ sets M"
have "(∫⇧+x. f x * indicator A x ∂M) = 0 ⟷ emeasure M {x ∈ space M. f x * indicator A x ≠ 0} = 0"
using f ‹A ∈ sets M› by (intro nn_integral_0_iff) auto
also have "… ⟷ (AE x in M. f x * indicator A x = 0)"
using f ‹A ∈ sets M› by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
also have "(AE x in M. f x * indicator A x = 0) ⟷ (AE x in M. x ∈ A ⟶ f x ≤ 0)"
by (auto simp add: indicator_def max_def split: if_split_asm)
finally have "(∫⇧+x. f x * indicator A x ∂M) = 0 ⟷ (AE x in M. x ∈ A ⟶ f x ≤ 0)" . }
with f show ?thesis
by (simp add: null_sets_def emeasure_density cong: conj_cong)
qed

lemma AE_density:
assumes f: "f ∈ borel_measurable M"
shows "(AE x in density M f. P x) ⟷ (AE x in M. 0 < f x ⟶ P x)"
proof
assume "AE x in density M f. P x"
with f obtain N where "{x ∈ space M. ¬ P x} ⊆ N" "N ∈ sets M" and ae: "AE x in M. x ∈ N ⟶ f x = 0"
by (auto simp: eventually_ae_filter null_sets_density_iff)
then have "AE x in M. x ∉ N ⟶ P x" by auto
with ae show "AE x in M. 0 < f x ⟶ P x"
by (rule eventually_elim2) auto
next
fix N assume ae: "AE x in M. 0 < f x ⟶ P x"
then obtain N where "{x ∈ space M. ¬ (0 < f x ⟶ P x)} ⊆ N" "N ∈ null_sets M"
by (auto simp: eventually_ae_filter)
then have *: "{x ∈ space (density M f). ¬ P x} ⊆ N ∪ {x∈space M. f x = 0}"
"N ∪ {x∈space M. f x = 0} ∈ sets M" and ae2: "AE x in M. x ∉ N"
using f by (auto simp: subset_eq zero_less_iff_neq_zero intro!: AE_not_in)
show "AE x in density M f. P x"
using ae2
unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
by (intro exI[of _ "N ∪ {x∈space M. f x = 0}"] conjI *) (auto elim: eventually_elim2)
qed

lemma✐‹tag important› nn_integral_density:
assumes f: "f ∈ borel_measurable M"
assumes g: "g ∈ borel_measurable M"
shows "integral⇧N (density M f) g = (∫⇧+ x. f x * g x ∂M)"
using g proof induct
case (cong u v)
then show ?case
apply (subst nn_integral_cong[OF cong(3)])
apply (simp_all cong: nn_integral_cong)
done
next
case (set A) then show ?case
next
case (mult u c)
moreover have "⋀x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
ultimately show ?case
using f by (simp add: nn_integral_cmult)
next
then have "⋀x. f x * (v x + u x) = f x * v x + f x * u x"
next
case (seq U)
have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
by eventually_elim (simp add: SUP_mult_left_ennreal seq)
from seq f show ?case
apply (subst nn_integral_cong_AE[OF eq])
apply (subst nn_integral_monotone_convergence_SUP_AE)
apply (auto simp: incseq_def le_fun_def intro!: mult_left_mono)
done
qed

lemma density_distr:
assumes [measurable]: "f ∈ borel_measurable N" "X ∈ measurable M N"
shows "density (distr M N X) f = distr (density M (λx. f (X x))) N X"
by (intro measure_eqI)
(auto simp add: emeasure_density nn_integral_distr emeasure_distr
split: split_indicator intro!: nn_integral_cong)

lemma emeasure_restricted:
assumes S: "S ∈ sets M" and X: "X ∈ sets M"
shows "emeasure (density M (indicator S)) X = emeasure M (S ∩ X)"
proof -
have "emeasure (density M (indicator S)) X = (∫⇧+x. indicator S x * indicator X x ∂M)"
using S X by (simp add: emeasure_density)
also have "… = (∫⇧+x. indicator (S ∩ X) x ∂M)"
by (auto intro!: nn_integral_cong simp: indicator_def)
also have "… = emeasure M (S ∩ X)"
using S X by (simp add: sets.Int)
finally show ?thesis .
qed

lemma measure_restricted:
"S ∈ sets M ⟹ X ∈ sets M ⟹ measure (density M (indicator S)) X = measure M (S ∩ X)"

lemma (in finite_measure) finite_measure_restricted:
"S ∈ sets M ⟹ finite_measure (density M (indicator S))"

lemma emeasure_density_const:
"A ∈ sets M ⟹ emeasure (density M (λ_. c)) A = c * emeasure M A"
by (auto simp: nn_integral_cmult_indicator emeasure_density)

lemma measure_density_const:
"A ∈ sets M ⟹ c ≠ ∞ ⟹ measure (density M (λ_. c)) A = enn2real c * measure M A"
by (auto simp: emeasure_density_const measure_def enn2real_mult)

lemma density_density_eq:
"f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹
density (density M f) g = density M (λx. f x * g x)"
by (auto intro!: measure_eqI simp: emeasure_density nn_integral_density ac_simps)

lemma distr_density_distr:
assumes T: "T ∈ measurable M M'" and T': "T' ∈ measurable M' M"
and inv: "∀x∈space M. T' (T x) = x"
assumes f: "f ∈ borel_measurable M'"
shows "distr (density (distr M M' T) f) M T' = density M (f ∘ T)" (is "?R = ?L")
proof (rule measure_eqI)
fix A assume A: "A ∈ sets ?R"
{ fix x assume "x ∈ space M"
with sets.sets_into_space[OF A]
have "indicator (T' -` A ∩ space M') (T x) = (indicator A x :: ennreal)"
using T inv by (auto simp: indicator_def measurable_space) }
with A T T' f show "emeasure ?R A = emeasure ?L A"
by (simp add: measurable_comp emeasure_density emeasure_distr
nn_integral_distr measurable_sets cong: nn_integral_cong)
qed simp

lemma density_density_divide:
fixes f g :: "'a ⇒ real"
assumes f: "f ∈ borel_measurable M" "AE x in M. 0 ≤ f x"
assumes g: "g ∈ borel_measurable M" "AE x in M. 0 ≤ g x"
assumes ac: "AE x in M. f x = 0 ⟶ g x = 0"
shows "density (density M f) (λx. g x / f x) = density M g"
proof -
have "density M g = density M (λx. ennreal (f x) * ennreal (g x / f x))"
using f g ac by (auto intro!: density_cong measurable_If simp: ennreal_mult[symmetric])
then show ?thesis
using f g by (subst density_density_eq) auto
qed

lemma density_1: "density M (λ_. 1) = M"
by (intro measure_eqI) (auto simp: emeasure_density)

assumes X: "X ∈ sets M"
assumes Mf[measurable]: "f ∈ borel_measurable M"
assumes Mg[measurable]: "g ∈ borel_measurable M"
shows "emeasure (density M f) X + emeasure (density M g) X =
emeasure (density M (λx. f x + g x)) X"
using assms
apply (subst (1 2 3) emeasure_density, simp_all) []
apply (intro nn_integral_cong, simp split: split_indicator)
done

subsubsection ‹Point measure›

definition✐‹tag important› point_measure :: "'a set ⇒ ('a ⇒ ennreal) ⇒ 'a measure" where
"point_measure A f = density (count_space A) f"

lemma
shows space_point_measure: "space (point_measure A f) = A"
and sets_point_measure: "sets (point_measure A f) = Pow A"
by (auto simp: point_measure_def)

lemma sets_point_measure_count_space[measurable_cong]: "sets (point_measure A f) = sets (count_space A)"

lemma measurable_point_measure_eq1[simp]:
"g ∈ measurable (point_measure A f) M ⟷ g ∈ A → space M"
unfolding point_measure_def by simp

lemma measurable_point_measure_eq2_finite[simp]:
"finite A ⟹
g ∈ measurable M (point_measure A f) ⟷
(g ∈ space M → A ∧ (∀a∈A. g -` {a} ∩ space M ∈ sets M))"
unfolding point_measure_def by (simp add: measurable_count_space_eq2)

lemma simple_function_point_measure[simp]:
"simple_function (point_measure A f) g ⟷ finite (g ` A)"

lemma emeasure_point_measure:
assumes A: "finite {a∈X. 0 < f a}" "X ⊆ A"
shows "emeasure (point_measure A f) X = (∑a|a∈X ∧ 0 < f a. f a)"
proof -
have "{a. (a ∈ X ⟶ a ∈ A ∧ 0 < f a) ∧ a ∈ X} = {a∈X. 0 < f a}"
using ‹X ⊆ A› by auto
with A show ?thesis
by (simp add: emeasure_density nn_integral_count_space point_measure_def indicator_def)
qed

lemma emeasure_point_measure_finite:
"finite A ⟹ X ⊆ A ⟹ emeasure (point_measure A f) X = (∑a∈X. f a)"
by (subst emeasure_point_measure) (auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le)

lemma emeasure_point_measure_finite2:
"X ⊆ A ⟹ finite X ⟹ emeasure (point_measure A f) X = (∑a∈X. f a)"
by (subst emeasure_point_measure)
(auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le)

lemma null_sets_point_measure_iff:
"X ∈ null_sets (point_measure A f) ⟷ X ⊆ A ∧ (∀x∈X. f x = 0)"
by (auto simp: AE_count_space null_sets_density_iff point_measure_def)

lemma AE_point_measure:
"(AE x in point_measure A f. P x) ⟷ (∀x∈A. 0 < f x ⟶ P x)"
unfolding point_measure_def
by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)

lemma nn_integral_point_measure:
"finite {a∈A. 0 < f a ∧ 0 < g a} ⟹
integral⇧N (point_measure A f) g = (∑a|a∈A ∧ 0 < f a ∧ 0 < g a. f a * g a)"
unfolding point_measure_def
by (subst nn_integral_density)

lemma nn_integral_point_measure_finite:
"finite A ⟹ integral⇧N (point_measure A f) g = (∑a∈A. f a * g a)"
by (subst nn_integral_point_measure) (auto intro!: sum.mono_neutral_left simp: less_le)

subsubsection ‹Uniform measure›

definition✐‹tag important› "uniform_measure M A = density M (λx. indicator A x / emeasure M A)"

lemma
shows sets_uniform_measure[simp, measurable_cong]: "sets (uniform_measure M A) = sets M"
and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
by (auto simp: uniform_measure_def)

lemma emeasure_uniform_measure[simp]:
assumes A: "A ∈ sets M" and B: "B ∈ sets M"
shows "emeasure (uniform_measure M A) B = emeasure M (A ∩ B) / emeasure M A"
proof -
from A B have "emeasure (uniform_measure M A) B = (∫⇧+x. (1 / emeasure M A) * indicator (A ∩ B) x ∂M)"
by (auto simp add: uniform_measure_def emeasure_density divide_ennreal_def split: split_indicator
intro!: nn_integral_cong)
also have "… = emeasure M (A ∩ B) / emeasure M A"
using A B
by (subst nn_integral_cmult_indicator) (simp_all add: sets.Int divide_ennreal_def mult.commute)
finally show ?thesis .
qed

lemma measure_uniform_measure[simp]:
assumes A: "emeasure M A ≠ 0" "emeasure M A ≠ ∞" and B: "B ∈ sets M"
shows "measure (uniform_measure M A) B = measure M (A ∩ B) / measure M A"
using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
by (cases "emeasure M A" "emeasure M (A ∩ B)" rule: ennreal2_cases)
(simp_all add: measure_def divide_ennreal top_ennreal.rep_eq top_ereal_def ennreal_top_divide)

lemma AE_uniform_measureI:
"A ∈ sets M ⟹ (AE x in M. x ∈ A ⟶ P x) ⟹ (AE x in uniform_measure M A. P x)"
unfolding uniform_measure_def by (auto simp: AE_density divide_ennreal_def)

lemma emeasure_uniform_measure_1:
"emeasure M S ≠ 0 ⟹ emeasure M S ≠ ∞ ⟹ emeasure (uniform_measure M S) S = 1"
by (subst emeasure_uniform_measure)
zero_less_iff_neq_zero[symmetric])

lemma nn_integral_uniform_measure:
assumes f[measurable]: "f ∈ borel_measurable M" and S[measurable]: "S ∈ sets M"
shows "(∫⇧+x. f x ∂uniform_measure M S) = (∫⇧+x. f x * indicator S x ∂M) / emeasure M S"
proof -
{ assume "emeasure M S = ∞"
then have ?thesis
by (simp add: uniform_measure_def nn_integral_density f) }
moreover
{ assume [simp]: "emeasure M S = 0"
then have ae: "AE x in M. x ∉ S"
using sets.sets_into_space[OF S]
by (subst AE_iff_measurable[OF _ refl]) (simp_all add: subset_eq cong: rev_conj_cong)
from ae have "(∫⇧+ x. indicator S x / 0 * f x ∂M) = 0"
by (subst nn_integral_0_iff_AE) auto
moreover from ae have "(∫⇧+ x. f x * indicator S x ∂M) = 0"
by (subst nn_integral_0_iff_AE) auto
ultimately have ?thesis
by (simp add: uniform_measure_def nn_integral_density f) }
moreover have "emeasure M S ≠ 0 ⟹ emeasure M S ≠ ∞ ⟹ ?thesis"
unfolding uniform_measure_def
by (subst nn_integral_density)
(auto simp: ennreal_times_divide f nn_integral_divide[symmetric] mult.commute)
ultimately show ?thesis by blast
qed

lemma AE_uniform_measure:
assumes "emeasure M A ≠ 0" "emeasure M A < ∞"
shows "(AE x in uniform_measure M A. P x) ⟷ (AE x in M. x ∈ A ⟶ P x)"
proof -
have "A ∈ sets M"
using ‹emeasure M A ≠ 0› by (metis emeasure_notin_sets)
moreover have "⋀x. 0 < indicator A x / emeasure M A ⟷ x ∈ A"
using assms
by (cases "emeasure M A") (auto split: split_indicator simp: ennreal_zero_less_divide)
ultimately show ?thesis
unfolding uniform_measure_def by (simp add: AE_density)
qed

subsubsection✐‹tag unimportant› ‹Null measure›

lemma null_measure_eq_density: "null_measure M = density M (λ_. 0)"
by (intro measure_eqI) (simp_all add: emeasure_density)

lemma nn_integral_null_measure[simp]: "(∫⇧+x. f x ∂null_measure M) = 0"
by (auto simp add: nn_integral_def simple_integral_def SUP_constant bot_ennreal_def le_fun_def
intro!: exI[of _ "λx. 0"])

lemma density_null_measure[simp]: "density (null_measure M) f = null_measure M"
proof (intro measure_eqI)
fix A show "emeasure (density (null_measure M) f) A = emeasure (null_measure M) A"
by (simp add: density_def) (simp only: null_measure_def[symmetric] emeasure_null_measure)
qed simp

subsubsection ‹Uniform count measure›

definition✐‹tag important› "uniform_count_measure A = point_measure A (λx. 1 / card A)"

lemma
shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)

lemma sets_uniform_count_measure_count_space[measurable_cong]:
"sets (uniform_count_measure A) = sets (count_space A)"

lemma emeasure_uniform_count_measure:
"finite A ⟹ X ⊆ A ⟹ emeasure (uniform_count_measure A) X = card X / card A"
by (simp add: emeasure_point_measure_finite uniform_count_measure_def divide_inverse ennreal_mult
ennreal_of_nat_eq_real_of_nat)

lemma measure_uniform_count_measure:
"finite A ⟹ X ⊆ A ⟹ measure (uniform_count_measure A) X = card X / card A"
by (simp add: emeasure_point_measure_finite uniform_count_measure_def measure_def enn2real_mult)

lemma space_uniform_count_measure_empty_iff [simp]:
"space (uniform_count_measure X) = {} ⟷ X = {}"

lemma sets_uniform_count_measure_eq_UNIV [simp]:
"sets (uniform_count_measure UNIV) = UNIV ⟷ True"
"UNIV = sets (uniform_count_measure UNIV) ⟷ True"

subsubsection✐‹tag unimportant› ‹Scaled measure›

lemma nn_integral_scale_measure:
assumes f: "f ∈ borel_measurable M"
shows "nn_integral (scale_measure r M) f = r * nn_integral M f"
using f
proof induction
case (cong f g)
thus ?case
by(simp add: cong.hyps space_scale_measure cong: nn_integral_cong_simp)
next
case (mult f c)
thus ?case
by(simp add: nn_integral_cmult max_def mult.assoc mult.left_commute)
next