Theory Nonnegative_Lebesgue_Integration

(*  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)
  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]
  by (simp add: comp_def)

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)
      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: 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"
      by (auto simp add: le_iff_add)
  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 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
        by (simp add: real)
    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"
    by (simp add: borel_measurable_simple_function)

  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 (simp add: sum.If_cases)
      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

lemma simple_integral_add[simp]:
  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)
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"
        by (simp add: less_le)
      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] obtain N
    where N: "{x ∈ space M. ¬ u x ≤ v x} ⊆ N" "emeasure M N = 0" "N ∈ sets M"
    by auto
  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"
    by (simp add: AE_all_countable)
  from this[THEN AE_E] obtain N
    where N: "{x ∈ space M. ¬ (∀i. f i x ≤ f (Suc i) x)} ⊆ N" "emeasure M N = 0" "N ∈ sets M"
    by auto
  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]
  by (simp add: nn_integral_eq_simple_integral[symmetric] borel_measurable_simple_function)

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 -
  obtain u
    where "⋀i. simple_function M (u i)" "incseq u" "⋀i x. u i x < top" "⋀x. (SUP i. u i x) = f x"
    using borel_measurable_implies_simple_function_sequence' f(1)
    by auto
  note u = nn_integral_monotone_convergence_simple[OF this(2,1)] this

  obtain v where
    "⋀i. simple_function M (v i)" "incseq v" "⋀i x. v i x < top" "⋀x. (SUP i. v i x) = g x"
    using borel_measurable_implies_simple_function_sequence' g(1)
    by auto
  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]
  obtain l where "⋀i. simple_function M (l i)" "incseq l" "⋀i x. l i x < top" "⋀x. (SUP i. l i x) = a * f x + g x"
    by auto
  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
      by (auto intro!: add_mono mult_left_mono)
    { 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
        by (auto intro!: ennreal_SUP_add simp: incseq_Suc_iff le_fun_def add_mono mult_left_mono) }
    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]
    by (simp add: ennreal_SUP_add[OF inc] v u SUP_mult_left_ennreal[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)

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
    by (simp add: nn_integral_cmult)
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)

lemma nn_integral_add:
  "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: "(λx. u x * indicator A x) ∈ borel_measurable M" and "A ∈ sets M"
  shows "(emeasure M) ({x∈A. 1 ≤ c * u x}) ≤ c * (∫⇧+ x. u x * indicator A x ∂M)"
    (is "(emeasure M) ?A ≤ _ * <