Theory Bochner_Integration

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

section ‹Bochner Integration for Vector-Valued Functions›

theory Bochner_Integration
  imports Finite_Product_Measure
begin

text ‹

In the following development of the Bochner integral we use second countable topologies instead
of separable spaces. A second countable topology is also separable.

›

proposition borel_measurable_implies_sequence_metric:
  fixes f :: "'a ⇒ 'b :: {metric_space, second_countable_topology}"
  assumes [measurable]: "f ∈ borel_measurable M"
  shows "∃F. (∀i. simple_function M (F i)) ∧ (∀x∈space M. (λi. F i x) ⇢ f x) ∧
    (∀i. ∀x∈space M. dist (F i x) z ≤ 2 * dist (f x) z)"
proof -
  obtain D :: "'b set" where "countable D" and D: "⋀X. open X ⟹ X ≠ {} ⟹ ∃d∈D. d ∈ X"
    by (erule countable_dense_setE)

  define e where "e = from_nat_into D"
  { fix n x
    obtain d where "d ∈ D" and d: "d ∈ ball x (1 / Suc n)"
      using D[of "ball x (1 / Suc n)"] by auto
    from ‹d ∈ D› D[of UNIV] ‹countable D› obtain i where "d = e i"
      unfolding e_def by (auto dest: from_nat_into_surj)
    with d have "∃i. dist x (e i) < 1 / Suc n"
      by auto }
  note e = this

  define A where [abs_def]: "A m n =
    {x∈space M. dist (f x) (e n) < 1 / (Suc m) ∧ 1 / (Suc m) ≤ dist (f x) z}" for m n
  define B where [abs_def]: "B m = disjointed (A m)" for m

  define m where [abs_def]: "m N x = Max {m. m ≤ N ∧ x ∈ (⋃n≤N. B m n)}" for N x
  define F where [abs_def]: "F N x =
    (if (∃m≤N. x ∈ (⋃n≤N. B m n)) ∧ (∃n≤N. x ∈ B (m N x) n)
     then e (LEAST n. x ∈ B (m N x) n) else z)" for N x

  have B_imp_A[intro, simp]: "⋀x m n. x ∈ B m n ⟹ x ∈ A m n"
    using disjointed_subset[of "A m" for m] unfolding B_def by auto

  { fix m
    have "⋀n. A m n ∈ sets M"
      by (auto simp: A_def)
    then have "⋀n. B m n ∈ sets M"
      using sets.range_disjointed_sets[of "A m" M] by (auto simp: B_def) }
  note this[measurable]

  { fix N i x assume "∃m≤N. x ∈ (⋃n≤N. B m n)"
    then have "m N x ∈ {m::nat. m ≤ N ∧ x ∈ (⋃n≤N. B m n)}"
      unfolding m_def by (intro Max_in) auto
    then have "m N x ≤ N" "∃n≤N. x ∈ B (m N x) n"
      by auto }
  note m = this

  { fix j N i x assume "j ≤ N" "i ≤ N" "x ∈ B j i"
    then have "j ≤ m N x"
      unfolding m_def by (intro Max_ge) auto }
  note m_upper = this

  show ?thesis
    unfolding simple_function_def
  proof (safe intro!: exI[of _ F])
    have [measurable]: "⋀i. F i ∈ borel_measurable M"
      unfolding F_def m_def by measurable
    show "⋀x i. F i -` {x} ∩ space M ∈ sets M"
      by measurable

    { fix i
      { fix n x assume "x ∈ B (m i x) n"
        then have "(LEAST n. x ∈ B (m i x) n) ≤ n"
          by (intro Least_le)
        also assume "n ≤ i"
        finally have "(LEAST n. x ∈ B (m i x) n) ≤ i" . }
      then have "F i ` space M ⊆ {z} ∪ e ` {.. i}"
        by (auto simp: F_def)
      then show "finite (F i ` space M)"
        by (rule finite_subset) auto }

    { fix N i n x assume "i ≤ N" "n ≤ N" "x ∈ B i n"
      then have 1: "∃m≤N. x ∈ (⋃n≤N. B m n)" by auto
      from m[OF this] obtain n where n: "m N x ≤ N" "n ≤ N" "x ∈ B (m N x) n" by auto
      moreover
      define L where "L = (LEAST n. x ∈ B (m N x) n)"
      have "dist (f x) (e L) < 1 / Suc (m N x)"
      proof -
        have "x ∈ B (m N x) L"
          using n(3) unfolding L_def by (rule LeastI)
        then have "x ∈ A (m N x) L"
          by auto
        then show ?thesis
          unfolding A_def by simp
      qed
      ultimately have "dist (f x) (F N x) < 1 / Suc (m N x)"
        by (auto simp add: F_def L_def) }
    note * = this

    fix x assume "x ∈ space M"
    show "(λi. F i x) ⇢ f x"
    proof cases
      assume "f x = z"
      then have "⋀i n. x ∉ A i n"
        unfolding A_def by auto
      then have "⋀i. F i x = z"
        by (auto simp: F_def)
      then show ?thesis
        using ‹f x = z› by auto
    next
      assume "f x ≠ z"

      show ?thesis
      proof (rule tendstoI)
        fix e :: real assume "0 < e"
        with ‹f x ≠ z› obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
          by (metis dist_nz order_less_trans neq_iff nat_approx_posE)
        with ‹x∈space M› ‹f x ≠ z› have "x ∈ (⋃i. B n i)"
          unfolding A_def B_def UN_disjointed_eq using e by auto
        then obtain i where i: "x ∈ B n i" by auto

        show "eventually (λi. dist (F i x) (f x) < e) sequentially"
          using eventually_ge_at_top[of "max n i"]
        proof eventually_elim
          fix j assume j: "max n i ≤ j"
          with i have "dist (f x) (F j x) < 1 / Suc (m j x)"
            by (intro *[OF _ _ i]) auto
          also have "… ≤ 1 / Suc n"
            using j m_upper[OF _ _ i]
            by (auto simp: field_simps)
          also note ‹1 / Suc n < e›
          finally show "dist (F j x) (f x) < e"
            by (simp add: less_imp_le dist_commute)
        qed
      qed
    qed
    fix i
    { fix n m assume "x ∈ A n m"
      then have "dist (e m) (f x) + dist (f x) z ≤ 2 * dist (f x) z"
        unfolding A_def by (auto simp: dist_commute)
      also have "dist (e m) z ≤ dist (e m) (f x) + dist (f x) z"
        by (rule dist_triangle)
      finally (xtrans) have "dist (e m) z ≤ 2 * dist (f x) z" . }
    then show "dist (F i x) z ≤ 2 * dist (f x) z"
      unfolding F_def
      apply auto
      apply (rule LeastI2)
      apply auto
      done
  qed
qed

lemma
  fixes f :: "'a ⇒ 'b::semiring_1" assumes "finite A"
  shows sum_mult_indicator[simp]: "(∑x ∈ A. f x * indicator (B x) (g x)) = (∑x∈{x∈A. g x ∈ B x}. f x)"
  and sum_indicator_mult[simp]: "(∑x ∈ A. indicator (B x) (g x) * f x) = (∑x∈{x∈A. g x ∈ B x}. f x)"
  unfolding indicator_def
  using assms by (auto intro!: sum.mono_neutral_cong_right split: if_split_asm)

lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
  fixes P :: "('a ⇒ real) ⇒ bool"
  assumes u: "u ∈ borel_measurable M" "⋀x. 0 ≤ u x"
  assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)"
  assumes mult: "⋀u c. 0 ≤ c ⟹ u ∈ borel_measurable M ⟹ (⋀x. 0 ≤ u x) ⟹ P u ⟹ P (λx. c * u x)"
  assumes add: "⋀u v. u ∈ borel_measurable M ⟹ (⋀x. 0 ≤ u x) ⟹ P u ⟹ v ∈ borel_measurable M ⟹ (⋀x. 0 ≤ v x) ⟹ (⋀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. 0 ≤ U i x) ⟹ (⋀i. P (U i)) ⟹ incseq U ⟹ (⋀x. x ∈ space M ⟹ (λi. U i x) ⇢ u x) ⟹ P u"
  shows "P u"
proof -
  have "(λx. ennreal (u x)) ∈ borel_measurable M" using u by auto
  from borel_measurable_implies_simple_function_sequence'[OF this]
  obtain U where U: "⋀i. simple_function M (U i)" "incseq U" "⋀i x. U i x < top" and
    sup: "⋀x. (SUP i. U i x) = ennreal (u x)"
    by blast

  define U' where [abs_def]: "U' i x = indicator (space M) x * enn2real (U i x)" for i x
  then have U'_sf[measurable]: "⋀i. simple_function M (U' i)"
    using U by (auto intro!: simple_function_compose1[where g=enn2real])

  show "P u"
  proof (rule seq)
    show U': "U' i ∈ borel_measurable M" "⋀x. 0 ≤ U' i x" for i
      using U by (auto
          intro: borel_measurable_simple_function
          intro!: borel_measurable_enn2real borel_measurable_times
          simp: U'_def zero_le_mult_iff)
    show "incseq U'"
      using U(2,3)
      by (auto simp: incseq_def le_fun_def image_iff eq_commute U'_def indicator_def enn2real_mono)

    fix x assume x: "x ∈ space M"
    have "(λi. U i x) ⇢ (SUP i. U i x)"
      using U(2) by (intro LIMSEQ_SUP) (auto simp: incseq_def le_fun_def)
    moreover have "(λi. U i x) = (λi. ennreal (U' i x))"
      using x U(3) by (auto simp: fun_eq_iff U'_def image_iff eq_commute)
    moreover have "(SUP i. U i x) = ennreal (u x)"
      using sup u(2) by (simp add: max_def)
    ultimately show "(λi. U' i x) ⇢ u x"
      using u U' by simp
  next
    fix i
    have "U' i ` space M ⊆ enn2real ` (U i ` space M)" "finite (U i ` space M)"
      unfolding U'_def using U(1) by (auto dest: simple_functionD)
    then have fin: "finite (U' i ` space M)"
      by (metis finite_subset finite_imageI)
    moreover have "⋀z. {y. U' i z = y ∧ y ∈ U' i ` space M ∧ z ∈ space M} = (if z ∈ space M then {U' i z} else {})"
      by auto
    ultimately have U': "(λz. ∑y∈U' i`space M. y * indicator {x∈space M. U' i x = y} z) = U' i"
      by (simp add: U'_def fun_eq_iff)
    have "⋀x. x ∈ U' i ` space M ⟹ 0 ≤ x"
      by (auto simp: U'_def)
    with fin have "P (λz. ∑y∈U' i`space M. y * indicator {x∈space M. U' i x = y} z)"
    proof induct
      case empty from set[of "{}"] show ?case
        by (simp add: indicator_def[abs_def])
    next
      case (insert x F)
      from insert.prems have nonneg: "x ≥ 0" "⋀y. y ∈ F ⟹ y ≥ 0"
        by simp_all
      hence *: "P (λxa. x * indicat_real {x' ∈ space M. U' i x' = x} xa)"
        by (intro mult set) auto
      have "P (λz. x * indicat_real {x' ∈ space M. U' i x' = x} z + 
                   (∑y∈F. y * indicat_real {x ∈ space M. U' i x = y} z))"
        using insert(1-3)
        by (intro add * sum_nonneg mult_nonneg_nonneg)
           (auto simp: nonneg indicator_def of_bool_def sum_nonneg_eq_0_iff)
      thus ?case 
        using insert.hyps by (subst sum.insert) auto
    qed
    with U' show "P (U' i)" by simp
  qed
qed

lemma scaleR_cong_right:
  fixes x :: "'a :: real_vector"
  shows "(x ≠ 0 ⟹ r = p) ⟹ r *⇩R x = p *⇩R x"
  by (cases "x = 0") auto

inductive simple_bochner_integrable :: "'a measure ⇒ ('a ⇒ 'b::real_vector) ⇒ bool" for M f where
  "simple_function M f ⟹ emeasure M {y∈space M. f y ≠ 0} ≠ ∞ ⟹
    simple_bochner_integrable M f"

lemma simple_bochner_integrable_compose2:
  assumes p_0: "p 0 0 = 0"
  shows "simple_bochner_integrable M f ⟹ simple_bochner_integrable M g ⟹
    simple_bochner_integrable M (λx. p (f x) (g x))"
proof (safe intro!: simple_bochner_integrable.intros elim!: simple_bochner_integrable.cases del: notI)
  assume sf: "simple_function M f" "simple_function M g"
  then show "simple_function M (λx. p (f x) (g x))"
    by (rule simple_function_compose2)

  from sf have [measurable]:
      "f ∈ measurable M (count_space UNIV)"
      "g ∈ measurable M (count_space UNIV)"
    by (auto intro: measurable_simple_function)

  assume fin: "emeasure M {y ∈ space M. f y ≠ 0} ≠ ∞" "emeasure M {y ∈ space M. g y ≠ 0} ≠ ∞"

  have "emeasure M {x∈space M. p (f x) (g x) ≠ 0} ≤
      emeasure M ({x∈space M. f x ≠ 0} ∪ {x∈space M. g x ≠ 0})"
    by (intro emeasure_mono) (auto simp: p_0)
  also have "… ≤ emeasure M {x∈space M. f x ≠ 0} + emeasure M {x∈space M. g x ≠ 0}"
    by (intro emeasure_subadditive) auto
  finally show "emeasure M {y ∈ space M. p (f y) (g y) ≠ 0} ≠ ∞"
    using fin by (auto simp: top_unique)
qed

lemma simple_function_finite_support:
  assumes f: "simple_function M f" and fin: "(∫⇧+x. f x ∂M) < ∞" and nn: "⋀x. 0 ≤ f x"
  shows "emeasure M {x∈space M. f x ≠ 0} ≠ ∞"
proof cases
  from f have meas[measurable]: "f ∈ borel_measurable M"
    by (rule borel_measurable_simple_function)

  assume non_empty: "∃x∈space M. f x ≠ 0"

  define m where "m = Min (f`space M - {0})"
  have "m ∈ f`space M - {0}"
    unfolding m_def using f non_empty by (intro Min_in) (auto simp: simple_function_def)
  then have m: "0 < m"
    using nn by (auto simp: less_le)

  from m have "m * emeasure M {x∈space M. 0 ≠ f x} =
    (∫⇧+x. m * indicator {x∈space M. 0 ≠ f x} x ∂M)"
    using f by (intro nn_integral_cmult_indicator[symmetric]) auto
  also have "… ≤ (∫⇧+x. f x ∂M)"
    using AE_space
  proof (intro nn_integral_mono_AE, eventually_elim)
    fix x assume "x ∈ space M"
    with nn show "m * indicator {x ∈ space M. 0 ≠ f x} x ≤ f x"
      using f by (auto split: split_indicator simp: simple_function_def m_def)
  qed
  also note ‹… < ∞›
  finally show ?thesis
    using m by (auto simp: ennreal_mult_less_top)
next
  assume "¬ (∃x∈space M. f x ≠ 0)"
  with nn have *: "{x∈space M. f x ≠ 0} = {}"
    by auto
  show ?thesis unfolding * by simp
qed

lemma simple_bochner_integrableI_bounded:
  assumes f: "simple_function M f" and fin: "(∫⇧+x. norm (f x) ∂M) < ∞"
  shows "simple_bochner_integrable M f"
proof
  have "emeasure M {y ∈ space M. ennreal (norm (f y)) ≠ 0} ≠ ∞"
  proof (rule simple_function_finite_support)
    show "simple_function M (λx. ennreal (norm (f x)))"
      using f by (rule simple_function_compose1)
    show "(∫⇧+ y. ennreal (norm (f y)) ∂M) < ∞" by fact
  qed simp
  then show "emeasure M {y ∈ space M. f y ≠ 0} ≠ ∞" by simp
qed fact

definition✐‹tag important› simple_bochner_integral :: "'a measure ⇒ ('a ⇒ 'b::real_vector) ⇒ 'b" where
  "simple_bochner_integral M f = (∑y∈f`space M. measure M {x∈space M. f x = y} *⇩R y)"

proposition simple_bochner_integral_partition:
  assumes f: "simple_bochner_integrable 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 "simple_bochner_integral M f = (∑y∈g ` space M. measure M {x∈space M. g x = y} *⇩R v y)"
    (is "_ = ?r")
proof -
  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
    by (auto simp: simple_function_def elim: simple_bochner_integrable.cases)

  from f have [measurable]: "f ∈ measurable M (count_space UNIV)"
    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)

  from g have [measurable]: "g ∈ measurable M (count_space UNIV)"
    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)

  { 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 "simple_bochner_integral M f =
    (∑y∈f`space M. (∑z∈g`space M.
      if ∃x∈space M. y = f x ∧ z = g x then measure M {x∈space M. g x = z} else 0) *⇩R y)"
    unfolding simple_bochner_integral_def
  proof (safe intro!: sum.cong scaleR_cong_right)
    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:"{x ∈ space M. f x = f y} =
        (⋃i∈{z. ∃x∈space M. f y = f x ∧ z = g x}. {x ∈ space M. g x = i})"
      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
    moreover
    { fix x assume "x ∈ space M" "f x = f y"
      then have "x ∈ space M" "f x ≠ 0"
        using y by auto
      then have "emeasure M {y ∈ space M. g y = g x} ≤ emeasure M {y ∈ space M. f y ≠ 0}"
        by (auto intro!: emeasure_mono cong: sub)
      then have "emeasure M {xa ∈ space M. g xa = g x} < ∞"
        using f by (auto simp: simple_bochner_integrable.simps less_top) }
    ultimately
    show "measure M {x ∈ space M. f x = f y} =
      (∑z∈g ` space M. if ∃x∈space M. f y = f x ∧ z = g x then measure M {x ∈ space M. g x = z} else 0)"
      apply (simp add: sum.If_cases eq)
      apply (subst measure_finite_Union[symmetric])
      apply (auto simp: disjoint_family_on_def less_top)
      done
  qed
  also have "… = (∑y∈f`space M. (∑z∈g`space M.
      if ∃x∈space M. y = f x ∧ z = g x then measure M {x∈space M. g x = z} *⇩R y else 0))"
    by (auto intro!: sum.cong simp: scaleR_sum_left)
  also have "… = ?r"
    by (subst sum.swap)
       (auto intro!: sum.cong simp: sum.If_cases scaleR_sum_right[symmetric] eq)
  finally show "simple_bochner_integral M f = ?r" .
qed

lemma simple_bochner_integral_add:
  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
  shows "simple_bochner_integral M (λx. f x + g x) =
    simple_bochner_integral M f + simple_bochner_integral M g"
proof -
  from f g have "simple_bochner_integral M (λx. f x + g x) =
    (∑y∈(λx. (f x, g x)) ` space M. measure M {x ∈ space M. (f x, g x) = y} *⇩R (fst y + snd y))"
    by (intro simple_bochner_integral_partition)
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
  moreover from f g have "simple_bochner_integral M f =
    (∑y∈(λx. (f x, g x)) ` space M. measure M {x ∈ space M. (f x, g x) = y} *⇩R fst y)"
    by (intro simple_bochner_integral_partition)
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
  moreover from f g have "simple_bochner_integral M g =
    (∑y∈(λx. (f x, g x)) ` space M. measure M {x ∈ space M. (f x, g x) = y} *⇩R snd y)"
    by (intro simple_bochner_integral_partition)
       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
  ultimately show ?thesis
    by (simp add: sum.distrib[symmetric] scaleR_add_right)
qed

lemma simple_bochner_integral_linear:
  assumes "linear f"
  assumes g: "simple_bochner_integrable M g"
  shows "simple_bochner_integral M (λx. f (g x)) = f (simple_bochner_integral M g)"
proof -
  interpret linear f by fact
  from g have "simple_bochner_integral M (λx. f (g x)) =
    (∑y∈g ` space M. measure M {x ∈ space M. g x = y} *⇩R f y)"
    by (intro simple_bochner_integral_partition)
       (auto simp: simple_bochner_integrable_compose2[where p="λx y. f x"]
             elim: simple_bochner_integrable.cases)
  also have "… = f (simple_bochner_integral M g)"
    by (simp add: simple_bochner_integral_def sum scale)
  finally show ?thesis .
qed

lemma simple_bochner_integral_minus:
  assumes f: "simple_bochner_integrable M f"
  shows "simple_bochner_integral M (λx. - f x) = - simple_bochner_integral M f"
proof -
  from linear_uminus f show ?thesis
    by (rule simple_bochner_integral_linear)
qed

lemma simple_bochner_integral_diff:
  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
  shows "simple_bochner_integral M (λx. f x - g x) =
    simple_bochner_integral M f - simple_bochner_integral M g"
  unfolding diff_conv_add_uminus using f g
  by (subst simple_bochner_integral_add)
     (auto simp: simple_bochner_integral_minus simple_bochner_integrable_compose2[where p="λx y. - y"])

lemma simple_bochner_integral_norm_bound:
  assumes f: "simple_bochner_integrable M f"
  shows "norm (simple_bochner_integral M f) ≤ simple_bochner_integral M (λx. norm (f x))"
proof -
  have "norm (simple_bochner_integral M f) ≤
    (∑y∈f ` space M. norm (measure M {x ∈ space M. f x = y} *⇩R y))"
    unfolding simple_bochner_integral_def by (rule norm_sum)
  also have "… = (∑y∈f ` space M. measure M {x ∈ space M. f x = y} *⇩R norm y)"
    by simp
  also have "… = simple_bochner_integral M (λx. norm (f x))"
    using f
    by (intro simple_bochner_integral_partition[symmetric])
       (auto intro: f simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
  finally show ?thesis .
qed

lemma simple_bochner_integral_nonneg[simp]:
  fixes f :: "'a ⇒ real"
  shows "(⋀x. 0 ≤ f x) ⟹ 0 ≤ simple_bochner_integral M f"
  by (force simp add: simple_bochner_integral_def intro: sum_nonneg)

lemma simple_bochner_integral_eq_nn_integral:
  assumes f: "simple_bochner_integrable M f" "⋀x. 0 ≤ f x"
  shows "simple_bochner_integral M f = (∫⇧+x. f x ∂M)"
proof -
  { fix x y z have "(x ≠ 0 ⟹ y = z) ⟹ ennreal x * y = ennreal x * z"
      by (cases "x = 0") (auto simp: zero_ennreal_def[symmetric]) }
  note ennreal_cong_mult = this

  have [measurable]: "f ∈ borel_measurable M"
    using f(1) by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)

  { fix y assume y: "y ∈ space M" "f y ≠ 0"
    have "ennreal (measure M {x ∈ space M. f x = f y}) = emeasure M {x ∈ space M. f x = f y}"
    proof (rule emeasure_eq_ennreal_measure[symmetric])
      have "emeasure M {x ∈ space M. f x = f y} ≤ emeasure M {x ∈ space M. f x ≠ 0}"
        using y by (intro emeasure_mono) auto
      with f show "emeasure M {x ∈ space M. f x = f y} ≠ top"
        by (auto simp: simple_bochner_integrable.simps top_unique)
    qed
    moreover have "{x ∈ space M. f x = f y} = (λx. ennreal (f x)) -` {ennreal (f y)} ∩ space M"
      using f by auto
    ultimately have "ennreal (measure M {x ∈ space M. f x = f y}) =
          emeasure M ((λx. ennreal (f x)) -` {ennreal (f y)} ∩ space M)" by simp }
  with f have "simple_bochner_integral M f = (∫⇧Sx. f x ∂M)"
    unfolding simple_integral_def
    by (subst simple_bochner_integral_partition[OF f(1), where g="λx. ennreal (f x)" and v=enn2real])
       (auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
             intro!: sum.cong ennreal_cong_mult
             simp: ac_simps ennreal_mult
             simp flip: sum_ennreal)
  also have "… = (∫⇧+x. f x ∂M)"
    using f
    by (intro nn_integral_eq_simple_integral[symmetric])
       (auto simp: simple_function_compose1 simple_bochner_integrable.simps)
  finally show ?thesis .
qed

lemma simple_bochner_integral_bounded:
  fixes f :: "'a ⇒ 'b::{real_normed_vector, second_countable_topology}"
  assumes f[measurable]: "f ∈ borel_measurable M"
  assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
  shows "ennreal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) ≤
    (∫⇧+ x. norm (f x - s x) ∂M) + (∫⇧+ x. norm (f x - t x) ∂M)"
    (is "ennreal (norm (?s - ?t)) ≤ ?S + ?T")
proof -
  have [measurable]: "s ∈ borel_measurable M" "t ∈ borel_measurable M"
    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)

  have "ennreal (norm (?s - ?t)) = norm (simple_bochner_integral M (λx. s x - t x))"
    using s t by (subst simple_bochner_integral_diff) auto
  also have "… ≤ simple_bochner_integral M (λx. norm (s x - t x))"
    using simple_bochner_integrable_compose2[of "(-)" M "s" "t"] s t
    by (auto intro!: simple_bochner_integral_norm_bound)
  also have "… = (∫⇧+x. norm (s x - t x) ∂M)"
    using simple_bochner_integrable_compose2[of "λx y. norm (x - y)" M "s" "t"] s t
    by (auto intro!: simple_bochner_integral_eq_nn_integral)
  also have "… ≤ (∫⇧+x. ennreal (norm (f x - s x)) + ennreal (norm (f x - t x)) ∂M)"
    by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
       (metis (erased, opaque_lifting) add_diff_cancel_left add_diff_eq diff_add_eq order_trans
              norm_minus_commute norm_triangle_ineq4 order_refl)
  also have "… = ?S + ?T"
   by (rule nn_integral_add) auto
  finally show ?thesis .
qed

inductive has_bochner_integral :: "'a measure ⇒ ('a ⇒ 'b) ⇒ 'b::{real_normed_vector, second_countable_topology} ⇒ bool"
  for M f x where
  "f ∈ borel_measurable M ⟹
    (⋀i. simple_bochner_integrable M (s i)) ⟹
    (λi. ∫⇧+x. norm (f x - s i x) ∂M) ⇢ 0 ⟹
    (λi. simple_bochner_integral M (s i)) ⇢ x ⟹
    has_bochner_integral M f x"

lemma has_bochner_integral_cong:
  assumes "M = N" "⋀x. x ∈ space N ⟹ f x = g x" "x = y"
  shows "has_bochner_integral M f x ⟷ has_bochner_integral N g y"
  unfolding has_bochner_integral.simps assms(1,3)
  using assms(2) by (simp cong: measurable_cong_simp nn_integral_cong_simp)

lemma has_bochner_integral_cong_AE:
  "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (AE x in M. f x = g x) ⟹
    has_bochner_integral M f x ⟷ has_bochner_integral M g x"
  unfolding has_bochner_integral.simps
  by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="λx. x ⇢ 0"]
            nn_integral_cong_AE)
     auto

lemma borel_measurable_has_bochner_integral:
  "has_bochner_integral M f x ⟹ f ∈ borel_measurable M"
  by (rule has_bochner_integral.cases)

lemma borel_measurable_has_bochner_integral'[measurable_dest]:
  "has_bochner_integral M f x ⟹ g ∈ measurable N M ⟹ (λx. f (g x)) ∈ borel_measurable N"
  using borel_measurable_has_bochner_integral[measurable] by measurable

lemma has_bochner_integral_simple_bochner_integrable:
  "simple_bochner_integrable M f ⟹ has_bochner_integral M f (simple_bochner_integral M f)"
  by (rule has_bochner_integral.intros[where s="λ_. f"])
     (auto intro: borel_measurable_simple_function
           elim: simple_bochner_integrable.cases
           simp: zero_ennreal_def[symmetric])

lemma has_bochner_integral_real_indicator:
  assumes [measurable]: "A ∈ sets M" and A: "emeasure M A < ∞"
  shows "has_bochner_integral M (indicator A) (measure M A)"
proof -
  have sbi: "simple_bochner_integrable M (indicator A::'a ⇒ real)"
  proof
    have "{y ∈ space M. (indicator A y::real) ≠ 0} = A"
      using sets.sets_into_space[OF ‹A∈sets M›] by (auto split: split_indicator)
    then show "emeasure M {y ∈ space M. (indicator A y::real) ≠ 0} ≠ ∞"
      using A by auto
  qed (rule simple_function_indicator assms)+
  moreover have "simple_bochner_integral M (indicator A) = measure M A"
    using simple_bochner_integral_eq_nn_integral[OF sbi] A
    by (simp add: ennreal_indicator emeasure_eq_ennreal_measure)
  ultimately show ?thesis
    by (metis has_bochner_integral_simple_bochner_integrable)
qed

lemma has_bochner_integral_add[intro]:
  "has_bochner_integral M f x ⟹ has_bochner_integral M g y ⟹
    has_bochner_integral M (λx. f x + g x) (x + y)"
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
  fix sf sg
  assume f_sf: "(λi. ∫⇧+ x. norm (f x - sf i x) ∂M) ⇢ 0"
  assume g_sg: "(λi. ∫⇧+ x. norm (g x - sg i x) ∂M) ⇢ 0"

  assume sf: "∀i. simple_bochner_integrable M (sf i)"
    and sg: "∀i. simple_bochner_integrable M (sg i)"
  then have [measurable]: "⋀i. sf i ∈ borel_measurable M" "⋀i. sg i ∈ borel_measurable M"
    by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
  assume [measurable]: "f ∈ borel_measurable M" "g ∈ borel_measurable M"

  show "⋀i. simple_bochner_integrable M (λx. sf i x + sg i x)"
    using sf sg by (simp add: simple_bochner_integrable_compose2)

  show "(λi. ∫⇧+ x. (norm (f x + g x - (sf i x + sg i x))) ∂M) ⇢ 0"
    (is "?f ⇢ 0")
  proof (rule tendsto_sandwich)
    show "eventually (λn. 0 ≤ ?f n) sequentially" "(λ_. 0) ⇢ 0"
      by auto
    show "eventually (λi. ?f i ≤ (∫⇧+ x. (norm (f x - sf i x)) ∂M) + ∫⇧+ x. (norm (g x - sg i x)) ∂M) sequentially"
      (is "eventually (λi. ?f i ≤ ?g i) sequentially")
    proof (intro always_eventually allI)
      fix i have "?f i ≤ (∫⇧+ x. (norm (f x - sf i x)) + ennreal (norm (g x - sg i x)) ∂M)"
        by (auto intro!: nn_integral_mono norm_diff_triangle_ineq
                 simp flip: ennreal_plus)
      also have "… = ?g i"
        by (intro nn_integral_add) auto
      finally show "?f i ≤ ?g i" .
    qed
    show "?g ⇢ 0"
      using tendsto_add[OF f_sf g_sg] by simp
  qed
qed (auto simp: simple_bochner_integral_add tendsto_add)

lemma has_bochner_integral_bounded_linear:
  assumes "bounded_linear T"
  shows "has_bochner_integral M f x ⟹ has_bochner_integral M (λx. T (f x)) (T x)"
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
  interpret T: bounded_linear T by fact
  have [measurable]: "T ∈ borel_measurable borel"
    by (intro borel_measurable_continuous_onI T.continuous_on continuous_on_id)
  assume [measurable]: "f ∈ borel_measurable M"
  then show "(λx. T (f x)) ∈ borel_measurable M"
    by auto

  fix s assume f_s: "(λi. ∫⇧+ x. norm (f x - s i x) ∂M) ⇢ 0"
  assume s: "∀i. simple_bochner_integrable M (s i)"
  then show "⋀i. simple_bochner_integrable M (λx. T (s i x))"
    by (auto intro: simple_bochner_integrable_compose2 T.zero)

  have [measurable]: "⋀i. s i ∈ borel_measurable M"
    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)

  obtain K where K: "K > 0" "⋀x i. norm (T (f x) - T (s i x)) ≤ norm (f x - s i x) * K"
    using T.pos_bounded by (auto simp: T.diff[symmetric])

  show "(λi. ∫⇧+ x. norm (T (f x) - T (s i x)) ∂M) ⇢ 0"
    (is "?f ⇢ 0")
  proof (rule tendsto_sandwich)
    show "eventually (λn. 0 ≤ ?f n) sequentially" "(λ_. 0) ⇢ 0"
      by auto

    show "eventually (λi. ?f i ≤ K * (∫⇧+ x. norm (f x - s i x) ∂M)) sequentially"
      (is "eventually (λi. ?f i ≤ ?g i) sequentially")
    proof (intro always_eventually allI)
      fix i have "?f i ≤ (∫⇧+ x. ennreal K * norm (f x - s i x) ∂M)"
        using K by (intro nn_integral_mono) (auto simp: ac_simps ennreal_mult[symmetric])
      also have "… = ?g i"
        using K by (intro nn_integral_cmult) auto
      finally show "?f i ≤ ?g i" .
    qed
    show "?g ⇢ 0"
      using ennreal_tendsto_cmult[OF _ f_s] by simp
  qed

  assume "(λi. simple_bochner_integral M (s i)) ⇢ x"
  with s show "(λi. simple_bochner_integral M (λx. T (s i x))) ⇢ T x"
    by (auto intro!: T.tendsto simp: simple_bochner_integral_linear T.linear_axioms)
qed

lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (λx. 0) 0"
  by (auto intro!: has_bochner_integral.intros[where s="λ_ _. 0"]
           simp: zero_ennreal_def[symmetric] simple_bochner_integrable.simps
                 simple_bochner_integral_def image_constant_conv)

lemma has_bochner_integral_scaleR_left[intro]:
  "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. f x *⇩R c) (x *⇩R c)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_left])

lemma has_bochner_integral_scaleR_right[intro]:
  "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. c *⇩R f x) (c *⇩R x)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_right])

lemma has_bochner_integral_mult_left[intro]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. f x * c) (x * c)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_left])

lemma has_bochner_integral_mult_right[intro]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. c * f x) (c * x)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_right])

lemmas has_bochner_integral_divide =
  has_bochner_integral_bounded_linear[OF bounded_linear_divide]

lemma has_bochner_integral_divide_zero[intro]:
  fixes c :: "_::{real_normed_field, field, second_countable_topology}"
  shows "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. f x / c) (x / c)"
  using has_bochner_integral_divide by (cases "c = 0") auto

lemma has_bochner_integral_inner_left[intro]:
  "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. f x ∙ c) (x ∙ c)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_left])

lemma has_bochner_integral_inner_right[intro]:
  "(c ≠ 0 ⟹ has_bochner_integral M f x) ⟹ has_bochner_integral M (λx. c ∙ f x) (c ∙ x)"
  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_right])

lemmas has_bochner_integral_minus =
  has_bochner_integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
lemmas has_bochner_integral_Re =
  has_bochner_integral_bounded_linear[OF bounded_linear_Re]
lemmas has_bochner_integral_Im =
  has_bochner_integral_bounded_linear[OF bounded_linear_Im]
lemmas has_bochner_integral_cnj =
  has_bochner_integral_bounded_linear[OF bounded_linear_cnj]
lemmas has_bochner_integral_of_real =
  has_bochner_integral_bounded_linear[OF bounded_linear_of_real]
lemmas has_bochner_integral_fst =
  has_bochner_integral_bounded_linear[OF bounded_linear_fst]
lemmas has_bochner_integral_snd =
  has_bochner_integral_bounded_linear[OF bounded_linear_snd]

lemma has_bochner_integral_indicator:
  "A ∈ sets M ⟹ emeasure M A < ∞ ⟹
    has_bochner_integral M (λx. indicator A x *⇩R c) (measure M A *⇩R c)"
  by (intro has_bochner_integral_scaleR_left has_bochner_integral_real_indicator)

lemma has_bochner_integral_diff:
  "has_bochner_integral M f x ⟹ has_bochner_integral M g y ⟹
    has_bochner_integral M (λx. f x - g x) (x - y)"
  unfolding diff_conv_add_uminus
  by (intro has_bochner_integral_add has_bochner_integral_minus)

lemma has_bochner_integral_sum:
  "(⋀i. i ∈ I ⟹ has_bochner_integral M (f i) (x i)) ⟹
    has_bochner_integral M (λx. ∑i∈I. f i x) (∑i∈I. x i)"
  by (induct I rule: infinite_finite_induct) auto

proposition has_bochner_integral_implies_finite_norm:
  "has_bochner_integral M f x ⟹ (∫⇧+x. norm (f x) ∂M) < ∞"
proof (elim has_bochner_integral.cases)
  fix s v
  assume [measurable]: "f ∈ borel_measurable M" and s: "⋀i. simple_bochner_integrable M (s i)" and
    lim_0: "(λi. ∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) ⇢ 0"
  from order_tendstoD[OF lim_0, of "∞"]
  obtain i where f_s_fin: "(∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) < ∞"
    by (auto simp: eventually_sequentially)

  have [measurable]: "⋀i. s i ∈ borel_measurable M"
    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)

  define m where "m = (if space M = {} then 0 else Max ((λx. norm (s i x))`space M))"
  have "finite (s i ` space M)"
    using s by (auto simp: simple_function_def simple_bochner_integrable.simps)
  then have "finite (norm ` s i ` space M)"
    by (rule finite_imageI)
  then have "⋀x. x ∈ space M ⟹ norm (s i x) ≤ m" "0 ≤ m"
    by (auto simp: m_def image_comp comp_def Max_ge_iff)
  then have "(∫⇧+x. norm (s i x) ∂M) ≤ (∫⇧+x. ennreal m * indicator {x∈space M. s i x ≠ 0} x ∂M)"
    by (auto split: split_indicator intro!: Max_ge nn_integral_mono simp:)
  also have "… < ∞"
    using s by (subst nn_integral_cmult_indicator) (auto simp: ‹0 ≤ m› simple_bochner_integrable.simps ennreal_mult_less_top less_top)
  finally have s_fin: "(∫⇧+x. norm (s i x) ∂M) < ∞" .

  have "(∫⇧+ x. norm (f x) ∂M) ≤ (∫⇧+ x. ennreal (norm (f x - s i x)) + ennreal (norm (s i x)) ∂M)"
    by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
       (metis add.commute norm_triangle_sub)
  also have "… = (∫⇧+x. norm (f x - s i x) ∂M) + (∫⇧+x. norm (s i x) ∂M)"
    by (rule nn_integral_add) auto
  also have "… < ∞"
    using s_fin f_s_fin by auto
  finally show "(∫⇧+ x. ennreal (norm (f x)) ∂M) < ∞" .
qed

proposition has_bochner_integral_norm_bound:
  assumes i: "has_bochner_integral M f x"
  shows "norm x ≤ (∫⇧+x. norm (f x) ∂M)"
using assms proof
  fix s assume
    x: "(λi. simple_bochner_integral M (s i)) ⇢ x" (is "?s ⇢ x") and
    s[simp]: "⋀i. simple_bochner_integrable M (s i)" and
    lim: "(λi. ∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) ⇢ 0" and
    f[measurable]: "f ∈ borel_measurable M"

  have [measurable]: "⋀i. s i ∈ borel_measurable M"
    using s by (auto simp: simple_bochner_integrable.simps intro: borel_measurable_simple_function)

  show "norm x ≤ (∫⇧+x. norm (f x) ∂M)"
  proof (rule LIMSEQ_le)
    show "(λi. ennreal (norm (?s i))) ⇢ norm x"
      using x by (auto simp: tendsto_ennreal_iff intro: tendsto_intros)
    show "∃N. ∀n≥N. norm (?s n) ≤ (∫⇧+x. norm (f x - s n x) ∂M) + (∫⇧+x. norm (f x) ∂M)"
      (is "∃N. ∀n≥N. _ ≤ ?t n")
    proof (intro exI allI impI)
      fix n
      have "ennreal (norm (?s n)) ≤ simple_bochner_integral M (λx. norm (s n x))"
        by (auto intro!: simple_bochner_integral_norm_bound)
      also have "… = (∫⇧+x. norm (s n x) ∂M)"
        by (intro simple_bochner_integral_eq_nn_integral)
           (auto intro: s simple_bochner_integrable_compose2)
      also have "… ≤ (∫⇧+x. ennreal (norm (f x - s n x)) + norm (f x) ∂M)"
        by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
           (metis add.commute norm_minus_commute norm_triangle_sub)
      also have "… = ?t n"
        by (rule nn_integral_add) auto
      finally show "norm (?s n) ≤ ?t n" .
    qed
    have "?t ⇢ 0 + (∫⇧+ x. ennreal (norm (f x)) ∂M)"
      using has_bochner_integral_implies_finite_norm[OF i]
      by (intro tendsto_add tendsto_const lim)
    then show "?t ⇢ ∫⇧+ x. ennreal (norm (f x)) ∂M"
      by simp
  qed
qed

lemma has_bochner_integral_eq:
  "has_bochner_integral M f x ⟹ has_bochner_integral M f y ⟹ x = y"
proof (elim has_bochner_integral.cases)
  assume f[measurable]: "f ∈ borel_measurable M"

  fix s t
  assume "(λi. ∫⇧+ x. norm (f x - s i x) ∂M) ⇢ 0" (is "?S ⇢ 0")
  assume "(λi. ∫⇧+ x. norm (f x - t i x) ∂M) ⇢ 0" (is "?T ⇢ 0")
  assume s: "⋀i. simple_bochner_integrable M (s i)"
  assume t: "⋀i. simple_bochner_integrable M (t i)"

  have [measurable]: "⋀i. s i ∈ borel_measurable M" "⋀i. t i ∈ borel_measurable M"
    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)

  let ?s = "λi. simple_bochner_integral M (s i)"
  let ?t = "λi. simple_bochner_integral M (t i)"
  assume "?s ⇢ x" "?t ⇢ y"
  then have "(λi. norm (?s i - ?t i)) ⇢ norm (x - y)"
    by (intro tendsto_intros)
  moreover
  have "(λi. ennreal (norm (?s i - ?t i))) ⇢ ennreal 0"
  proof (rule tendsto_sandwich)
    show "eventually (λi. 0 ≤ ennreal (norm (?s i - ?t i))) sequentially" "(λ_. 0) ⇢ ennreal 0"
      by auto

    show "eventually (λi. norm (?s i - ?t i) ≤ ?S i + ?T i) sequentially"
      by (intro always_eventually allI simple_bochner_integral_bounded s t f)
    show "(λi. ?S i + ?T i) ⇢ ennreal 0"
      using tendsto_add[OF ‹?S ⇢ 0› ‹?T ⇢ 0›] by simp
  qed
  then have "(λi. norm (?s i - ?t i)) ⇢ 0"
    by (simp flip: ennreal_0)
  ultimately have "norm (x - y) = 0"
    by (rule LIMSEQ_unique)
  then show "x = y" by simp
qed

lemma has_bochner_integralI_AE:
  assumes f: "has_bochner_integral M f x"
    and g: "g ∈ borel_measurable M"
    and ae: "AE x in M. f x = g x"
  shows "has_bochner_integral M g x"
  using f
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
  fix s assume "(λi. ∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) ⇢ 0"
  also have "(λi. ∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) = (λi. ∫⇧+ x. ennreal (norm (g x - s i x)) ∂M)"
    using ae
    by (intro ext nn_integral_cong_AE, eventually_elim) simp
  finally show "(λi. ∫⇧+ x. ennreal (norm (g x - s i x)) ∂M) ⇢ 0" .
qed (auto intro: g)

lemma has_bochner_integral_eq_AE:
  assumes f: "has_bochner_integral M f x"
    and g: "has_bochner_integral M g y"
    and ae: "AE x in M. f x = g x"
  shows "x = y"
proof -
  from assms have "has_bochner_integral M g x"
    by (auto intro: has_bochner_integralI_AE)
  from this g show "x = y"
    by (rule has_bochner_integral_eq)
qed

lemma simple_bochner_integrable_restrict_space:
  fixes f :: "_ ⇒ 'b::real_normed_vector"
  assumes Ω: "Ω ∩ space M ∈ sets M"
  shows "simple_bochner_integrable (restrict_space M Ω) f ⟷
    simple_bochner_integrable M (λx. indicator Ω x *⇩R f x)"
  by (simp add: simple_bochner_integrable.simps space_restrict_space
    simple_function_restrict_space[OF Ω] emeasure_restrict_space[OF Ω] Collect_restrict
    indicator_eq_0_iff conj_left_commute)

lemma simple_bochner_integral_restrict_space:
  fixes f :: "_ ⇒ 'b::real_normed_vector"
  assumes Ω: "Ω ∩ space M ∈ sets M"
  assumes f: "simple_bochner_integrable (restrict_space M Ω) f"
  shows "simple_bochner_integral (restrict_space M Ω) f =
    simple_bochner_integral M (λx. indicator Ω x *⇩R f x)"
proof -
  have "finite ((λx. indicator Ω x *⇩R f x)`space M)"
    using f simple_bochner_integrable_restrict_space[OF Ω, of f]
    by (simp add: simple_bochner_integrable.simps simple_function_def)
  then show ?thesis
    by (auto simp: space_restrict_space measure_restrict_space[OF Ω(1)] le_infI2
                   simple_bochner_integral_def Collect_restrict
             split: split_indicator split_indicator_asm
             intro!: sum.mono_neutral_cong_left arg_cong2[where f=measure])
qed

context
  notes [[inductive_internals]]
begin

inductive integrable for M f where
  "has_bochner_integral M f x ⟹ integrable M f"

end

definition✐‹tag important› lebesgue_integral ("integral⇧L") where
  "integral⇧L M f = (if ∃x. has_bochner_integral M f x then THE x. has_bochner_integral M f x else 0)"

syntax
  "_lebesgue_integral" :: "pttrn ⇒ real ⇒ 'a measure ⇒ real" ("∫((2 _./ _)/ ∂_)" [60,61] 110)

translations
  "∫ x. f ∂M" == "CONST lebesgue_integral M (λx. f)"

syntax
  "_ascii_lebesgue_integral" :: "pttrn ⇒ 'a measure ⇒ real ⇒ real" ("(3LINT (1_)/|(_)./ _)" [0,110,60] 60)

translations
  "LINT x|M. f" == "CONST lebesgue_integral M (λx. f)"

lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x ⟹ integral⇧L M f = x"
  by (metis the_equality has_bochner_integral_eq lebesgue_integral_def)

lemma has_bochner_integral_integrable:
  "integrable M f ⟹ has_bochner_integral M f (integral⇧L M f)"
  by (auto simp: has_bochner_integral_integral_eq integrable.simps)

lemma has_bochner_integral_iff:
  "has_bochner_integral M f x ⟷ integrable M f ∧ integral⇧L M f = x"
  by (metis has_bochner_integral_integrable has_bochner_integral_integral_eq integrable.intros)

lemma simple_bochner_integrable_eq_integral:
  "simple_bochner_integrable M f ⟹ simple_bochner_integral M f = integral⇧L M f"
  using has_bochner_integral_simple_bochner_integrable[of M f]
  by (simp add: has_bochner_integral_integral_eq)

lemma not_integrable_integral_eq: "¬ integrable M f ⟹ integral⇧L M f = 0"
  unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])

lemma integral_eq_cases:
  "integrable M f ⟷ integrable N g ⟹
    (integrable M f ⟹ integrable N g ⟹ integral⇧L M f = integral⇧L N g) ⟹
    integral⇧L M f = integral⇧L N g"
  by (metis not_integrable_integral_eq)

lemma borel_measurable_integrable[measurable_dest]: "integrable M f ⟹ f ∈ borel_measurable M"
  by (auto elim: integrable.cases has_bochner_integral.cases)

lemma borel_measurable_integrable'[measurable_dest]:
  "integrable M f ⟹ g ∈ measurable N M ⟹ (λx. f (g x)) ∈ borel_measurable N"
  using borel_measurable_integrable[measurable] by measurable

lemma integrable_cong:
  "M = N ⟹ (⋀x. x ∈ space N ⟹ f x = g x) ⟹ integrable M f ⟷ integrable N g"
  by (simp cong: has_bochner_integral_cong add: integrable.simps)

lemma integrable_cong_AE:
  "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ AE x in M. f x = g x ⟹
    integrable M f ⟷ integrable M g"
  unfolding integrable.simps
  by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)

lemma integrable_cong_AE_imp:
  "integrable M g ⟹ f ∈ borel_measurable M ⟹ (AE x in M. g x = f x) ⟹ integrable M f"
  using integrable_cong_AE[of f M g] by (auto simp: eq_commute)

lemma integral_cong:
  "M = N ⟹ (⋀x. x ∈ space N ⟹ f x = g x) ⟹ integral⇧L M f = integral⇧L N g"
  by (simp cong: has_bochner_integral_cong cong del: if_weak_cong add: lebesgue_integral_def)

lemma integral_cong_AE:
  "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ AE x in M. f x = g x ⟹
    integral⇧L M f = integral⇧L M g"
  unfolding lebesgue_integral_def
  by (rule arg_cong[where x="has_bochner_integral M f"]) (intro has_bochner_integral_cong_AE ext)

lemma integrable_add[simp, intro]: "integrable M f ⟹ integrable M g ⟹ integrable M (λx. f x + g x)"
  by (auto simp: integrable.simps)

lemma integrable_zero[simp, intro]: "integrable M (λx. 0)"
  by (metis has_bochner_integral_zero integrable.simps)

lemma integrable_sum[simp, intro]: "(⋀i. i ∈ I ⟹ integrable M (f i)) ⟹ integrable M (λx. ∑i∈I. f i x)"
  by (metis has_bochner_integral_sum integrable.simps)

lemma integrable_indicator[simp, intro]: "A ∈ sets M ⟹ emeasure M A < ∞ ⟹
  integrable M (λx. indicator A x *⇩R c)"
  by (metis has_bochner_integral_indicator integrable.simps)

lemma integrable_real_indicator[simp, intro]: "A ∈ sets M ⟹ emeasure M A < ∞ ⟹
  integrable M (indicator A :: 'a ⇒ real)"
  by (metis has_bochner_integral_real_indicator integrable.simps)

lemma integrable_diff[simp, intro]: "integrable M f ⟹ integrable M g ⟹ integrable M (λx. f x - g x)"
  by (auto simp: integrable.simps intro: has_bochner_integral_diff)

lemma integrable_bounded_linear: "bounded_linear T ⟹ integrable M f ⟹ integrable M (λx. T (f x))"
  by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)

lemma integrable_scaleR_left[simp, intro]: "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. f x *⇩R c)"
  unfolding integrable.simps by fastforce

lemma integrable_scaleR_right[simp, intro]: "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. c *⇩R f x)"
  unfolding integrable.simps by fastforce

lemma integrable_mult_left[simp, intro]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. f x * c)"
  unfolding integrable.simps by fastforce

lemma integrable_mult_right[simp, intro]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. c * f x)"
  unfolding integrable.simps by fastforce

lemma integrable_divide_zero[simp, intro]:
  fixes c :: "_::{real_normed_field, field, second_countable_topology}"
  shows "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. f x / c)"
  unfolding integrable.simps by fastforce

lemma integrable_inner_left[simp, intro]:
  "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. f x ∙ c)"
  unfolding integrable.simps by fastforce

lemma integrable_inner_right[simp, intro]:
  "(c ≠ 0 ⟹ integrable M f) ⟹ integrable M (λx. c ∙ f x)"
  unfolding integrable.simps by fastforce

lemmas integrable_minus[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
lemmas integrable_divide[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_divide]
lemmas integrable_Re[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_Re]
lemmas integrable_Im[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_Im]
lemmas integrable_cnj[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_cnj]
lemmas integrable_of_real[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_of_real]
lemmas integrable_fst[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_fst]
lemmas integrable_snd[simp, intro] =
  integrable_bounded_linear[OF bounded_linear_snd]

lemma integral_zero[simp]: "integral⇧L M (λx. 0) = 0"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)

lemma integral_add[simp]: "integrable M f ⟹ integrable M g ⟹
    integral⇧L M (λx. f x + g x) = integral⇧L M f + integral⇧L M g"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_add has_bochner_integral_integrable)

lemma integral_diff[simp]: "integrable M f ⟹ integrable M g ⟹
    integral⇧L M (λx. f x - g x) = integral⇧L M f - integral⇧L M g"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)

lemma integral_sum: "(⋀i. i ∈ I ⟹ integrable M (f i)) ⟹
  integral⇧L M (λx. ∑i∈I. f i x) = (∑i∈I. integral⇧L M (f i))"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_sum has_bochner_integral_integrable)

lemma integral_sum'[simp]: "(⋀i. i ∈ I =simp=> integrable M (f i)) ⟹
  integral⇧L M (λx. ∑i∈I. f i x) = (∑i∈I. integral⇧L M (f i))"
  unfolding simp_implies_def by (rule integral_sum)

lemma integral_bounded_linear: "bounded_linear T ⟹ integrable M f ⟹
    integral⇧L M (λx. T (f x)) = T (integral⇧L M f)"
  by (metis has_bochner_integral_bounded_linear has_bochner_integral_integrable has_bochner_integral_integral_eq)

lemma integral_bounded_linear':
  assumes T: "bounded_linear T" and T': "bounded_linear T'"
  assumes *: "¬ (∀x. T x = 0) ⟹ (∀x. T' (T x) = x)"
  shows "integral⇧L M (λx. T (f x)) = T (integral⇧L M f)"
proof cases
  assume "(∀x. T x = 0)" then show ?thesis
    by simp
next
  assume **: "¬ (∀x. T x = 0)"
  show ?thesis
  proof cases
    assume "integrable M f" with T show ?thesis
      by (rule integral_bounded_linear)
  next
    assume not: "¬ integrable M f"
    moreover have "¬ integrable M (λx. T (f x))"
    proof
      assume "integrable M (λx. T (f x))"
      from integrable_bounded_linear[OF T' this] not *[OF **]
      show False
        by auto
    qed
    ultimately show ?thesis
      using T by (simp add: not_integrable_integral_eq linear_simps)
  qed
qed

lemma integral_scaleR_left[simp]: "(c ≠ 0 ⟹ integrable M f) ⟹ (∫ x. f x *⇩R c ∂M) = integral⇧L M f *⇩R c"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_left)

lemma integral_scaleR_right[simp]: "(∫ x. c *⇩R f x ∂M) = c *⇩R integral⇧L M f"
  by (rule integral_bounded_linear'[OF bounded_linear_scaleR_right bounded_linear_scaleR_right[of "1 / c"]]) simp

lemma integral_mult_left[simp]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ integrable M f) ⟹ (∫ x. f x * c ∂M) = integral⇧L M f * c"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_left)

lemma integral_mult_right[simp]:
  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
  shows "(c ≠ 0 ⟹ integrable M f) ⟹ (∫ x. c * f x ∂M) = c * integral⇧L M f"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_right)

lemma integral_mult_left_zero[simp]:
  fixes c :: "_::{real_normed_field,second_countable_topology}"
  shows "(∫ x. f x * c ∂M) = integral⇧L M f * c"
  by (rule integral_bounded_linear'[OF bounded_linear_mult_left bounded_linear_mult_left[of "1 / c"]]) simp

lemma integral_mult_right_zero[simp]:
  fixes c :: "_::{real_normed_field,second_countable_topology}"
  shows "(∫ x. c * f x ∂M) = c * integral⇧L M f"
  by (rule integral_bounded_linear'[OF bounded_linear_mult_right bounded_linear_mult_right[of "1 / c"]]) simp

lemma integral_inner_left[simp]: "(c ≠ 0 ⟹ integrable M f) ⟹ (∫ x. f x ∙ c ∂M) = integral⇧L M f ∙ c"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)

lemma integral_inner_right[simp]: "(c ≠ 0 ⟹ integrable M f) ⟹ (∫ x. c ∙ f x ∂M) = c ∙ integral⇧L M f"
  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_right)

lemma integral_divide_zero[simp]:
  fixes c :: "_::{real_normed_field, field, second_countable_topology}"
  shows "integral⇧L M (λx. f x / c) = integral⇧L M f / c"
  by (rule integral_bounded_linear'[OF bounded_linear_divide bounded_linear_mult_left[of c]]) simp

lemma integral_minus[simp]: "integral⇧L M (λx. - f x) = - integral⇧L M f"
  by (rule integral_bounded_linear'[OF bounded_linear_minus[OF bounded_linear_ident] bounded_linear_minus[OF bounded_linear_ident]]) simp

lemma integral_complex_of_real[simp]: "integral⇧L M (λx. complex_of_real (f x)) = of_real (integral⇧L M f)"
  by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_Re]) simp

lemma integral_cnj[simp]: "integral⇧L M (λx. cnj (f x)) = cnj (integral⇧L M f)"
  by (rule integral_bounded_linear'[OF bounded_linear_cnj bounded_linear_cnj]) simp

lemmas integral_divide[simp] =
  integral_bounded_linear[OF bounded_linear_divide]
lemmas integral_Re[simp] =
  integral_bounded_linear[OF bounded_linear_Re]
lemmas integral_Im[simp] =
  integral_bounded_linear[OF bounded_linear_Im]
lemmas integral_of_real[simp] =
  integral_bounded_linear[OF bounded_linear_of_real]
lemmas integral_fst[simp] =
  integral_bounded_linear[OF bounded_linear_fst]
lemmas integral_snd[simp] =
  integral_bounded_linear[OF bounded_linear_snd]

lemma integral_norm_bound_ennreal:
  "integrable M f ⟹ norm (integral⇧L M f) ≤ (∫⇧+x. norm (f x) ∂M)"
  by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)

lemma integrableI_sequence:
  fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}"
  assumes f[measurable]: "f ∈ borel_measurable M"
  assumes s: "⋀i. simple_bochner_integrable M (s i)"
  assumes lim: "(λi. ∫⇧+x. norm (f x - s i x) ∂M) ⇢ 0" (is "?S ⇢ 0")
  shows "integrable M f"
proof -
  let ?s = "λn. simple_bochner_integral M (s n)"

  have "∃x. ?s ⇢ x"
    unfolding convergent_eq_Cauchy
  proof (rule metric_CauchyI)
    fix e :: real assume "0 < e"
    then have "0 < ennreal (e / 2)" by auto
    from order_tendstoD(2)[OF lim this]
    obtain M where M: "⋀n. M ≤ n ⟹ ?S n < e / 2"
      by (auto simp: eventually_sequentially)
    show "∃M. ∀m≥M. ∀n≥M. dist (?s m) (?s n) < e"
    proof (intro exI allI impI)
      fix m n assume m: "M ≤ m" and n: "M ≤ n"
      have "?S n ≠ ∞"
        using M[OF n] by auto
      have "norm (?s n - ?s m) ≤ ?S n + ?S m"
        by (intro simple_bochner_integral_bounded s f)
      also have "… < ennreal (e / 2) + e / 2"
        by (intro add_strict_mono M n m)
      also have "… = e" using ‹0<e› by (simp flip: ennreal_plus)
      finally show "dist (?s n) (?s m) < e"
        using ‹0<e› by (simp add: dist_norm ennreal_less_iff)
    qed
  qed
  then obtain x where "?s ⇢ x" ..
  show ?thesis
    by (rule, rule) fact+
qed

proposition nn_integral_dominated_convergence_norm:
  fixes u' :: "_ ⇒ _::{real_normed_vector, second_countable_topology}"
  assumes [measurable]:
       "⋀i. u i ∈ borel_measurable M" "u' ∈ borel_measurable M" "w ∈ borel_measurable M"
    and bound: "⋀j. AE x in M. norm (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. norm (u' x - u i x) ∂M)) ⇢ 0"
proof -
  have "AE x in M. ∀j. norm (u j x) ≤ w x"
    unfolding AE_all_countable by rule fact
  with u' have bnd: "AE x in M. ∀j. norm (u' x - u j x) ≤ 2 * w x"
  proof (eventually_elim, intro allI)
    fix i x assume "(λi. u i x) ⇢ u' x" "∀j. norm (u j x) ≤ w x" "∀j. norm (u j x) ≤ w x"
    then have "norm (u' x) ≤ w x" "norm (u i x) ≤ w x"
      by (auto intro: LIMSEQ_le_const2 tendsto_norm)
    then have "norm (u' x) + norm (u i x) ≤ 2 * w x"
      by simp
    also have "norm (u' x - u i x) ≤ norm (u' x) + norm (u i x)"
      by (rule norm_triangle_ineq4)
    finally (xtrans) show "norm (u' x - u i x) ≤ 2 * w x" .
  qed
  have w_nonneg: "AE x in M. 0 ≤ w x"
    using bound[of 0] by (auto intro: order_trans[OF norm_ge_zero])

  have "(λi. (∫⇧+x. norm (u' x - u i x) ∂M)) ⇢ (∫⇧+x. 0 ∂M)"
  proof (rule nn_integral_dominated_convergence)
    show "(∫⇧+x. 2 * w x ∂M) < ∞"
      by (rule nn_integral_mult_bounded_inf[OF _ w, of 2]) (insert w_nonneg, auto simp: ennreal_mult )
    show "AE x in M. (λi. ennreal (norm (u' x - u i x))) ⇢ 0"
      using u'
    proof eventually_elim
      fix x assume "(λi. u i x) ⇢ u' x"
      from tendsto_diff[OF tendsto_const[of "u' x"] this]
      show "(λi. ennreal (norm (u' x - u i x))) ⇢ 0"
        by (simp add: tendsto_norm_zero_iff flip: ennreal_0)
    qed
  qed (insert bnd w_nonneg, auto)
  then show ?thesis by simp
qed

proposition integrableI_bounded:
  fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}"
  assumes f[measurable]: "f ∈ borel_measurable M" and fin: "(∫⇧+x. norm (f x) ∂M) < ∞"
  shows "integrable M f"
proof -
  from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
    s: "⋀i. simple_function M (s i)" and
    pointwise: "⋀x. x ∈ space M ⟹ (λi. s i x) ⇢ f x" and
    bound: "⋀i x. x ∈ space M ⟹ norm (s i x) ≤ 2 * norm (f x)"
    by simp metis

  show ?thesis
  proof (rule integrableI_sequence)
    { fix i
      have "(∫⇧+x. norm (s i x) ∂M) ≤ (∫⇧+x. ennreal (2 * norm (f x)) ∂M)"
        by (intro nn_integral_mono) (simp add: bound)
      also have "… = 2 * (∫⇧+x. ennreal (norm (f x)) ∂M)"
        by (simp add: ennreal_mult nn_integral_cmult)
      also have "… < top"
        using fin by (simp add: ennreal_mult_less_top)
      finally have "(∫⇧+x. norm (s i x) ∂M) < ∞"
        by simp }
    note fin_s = this

    show "⋀i. simple_bochner_integrable M (s i)"
      by (rule simple_bochner_integrableI_bounded) fact+

    show "(λi. ∫⇧+ x. ennreal (norm (f x - s i x)) ∂M) ⇢ 0"
    proof (rule nn_integral_dominated_convergence_norm)
      show "⋀j. AE x in M. norm (s j x) ≤ 2 * norm (f x)"
        using bound by auto
      show "⋀i. s i ∈ borel_measurable M" "(λx. 2 * norm (f x)) ∈ borel_measurable M"
        using s by (auto intro: borel_measurable_simple_function)
      show "(∫⇧+ x. ennreal (2 * norm (f x)) ∂M) < ∞"
        using fin by (simp add: nn_integral_cmult ennreal_mult ennreal_mult_less_top)
      show "AE x in M. (λi. s i x) ⇢ f x"
        using pointwise by auto
    qed fact
  qed fact
qed

lemma integrableI_bounded_set:
  fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}"
  assumes [measurable]: "A ∈ sets M" "f ∈ borel_measurable M"
  assumes finite: "emeasure M A < ∞"
    and bnd: "AE x in M. x ∈ A ⟶ norm (f x) ≤ B"
    and null: "AE x in M. x ∉ A ⟶ f x = 0"
  shows "integrable M f"
proof (rule integrableI_bounded)
  { fix x :: 'b have "norm x ≤ B ⟹