# Theory Lebesgue_Measure

theory Lebesgue_Measure
imports Caratheodory Complete_Measure Regularity
```(*  Title:      HOL/Analysis/Lebesgue_Measure.thy
Author:     Johannes Hölzl, TU München
Author:     Robert Himmelmann, TU München
Author:     Luke Serafin
*)

section ‹Lebesgue measure›

theory Lebesgue_Measure
imports Finite_Product_Measure Bochner_Integration Caratheodory Complete_Measure Summation_Tests Regularity
begin

lemma measure_eqI_lessThan:
fixes M N :: "real measure"
assumes sets: "sets M = sets borel" "sets N = sets borel"
assumes fin: "⋀x. emeasure M {x <..} < ∞"
assumes "⋀x. emeasure M {x <..} = emeasure N {x <..}"
shows "M = N"
proof (rule measure_eqI_generator_eq_countable)
let ?LT = "λa::real. {a <..}" let ?E = "range ?LT"
show "Int_stable ?E"
by (auto simp: Int_stable_def lessThan_Int_lessThan)

show "?E ⊆ Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
unfolding sets borel_Ioi by auto

show "?LT`Rats ⊆ ?E" "(⋃i∈Rats. ?LT i) = UNIV" "⋀a. a ∈ ?LT`Rats ⟹ emeasure M a ≠ ∞"
using fin by (auto intro: Rats_no_bot_less simp: less_top)
qed (auto intro: assms countable_rat)

subsection ‹Every right continuous and nondecreasing function gives rise to a measure›

definition interval_measure :: "(real ⇒ real) ⇒ real measure" where
"interval_measure F = extend_measure UNIV {(a, b). a ≤ b} (λ(a, b). {a <.. b}) (λ(a, b). ennreal (F b - F a))"

lemma emeasure_interval_measure_Ioc:
assumes "a ≤ b"
assumes mono_F: "⋀x y. x ≤ y ⟹ F x ≤ F y"
assumes right_cont_F : "⋀a. continuous (at_right a) F"
shows "emeasure (interval_measure F) {a <.. b} = F b - F a"
proof (rule extend_measure_caratheodory_pair[OF interval_measure_def ‹a ≤ b›])
show "semiring_of_sets UNIV {{a<..b} |a b :: real. a ≤ b}"
proof (unfold_locales, safe)
fix a b c d :: real assume *: "a ≤ b" "c ≤ d"
then show "∃C⊆{{a<..b} |a b. a ≤ b}. finite C ∧ disjoint C ∧ {a<..b} - {c<..d} = ⋃C"
proof cases
let ?C = "{{a<..b}}"
assume "b < c ∨ d ≤ a ∨ d ≤ c"
with * have "?C ⊆ {{a<..b} |a b. a ≤ b} ∧ finite ?C ∧ disjoint ?C ∧ {a<..b} - {c<..d} = ⋃?C"
thus ?thesis ..
next
let ?C = "{{a<..c}, {d<..b}}"
assume "¬ (b < c ∨ d ≤ a ∨ d ≤ c)"
with * have "?C ⊆ {{a<..b} |a b. a ≤ b} ∧ finite ?C ∧ disjoint ?C ∧ {a<..b} - {c<..d} = ⋃?C"
by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
thus ?thesis ..
qed
qed (auto simp: Ioc_inj, metis linear)
next
fix l r :: "nat ⇒ real" and a b :: real
assume l_r[simp]: "⋀n. l n ≤ r n" and "a ≤ b" and disj: "disjoint_family (λn. {l n<..r n})"
assume lr_eq_ab: "(⋃i. {l i<..r i}) = {a<..b}"

have [intro, simp]: "⋀a b. a ≤ b ⟹ F a ≤ F b"
by (auto intro!: l_r mono_F)

{ fix S :: "nat set" assume "finite S"
moreover note ‹a ≤ b›
moreover have "⋀i. i ∈ S ⟹ {l i <.. r i} ⊆ {a <.. b}"
unfolding lr_eq_ab[symmetric] by auto
ultimately have "(∑i∈S. F (r i) - F (l i)) ≤ F b - F a"
proof (induction S arbitrary: a rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "∃i∈S. l i < r i"
with ‹finite S› have "Min (l ` {i∈S. l i < r i}) ∈ l ` {i∈S. l i < r i}"
by (intro Min_in) auto
then obtain m where m: "m ∈ S" "l m < r m" "l m = Min (l ` {i∈S. l i < r i})"
by fastforce

have "(∑i∈S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (∑i∈S - {m}. F (r i) - F (l i))"
using m psubset by (intro sum.remove) auto
also have "(∑i∈S - {m}. F (r i) - F (l i)) ≤ F b - F (r m)"
proof (intro psubset.IH)
show "S - {m} ⊂ S"
using ‹m∈S› by auto
show "r m ≤ b"
using psubset.prems(2)[OF ‹m∈S›] ‹l m < r m› by auto
next
fix i assume "i ∈ S - {m}"
then have i: "i ∈ S" "i ≠ m" by auto
{ assume i': "l i < r i" "l i < r m"
with ‹finite S› i m have "l m ≤ l i"
by auto
with i' have "{l i <.. r i} ∩ {l m <.. r m} ≠ {}"
by auto
then have False
using disjoint_family_onD[OF disj, of i m] i by auto }
then have "l i ≠ r i ⟹ r m ≤ l i"
unfolding not_less[symmetric] using l_r[of i] by auto
then show "{l i <.. r i} ⊆ {r m <.. b}"
using psubset.prems(2)[OF ‹i∈S›] by auto
qed
also have "F (r m) - F (l m) ≤ F (r m) - F a"
using psubset.prems(2)[OF ‹m ∈ S›] ‹l m < r m›
by (auto simp add: Ioc_subset_iff intro!: mono_F)
finally show ?case
qed (auto simp add: ‹a ≤ b› less_le)
qed }
note claim1 = this

(* second key induction: a lower bound on the measures of any finite collection of Ai's
that cover an interval {u..v} *)

{ fix S u v and l r :: "nat ⇒ real"
assume "finite S" "⋀i. i∈S ⟹ l i < r i" "{u..v} ⊆ (⋃i∈S. {l i<..< r i})"
then have "F v - F u ≤ (∑i∈S. F (r i) - F (l i))"
proof (induction arbitrary: v u rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "S = {}" then show ?case
using psubset by (simp add: mono_F)
next
assume "S ≠ {}"
then obtain j where "j ∈ S"
by auto

let ?R = "r j < u ∨ l j > v ∨ (∃i∈S-{j}. l i ≤ l j ∧ r j ≤ r i)"
show ?case
proof cases
assume "?R"
with ‹j ∈ S› psubset.prems have "{u..v} ⊆ (⋃i∈S-{j}. {l i<..< r i})"
apply (auto simp: subset_eq Ball_def)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis order_trans less_le_not_le linear)
done
with ‹j ∈ S› have "F v - F u ≤ (∑i∈S - {j}. F (r i) - F (l i))"
by (intro psubset) auto
also have "… ≤ (∑i∈S. F (r i) - F (l i))"
using psubset.prems
by (intro sum_mono2 psubset) (auto intro: less_imp_le)
finally show ?thesis .
next
assume "¬ ?R"
then have j: "u ≤ r j" "l j ≤ v" "⋀i. i ∈ S - {j} ⟹ r i < r j ∨ l i > l j"
by (auto simp: not_less)
let ?S1 = "{i ∈ S. l i < l j}"
let ?S2 = "{i ∈ S. r i > r j}"

have "(∑i∈S. F (r i) - F (l i)) ≥ (∑i∈?S1 ∪ ?S2 ∪ {j}. F (r i) - F (l i))"
using ‹j ∈ S› ‹finite S› psubset.prems j
by (intro sum_mono2) (auto intro: less_imp_le)
also have "(∑i∈?S1 ∪ ?S2 ∪ {j}. F (r i) - F (l i)) =
(∑i∈?S1. F (r i) - F (l i)) + (∑i∈?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
using psubset(1) psubset.prems(1) j
apply (subst sum.union_disjoint)
apply simp_all
apply (subst sum.union_disjoint)
apply auto
apply (metis less_le_not_le)
done
also (xtrans) have "(∑i∈?S1. F (r i) - F (l i)) ≥ F (l j) - F u"
using ‹j ∈ S› ‹finite S› psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis less_le_trans not_le)
done
also (xtrans) have "(∑i∈?S2. F (r i) - F (l i)) ≥ F v - F (r j)"
using ‹j ∈ S› ‹finite S› psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis le_less_trans not_le)
done
finally (xtrans) show ?case
qed
qed
qed }
note claim2 = this

(* now prove the inequality going the other way *)
have "ennreal (F b - F a) ≤ (∑i. ennreal (F (r i) - F (l i)))"
proof (rule ennreal_le_epsilon)
fix epsilon :: real assume egt0: "epsilon > 0"
have "∀i. ∃d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
proof
fix i
note right_cont_F [of "r i"]
thus "∃d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
apply -
apply (subst (asm) continuous_at_right_real_increasing)
apply (rule mono_F, assumption)
apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
apply (erule impE)
using egt0 by (auto simp add: field_simps)
qed
then obtain delta where
deltai_gt0: "⋀i. delta i > 0" and
deltai_prop: "⋀i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
by metis
have "∃a' > a. F a' - F a < epsilon / 2"
apply (insert right_cont_F [of a])
apply (subst (asm) continuous_at_right_real_increasing)
using mono_F apply force
apply (drule_tac x = "epsilon / 2" in spec)
using egt0 unfolding mult.commute [of 2] by force
then obtain a' where a'lea [arith]: "a' > a" and
a_prop: "F a' - F a < epsilon / 2"
by auto
define S' where "S' = {i. l i < r i}"
obtain S :: "nat set" where
"S ⊆ S'" and finS: "finite S" and
Sprop: "{a'..b} ⊆ (⋃i ∈ S. {l i<..<r i + delta i})"
proof (rule compactE_image)
show "compact {a'..b}"
by (rule compact_Icc)
show "⋀i. i ∈ S' ⟹ open ({l i<..<r i + delta i})" by auto
have "{a'..b} ⊆ {a <.. b}"
by auto
also have "{a <.. b} = (⋃i∈S'. {l i<..r i})"
unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
also have "… ⊆ (⋃i ∈ S'. {l i<..<r i + delta i})"
apply (intro UN_mono)
apply (auto simp: S'_def)
apply (cut_tac i=i in deltai_gt0)
apply simp
done
finally show "{a'..b} ⊆ (⋃i ∈ S'. {l i<..<r i + delta i})" .
qed
with S'_def have Sprop2: "⋀i. i ∈ S ⟹ l i < r i" by auto
from finS have "∃n. ∀i ∈ S. i ≤ n"
by (subst finite_nat_set_iff_bounded_le [symmetric])
then obtain n where Sbound [rule_format]: "∀i ∈ S. i ≤ n" ..
have "F b - F a' ≤ (∑i∈S. F (r i + delta i) - F (l i))"
apply (rule claim2 [rule_format])
using finS Sprop apply auto
apply (frule Sprop2)
apply (subgoal_tac "delta i > 0")
apply arith
by (rule deltai_gt0)
also have "... ≤ (∑i ∈ S. F(r i) - F(l i) + epsilon / 2^(i+2))"
apply (rule sum_mono)
apply simp
apply (rule order_trans)
apply (rule less_imp_le)
apply (rule deltai_prop)
by auto
also have "... = (∑i ∈ S. F(r i) - F(l i)) +
(epsilon / 4) * (∑i ∈ S. (1 / 2)^i)" (is "_ = ?t + _")
by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
also have "... ≤ ?t + (epsilon / 4) * (∑ i < Suc n. (1 / 2)^i)"
apply (rule mult_left_mono)
apply (rule sum_mono2)
using egt0 apply auto
by (frule Sbound, auto)
also have "... ≤ ?t + (epsilon / 2)"
apply (subst geometric_sum)
apply auto
apply (rule mult_left_mono)
using egt0 apply auto
done
finally have aux2: "F b - F a' ≤ (∑i∈S. F (r i) - F (l i)) + epsilon / 2"
by simp

have "F b - F a = (F b - F a') + (F a' - F a)"
by auto
also have "... ≤ (F b - F a') + epsilon / 2"
using a_prop by (intro add_left_mono) simp
also have "... ≤ (∑i∈S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
apply (rule aux2)
done
also have "... = (∑i∈S. F (r i) - F (l i)) + epsilon"
by auto
also have "... ≤ (∑i≤n. F (r i) - F (l i)) + epsilon"
using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono2)
finally have "ennreal (F b - F a) ≤ (∑i≤n. ennreal (F (r i) - F (l i))) + epsilon"
using egt0 by (simp add: sum_nonneg flip: ennreal_plus)
then show "ennreal (F b - F a) ≤ (∑i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
qed
moreover have "(∑i. ennreal (F (r i) - F (l i))) ≤ ennreal (F b - F a)"
using ‹a ≤ b› by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
ultimately show "(∑n. ennreal (F (r n) - F (l n))) = ennreal (F b - F a)"
by (rule antisym[rotated])
qed (auto simp: Ioc_inj mono_F)

lemma measure_interval_measure_Ioc:
assumes "a ≤ b"
assumes mono_F: "⋀x y. x ≤ y ⟹ F x ≤ F y"
assumes right_cont_F : "⋀a. continuous (at_right a) F"
shows "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def
apply (subst emeasure_interval_measure_Ioc)
apply fact+
done

lemma emeasure_interval_measure_Ioc_eq:
"(⋀x y. x ≤ y ⟹ F x ≤ F y) ⟹ (⋀a. continuous (at_right a) F) ⟹
emeasure (interval_measure F) {a <.. b} = (if a ≤ b then F b - F a else 0)"
using emeasure_interval_measure_Ioc[of a b F] by auto

lemma sets_interval_measure [simp, measurable_cong]: "sets (interval_measure F) = sets borel"
apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
apply (rule sigma_sets_eqI)
apply auto
apply (case_tac "a ≤ ba")
apply (auto intro: sigma_sets.Empty)
done

lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"

lemma emeasure_interval_measure_Icc:
assumes "a ≤ b"
assumes mono_F: "⋀x y. x ≤ y ⟹ F x ≤ F y"
assumes cont_F : "continuous_on UNIV F"
shows "emeasure (interval_measure F) {a .. b} = F b - F a"
proof (rule tendsto_unique)
{ fix a b :: real assume "a ≤ b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
using cont_F
by (subst emeasure_interval_measure_Ioc)
(auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
note * = this

let ?F = "interval_measure F"
show "((λa. F b - F a) ⤏ emeasure ?F {a..b}) (at_left a)"
proof (rule tendsto_at_left_sequentially)
show "a - 1 < a" by simp
fix X assume "⋀n. X n < a" "incseq X" "X ⇢ a"
with ‹a ≤ b› have "(λn. emeasure ?F {X n<..b}) ⇢ emeasure ?F (⋂n. {X n <..b})"
apply (intro Lim_emeasure_decseq)
apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
apply force
apply (subst (asm ) *)
apply (auto intro: less_le_trans less_imp_le)
done
also have "(⋂n. {X n <..b}) = {a..b}"
using ‹⋀n. X n < a›
apply auto
apply (rule LIMSEQ_le_const2[OF ‹X ⇢ a›])
apply (auto intro: less_imp_le)
apply (auto intro: less_le_trans)
done
also have "(λn. emeasure ?F {X n<..b}) = (λn. F b - F (X n))"
using ‹⋀n. X n < a› ‹a ≤ b› by (subst *) (auto intro: less_imp_le less_le_trans)
finally show "(λn. F b - F (X n)) ⇢ emeasure ?F {a..b}" .
qed
show "((λa. ennreal (F b - F a)) ⤏ F b - F a) (at_left a)"
by (rule continuous_on_tendsto_compose[where g="λx. x" and s=UNIV])
(auto simp: continuous_on_ennreal continuous_on_diff cont_F continuous_on_const)
qed (rule trivial_limit_at_left_real)

lemma sigma_finite_interval_measure:
assumes mono_F: "⋀x y. x ≤ y ⟹ F x ≤ F y"
assumes right_cont_F : "⋀a. continuous (at_right a) F"
shows "sigma_finite_measure (interval_measure F)"
apply unfold_locales
apply (intro exI[of _ "(λ(a, b). {a <.. b}) ` (ℚ × ℚ)"])
apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
done

subsection ‹Lebesgue-Borel measure›

definition lborel :: "('a :: euclidean_space) measure" where
"lborel = distr (Π⇩M b∈Basis. interval_measure (λx. x)) borel (λf. ∑b∈Basis. f b *⇩R b)"

abbreviation lebesgue :: "'a::euclidean_space measure"
where "lebesgue ≡ completion lborel"

abbreviation lebesgue_on :: "'a set ⇒ 'a::euclidean_space measure"
where "lebesgue_on Ω ≡ restrict_space (completion lborel) Ω"

lemma
shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
and space_lborel[simp]: "space lborel = space borel"
and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"

lemma sets_lebesgue_on_refl [iff]: "S ∈ sets (lebesgue_on S)"
by (metis inf_top.right_neutral sets.top space_borel space_completion space_lborel space_restrict_space)

lemma Compl_in_sets_lebesgue: "-A ∈ sets lebesgue ⟷ A  ∈ sets lebesgue"
by (metis Compl_eq_Diff_UNIV double_compl space_borel space_completion space_lborel Sigma_Algebra.sets.compl_sets)

lemma measurable_lebesgue_cong:
assumes "⋀x. x ∈ S ⟹ f x = g x"
shows "f ∈ measurable (lebesgue_on S) M ⟷ g ∈ measurable (lebesgue_on S) M"
by (metis (mono_tags, lifting) IntD1 assms measurable_cong_strong space_restrict_space)

text‹Measurability of continuous functions›
lemma continuous_imp_measurable_on_sets_lebesgue:
assumes f: "continuous_on S f" and S: "S ∈ sets lebesgue"
shows "f ∈ borel_measurable (lebesgue_on S)"
proof -
have "sets (restrict_space borel S) ⊆ sets (lebesgue_on S)"
then show ?thesis
by (simp add: borel_measurable_continuous_on_restrict [OF f] borel_measurable_subalgebra
space_restrict_space)
qed

context
begin

interpretation sigma_finite_measure "interval_measure (λx. x)"
by (rule sigma_finite_interval_measure) auto
interpretation finite_product_sigma_finite "λ_. interval_measure (λx. x)" Basis
proof qed simp

lemma lborel_eq_real: "lborel = interval_measure (λx. x)"
unfolding lborel_def Basis_real_def
using distr_id[of "interval_measure (λx. x)"]
by (subst distr_component[symmetric])
(simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)

lemma lborel_eq: "lborel = distr (Π⇩M b∈Basis. lborel) borel (λf. ∑b∈Basis. f b *⇩R b)"
by (subst lborel_def) (simp add: lborel_eq_real)

lemma nn_integral_lborel_prod:
assumes [measurable]: "⋀b. b ∈ Basis ⟹ f b ∈ borel_measurable borel"
assumes nn[simp]: "⋀b x. b ∈ Basis ⟹ 0 ≤ f b x"
shows "(∫⇧+x. (∏b∈Basis. f b (x ∙ b)) ∂lborel) = (∏b∈Basis. (∫⇧+x. f b x ∂lborel))"
by (simp add: lborel_def nn_integral_distr product_nn_integral_prod
product_nn_integral_singleton)

lemma emeasure_lborel_Icc[simp]:
fixes l u :: real
assumes [simp]: "l ≤ u"
shows "emeasure lborel {l .. u} = u - l"
proof -
have "((λf. f 1) -` {l..u} ∩ space (Pi⇩M {1} (λb. interval_measure (λx. x)))) = {1::real} →⇩E {l..u}"
by (auto simp: space_PiM)
then show ?thesis
by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
qed

lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l ≤ u then u - l else 0)"
by simp

lemma emeasure_lborel_cbox[simp]:
assumes [simp]: "⋀b. b ∈ Basis ⟹ l ∙ b ≤ u ∙ b"
shows "emeasure lborel (cbox l u) = (∏b∈Basis. (u - l) ∙ b)"
proof -
have "(λx. ∏b∈Basis. indicator {l∙b .. u∙b} (x ∙ b) :: ennreal) = indicator (cbox l u)"
by (auto simp: fun_eq_iff cbox_def split: split_indicator)
then have "emeasure lborel (cbox l u) = (∫⇧+x. (∏b∈Basis. indicator {l∙b .. u∙b} (x ∙ b)) ∂lborel)"
by simp
also have "… = (∏b∈Basis. (u - l) ∙ b)"
by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
finally show ?thesis .
qed

lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x ≠ c"
using SOME_Basis AE_discrete_difference [of "{c}" lborel] emeasure_lborel_cbox [of c c]

lemma emeasure_lborel_Ioo[simp]:
assumes [simp]: "l ≤ u"
shows "emeasure lborel {l <..< u} = ennreal (u - l)"
proof -
have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed

lemma emeasure_lborel_Ioc[simp]:
assumes [simp]: "l ≤ u"
shows "emeasure lborel {l <.. u} = ennreal (u - l)"
proof -
have "emeasure lborel {l <.. u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed

lemma emeasure_lborel_Ico[simp]:
assumes [simp]: "l ≤ u"
shows "emeasure lborel {l ..< u} = ennreal (u - l)"
proof -
have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed

lemma emeasure_lborel_box[simp]:
assumes [simp]: "⋀b. b ∈ Basis ⟹ l ∙ b ≤ u ∙ b"
shows "emeasure lborel (box l u) = (∏b∈Basis. (u - l) ∙ b)"
proof -
have "(λx. ∏b∈Basis. indicator {l∙b <..< u∙b} (x ∙ b) :: ennreal) = indicator (box l u)"
by (auto simp: fun_eq_iff box_def split: split_indicator)
then have "emeasure lborel (box l u) = (∫⇧+x. (∏b∈Basis. indicator {l∙b <..< u∙b} (x ∙ b)) ∂lborel)"
by simp
also have "… = (∏b∈Basis. (u - l) ∙ b)"
by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
finally show ?thesis .
qed

lemma emeasure_lborel_cbox_eq:
"emeasure lborel (cbox l u) = (if ∀b∈Basis. l ∙ b ≤ u ∙ b then ∏b∈Basis. (u - l) ∙ b else 0)"
using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)

lemma emeasure_lborel_box_eq:
"emeasure lborel (box l u) = (if ∀b∈Basis. l ∙ b ≤ u ∙ b then ∏b∈Basis. (u - l) ∙ b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force

lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
using emeasure_lborel_cbox[of x x] nonempty_Basis
by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: prod_constant)

lemma fmeasurable_cbox [iff]: "cbox a b ∈ fmeasurable lborel"
and fmeasurable_box [iff]: "box a b ∈ fmeasurable lborel"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)

lemma
fixes l u :: real
assumes [simp]: "l ≤ u"
shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"

lemma
assumes [simp]: "⋀b. b ∈ Basis ⟹ l ∙ b ≤ u ∙ b"
shows measure_lborel_box[simp]: "measure lborel (box l u) = (∏b∈Basis. (u - l) ∙ b)"
and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (∏b∈Basis. (u - l) ∙ b)"
by (simp_all add: measure_def inner_diff_left prod_nonneg)

lemma measure_lborel_cbox_eq:
"measure lborel (cbox l u) = (if ∀b∈Basis. l ∙ b ≤ u ∙ b then ∏b∈Basis. (u - l) ∙ b else 0)"
using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)

lemma measure_lborel_box_eq:
"measure lborel (box l u) = (if ∀b∈Basis. l ∙ b ≤ u ∙ b then ∏b∈Basis. (u - l) ∙ b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force

lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"

lemma sigma_finite_lborel: "sigma_finite_measure lborel"
proof
show "∃A::'a set set. countable A ∧ A ⊆ sets lborel ∧ ⋃A = space lborel ∧ (∀a∈A. emeasure lborel a ≠ ∞)"
by (intro exI[of _ "range (λn::nat. box (- real n *⇩R One) (real n *⇩R One))"])
(auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
qed

end

lemma emeasure_lborel_UNIV [simp]: "emeasure lborel (UNIV::'a::euclidean_space set) = ∞"
proof -
{ fix n::nat
let ?Ba = "Basis :: 'a set"
have "real n ≤ (2::real) ^ card ?Ba * real n"
also
have "... ≤ (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
apply (rule mult_left_mono)
apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
done
finally have "real n ≤ (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
} note [intro!] = this
show ?thesis
unfolding UN_box_eq_UNIV[symmetric]
apply (subst SUP_emeasure_incseq[symmetric])
apply (auto simp: incseq_def subset_box inner_add_left prod_constant
simp del: Sup_eq_top_iff SUP_eq_top_iff
intro!: ennreal_SUP_eq_top)
done
qed

lemma emeasure_lborel_countable:
fixes A :: "'a::euclidean_space set"
assumes "countable A"
shows "emeasure lborel A = 0"
proof -
have "A ⊆ (⋃i. {from_nat_into A i})" using from_nat_into_surj assms by force
then have "emeasure lborel A ≤ emeasure lborel (⋃i. {from_nat_into A i})"
by (intro emeasure_mono) auto
also have "emeasure lborel (⋃i. {from_nat_into A i}) = 0"
by (rule emeasure_UN_eq_0) auto
finally show ?thesis
qed

lemma countable_imp_null_set_lborel: "countable A ⟹ A ∈ null_sets lborel"
by (simp add: null_sets_def emeasure_lborel_countable sets.countable)

lemma finite_imp_null_set_lborel: "finite A ⟹ A ∈ null_sets lborel"
by (intro countable_imp_null_set_lborel countable_finite)

lemma lborel_neq_count_space[simp]: "lborel ≠ count_space (A::('a::ordered_euclidean_space) set)"
proof
assume asm: "lborel = count_space A"
have "space lborel = UNIV" by simp
hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
have "emeasure lborel {undefined::'a} = 1"
by (subst asm, subst emeasure_count_space_finite) auto
moreover have "emeasure lborel {undefined} ≠ 1" by simp
qed

lemma mem_closed_if_AE_lebesgue_open:
assumes "open S" "closed C"
assumes "AE x ∈ S in lebesgue. x ∈ C"
assumes "x ∈ S"
shows "x ∈ C"
proof (rule ccontr)
assume xC: "x ∉ C"
with openE[of "S - C"] assms
obtain e where e: "0 < e" "ball x e ⊆ S - C"
by blast
then obtain a b where box: "x ∈ box a b" "box a b ⊆ S - C"
by (metis rational_boxes order_trans)
then have "0 < emeasure lebesgue (box a b)"
by (auto simp: emeasure_lborel_box_eq mem_box algebra_simps intro!: prod_pos)
also have "… ≤ emeasure lebesgue (S - C)"
using assms box
by (auto intro!: emeasure_mono)
also have "… = 0"
using assms
by (auto simp: eventually_ae_filter completion.complete2 set_diff_eq null_setsD1)
finally show False by simp
qed

lemma mem_closed_if_AE_lebesgue: "closed C ⟹ (AE x in lebesgue. x ∈ C) ⟹ x ∈ C"
using mem_closed_if_AE_lebesgue_open[OF open_UNIV] by simp

subsection ‹Affine transformation on the Lebesgue-Borel›

lemma lborel_eqI:
fixes M :: "'a::euclidean_space measure"
assumes emeasure_eq: "⋀l u. (⋀b. b ∈ Basis ⟹ l ∙ b ≤ u ∙ b) ⟹ emeasure M (box l u) = (∏b∈Basis. (u - l) ∙ b)"
assumes sets_eq: "sets M = sets borel"
shows "lborel = M"
proof (rule measure_eqI_generator_eq)
let ?E = "range (λ(a, b). box a b::'a set)"
show "Int_stable ?E"
by (auto simp: Int_stable_def box_Int_box)

show "?E ⊆ Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"

let ?A = "λn::nat. box (- (real n *⇩R One)) (real n *⇩R One) :: 'a set"
show "range ?A ⊆ ?E" "(⋃i. ?A i) = UNIV"
unfolding UN_box_eq_UNIV by auto

{ fix i show "emeasure lborel (?A i) ≠ ∞" by auto }
{ fix X assume "X ∈ ?E" then show "emeasure lborel X = emeasure M X"
apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
apply (subst box_eq_empty(1)[THEN iffD2])
apply (auto intro: less_imp_le simp: not_le)
done }
qed

lemma lborel_affine_euclidean:
fixes c :: "'a::euclidean_space ⇒ real" and t
defines "T x ≡ t + (∑j∈Basis. (c j * (x ∙ j)) *⇩R j)"
assumes c: "⋀j. j ∈ Basis ⟹ c j ≠ 0"
shows "lborel = density (distr lborel borel T) (λ_. (∏j∈Basis. ¦c j¦))" (is "_ = ?D")
proof (rule lborel_eqI)
let ?B = "Basis :: 'a set"
fix l u assume le: "⋀b. b ∈ ?B ⟹ l ∙ b ≤ u ∙ b"
have [measurable]: "T ∈ borel →⇩M borel"
have eq: "T -` box l u = box
(∑j∈Basis. (((if 0 < c j then l - t else u - t) ∙ j) / c j) *⇩R j)
(∑j∈Basis. (((if 0 < c j then u - t else l - t) ∙ j) / c j) *⇩R j)"
using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
with le c show "emeasure ?D (box l u) = (∏b∈?B. (u - l) ∙ b)"
by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
field_simps divide_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
intro!: prod.cong)
qed simp

lemma lborel_affine:
fixes t :: "'a::euclidean_space"
shows "c ≠ 0 ⟹ lborel = density (distr lborel borel (λx. t + c *⇩R x)) (λ_. ¦c¦^DIM('a))"
using lborel_affine_euclidean[where c="λ_::'a. c" and t=t]
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation prod_constant by simp

lemma lborel_real_affine:
"c ≠ 0 ⟹ lborel = density (distr lborel borel (λx. t + c * x)) (λ_. ennreal (abs c))"
using lborel_affine[of c t] by simp

lemma AE_borel_affine:
fixes P :: "real ⇒ bool"
shows "c ≠ 0 ⟹ Measurable.pred borel P ⟹ AE x in lborel. P x ⟹ AE x in lborel. P (t + c * x)"
by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])

lemma nn_integral_real_affine:
fixes c :: real assumes [measurable]: "f ∈ borel_measurable borel" and c: "c ≠ 0"
shows "(∫⇧+x. f x ∂lborel) = ¦c¦ * (∫⇧+x. f (t + c * x) ∂lborel)"
by (subst lborel_real_affine[OF c, of t])

lemma lborel_integrable_real_affine:
fixes f :: "real ⇒ 'a :: {banach, second_countable_topology}"
assumes f: "integrable lborel f"
shows "c ≠ 0 ⟹ integrable lborel (λx. f (t + c * x))"
using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
by (subst (asm) nn_integral_real_affine[where c=c and t=t]) (auto simp: ennreal_mult_less_top)

lemma lborel_integrable_real_affine_iff:
fixes f :: "real ⇒ 'a :: {banach, second_countable_topology}"
shows "c ≠ 0 ⟹ integrable lborel (λx. f (t + c * x)) ⟷ integrable lborel f"
using
lborel_integrable_real_affine[of f c t]
lborel_integrable_real_affine[of "λx. f (t + c * x)" "1/c" "-t/c"]

lemma lborel_integral_real_affine:
fixes f :: "real ⇒ 'a :: {banach, second_countable_topology}" and c :: real
assumes c: "c ≠ 0" shows "(∫x. f x ∂ lborel) = ¦c¦ *⇩R (∫x. f (t + c * x) ∂lborel)"
proof cases
assume f[measurable]: "integrable lborel f" then show ?thesis
using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
by (subst lborel_real_affine[OF c, of t])
next
assume "¬ integrable lborel f" with c show ?thesis
qed

lemma
fixes c :: "'a::euclidean_space ⇒ real" and t
assumes c: "⋀j. j ∈ Basis ⟹ c j ≠ 0"
defines "T == (λx. t + (∑j∈Basis. (c j * (x ∙ j)) *⇩R j))"
shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (λ_. (∏j∈Basis. ¦c j¦))" (is "_ = ?D")
and lebesgue_affine_measurable: "T ∈ lebesgue →⇩M lebesgue"
proof -
have T_borel[measurable]: "T ∈ borel →⇩M borel"
by (auto simp: T_def[abs_def])
{ fix A :: "'a set" assume A: "A ∈ sets borel"
then have "emeasure lborel A = 0 ⟷ emeasure (density (distr lborel borel T) (λ_. (∏j∈Basis. ¦c j¦))) A = 0"
unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
also have "… ⟷ emeasure (distr lebesgue lborel T) A = 0"
using A c by (simp add: distr_completion emeasure_density nn_integral_cmult prod_nonneg cong: distr_cong)
finally have "emeasure lborel A = 0 ⟷ emeasure (distr lebesgue lborel T) A = 0" . }
then have eq: "null_sets lborel = null_sets (distr lebesgue lborel T)"
by (auto simp: null_sets_def)

show "T ∈ lebesgue →⇩M lebesgue"
by (rule completion.measurable_completion2) (auto simp: eq measurable_completion)

have "lebesgue = completion (density (distr lborel borel T) (λ_. (∏j∈Basis. ¦c j¦)))"
using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
also have "… = density (completion (distr lebesgue lborel T)) (λ_. (∏j∈Basis. ¦c j¦))"
using c by (auto intro!: always_eventually prod_pos completion_density_eq simp: distr_completion cong: distr_cong)
also have "… = density (distr lebesgue lebesgue T) (λ_. (∏j∈Basis. ¦c j¦))"
by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
finally show "lebesgue = density (distr lebesgue lebesgue T) (λ_. (∏j∈Basis. ¦c j¦))" .
qed

lemma lebesgue_measurable_scaling[measurable]: "( *⇩R) x ∈ lebesgue →⇩M lebesgue"
proof cases
assume "x = 0"
then have "( *⇩R) x = (λx. 0::'a)"
by (auto simp: fun_eq_iff)
then show ?thesis by auto
next
assume "x ≠ 0" then show ?thesis
using lebesgue_affine_measurable[of "λ_. x" 0]
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
qed

lemma
fixes m :: real and δ :: "'a::euclidean_space"
defines "T r d x ≡ r *⇩R x + d"
shows emeasure_lebesgue_affine: "emeasure lebesgue (T m δ ` S) = ¦m¦ ^ DIM('a) * emeasure lebesgue S" (is ?e)
and measure_lebesgue_affine: "measure lebesgue (T m δ ` S) = ¦m¦ ^ DIM('a) * measure lebesgue S" (is ?m)
proof -
show ?e
proof cases
assume "m = 0" then show ?thesis
next
let ?T = "T m δ" and ?T' = "T (1 / m) (- ((1/m) *⇩R δ))"
assume "m ≠ 0"
then have s_comp_s: "?T' ∘ ?T = id" "?T ∘ ?T' = id"
by (auto simp: T_def[abs_def] fun_eq_iff scaleR_add_right scaleR_diff_right)
then have "inv ?T' = ?T" "bij ?T'"
by (auto intro: inv_unique_comp o_bij)
then have eq: "T m δ ` S = T (1 / m) ((-1/m) *⇩R δ) -` S ∩ space lebesgue"
using bij_vimage_eq_inv_image[OF ‹bij ?T'›, of S] by auto

have trans_eq_T: "(λx. δ + (∑j∈Basis. (m * (x ∙ j)) *⇩R j)) = T m δ" for m δ
unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
by (auto simp add: euclidean_representation ac_simps)

have T[measurable]: "T r d ∈ lebesgue →⇩M lebesgue" for r d
using lebesgue_affine_measurable[of "λ_. r" d]
by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])

show ?thesis
proof cases
assume "S ∈ sets lebesgue" with ‹m ≠ 0› show ?thesis
unfolding eq
apply (subst lebesgue_affine_euclidean[of "λ_. m" δ])
apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
del: space_completion emeasure_completion)
apply (simp add: vimage_comp s_comp_s prod_constant)
done
next
assume "S ∉ sets lebesgue"
moreover have "?T ` S ∉ sets lebesgue"
proof
assume "?T ` S ∈ sets lebesgue"
then have "?T -` (?T ` S) ∩ space lebesgue ∈ sets lebesgue"
by (rule measurable_sets[OF T])
also have "?T -` (?T ` S) ∩ space lebesgue = S"
by (simp add: vimage_comp s_comp_s eq)
finally show False using ‹S ∉ sets lebesgue› by auto
qed
ultimately show ?thesis
qed
qed
show ?m
unfolding measure_def ‹?e› by (simp add: enn2real_mult prod_nonneg)
qed

lemma lebesgue_real_scale:
assumes "c ≠ 0"
shows   "lebesgue = density (distr lebesgue lebesgue (λx. c * x)) (λx. ennreal ¦c¦)"
using assms by (subst lebesgue_affine_euclidean[of "λ_. c" 0]) simp_all

lemma divideR_right:
fixes x y :: "'a::real_normed_vector"
shows "r ≠ 0 ⟹ y = x /⇩R r ⟷ r *⇩R y = x"
using scaleR_cancel_left[of r y "x /⇩R r"] by simp

lemma lborel_has_bochner_integral_real_affine_iff:
fixes x :: "'a :: {banach, second_countable_topology}"
shows "c ≠ 0 ⟹
has_bochner_integral lborel f x ⟷
has_bochner_integral lborel (λx. f (t + c * x)) (x /⇩R ¦c¦)"
unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)

lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
by (subst lborel_real_affine[of "-1" 0])
(auto simp: density_1 one_ennreal_def[symmetric])

lemma lborel_distr_mult:
assumes "(c::real) ≠ 0"
shows "distr lborel borel (( * ) c) = density lborel (λ_. inverse ¦c¦)"
proof-
have "distr lborel borel (( * ) c) = distr lborel lborel (( * ) c)" by (simp cong: distr_cong)
also from assms have "... = density lborel (λ_. inverse ¦c¦)"
by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
finally show ?thesis .
qed

lemma lborel_distr_mult':
assumes "(c::real) ≠ 0"
shows "lborel = density (distr lborel borel (( * ) c)) (λ_. ¦c¦)"
proof-
have "lborel = density lborel (λ_. 1)" by (rule density_1[symmetric])
also from assms have "(λ_. 1 :: ennreal) = (λ_. inverse ¦c¦ * ¦c¦)" by (intro ext) simp
also have "density lborel ... = density (density lborel (λ_. inverse ¦c¦)) (λ_. ¦c¦)"
by (subst density_density_eq) (auto simp: ennreal_mult)
also from assms have "density lborel (λ_. inverse ¦c¦) = distr lborel borel (( * ) c)"
by (rule lborel_distr_mult[symmetric])
finally show ?thesis .
qed

lemma lborel_distr_plus: "distr lborel borel ((+) c) = (lborel :: real measure)"
by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])

interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)

interpretation lborel_pair: pair_sigma_finite lborel lborel ..

lemma lborel_prod:
"lborel ⨂⇩M lborel = (lborel :: ('a::euclidean_space × 'b::euclidean_space) measure)"
proof (rule lborel_eqI[symmetric], clarify)
fix la ua :: 'a and lb ub :: 'b
assume lu: "⋀a b. (a, b) ∈ Basis ⟹ (la, lb) ∙ (a, b) ≤ (ua, ub) ∙ (a, b)"
have [simp]:
"⋀b. b ∈ Basis ⟹ la ∙ b ≤ ua ∙ b"
"⋀b. b ∈ Basis ⟹ lb ∙ b ≤ ub ∙ b"
"inj_on (λu. (u, 0)) Basis" "inj_on (λu. (0, u)) Basis"
"(λu. (u, 0)) ` Basis ∩ (λu. (0, u)) ` Basis = {}"
"box (la, lb) (ua, ub) = box la ua × box lb ub"
using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
show "emeasure (lborel ⨂⇩M lborel) (box (la, lb) (ua, ub)) =
ennreal (prod ((∙) ((ua, ub) - (la, lb))) Basis)"
by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def prod.union_disjoint
prod.reindex ennreal_mult inner_diff_left prod_nonneg)

(* FIXME: conversion in measurable prover *)
lemma lborelD_Collect[measurable (raw)]: "{x∈space borel. P x} ∈ sets borel ⟹ {x∈space lborel. P x} ∈ sets lborel"
by simp

lemma lborelD[measurable (raw)]: "A ∈ sets borel ⟹ A ∈ sets lborel"
by simp

lemma emeasure_bounded_finite:
assumes "bounded A" shows "emeasure lborel A < ∞"
proof -
obtain a b where "A ⊆ cbox a b"
by (meson bounded_subset_cbox_symmetric ‹bounded A›)
then have "emeasure lborel A ≤ emeasure lborel (cbox a b)"
by (intro emeasure_mono) auto
then show ?thesis
by (auto simp: emeasure_lborel_cbox_eq prod_nonneg less_top[symmetric] top_unique split: if_split_asm)
qed

lemma emeasure_compact_finite: "compact A ⟹ emeasure lborel A < ∞"
using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)

lemma borel_integrable_compact:
fixes f :: "'a::euclidean_space ⇒ 'b::{banach, second_countable_topology}"
assumes "compact S" "continuous_on S f"
shows "integrable lborel (λx. indicator S x *⇩R f x)"
proof cases
assume "S ≠ {}"
have "continuous_on S (λx. norm (f x))"
using assms by (intro continuous_intros)
from continuous_attains_sup[OF ‹compact S› ‹S ≠ {}› this]
obtain M where M: "⋀x. x ∈ S ⟹ norm (f x) ≤ M"
by auto
show ?thesis
proof (rule integrable_bound)
show "integrable lborel (λx. indicator S x * M)"
using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
show "(λx. indicator S x *⇩R f x) ∈ borel_measurable lborel"
using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact)
show "AE x in lborel. norm (indicator S x *⇩R f x) ≤ norm (indicator S x * M)"
by (auto split: split_indicator simp: abs_real_def dest!: M)
qed
qed simp

lemma borel_integrable_atLeastAtMost:
fixes f :: "real ⇒ real"
assumes f: "⋀x. a ≤ x ⟹ x ≤ b ⟹ isCont f x"
shows "integrable lborel (λx. f x * indicator {a .. b} x)" (is "integrable _ ?f")
proof -
have "integrable lborel (λx. indicator {a .. b} x *⇩R f x)"
proof (rule borel_integrable_compact)
from f show "continuous_on {a..b} f"
by (auto intro: continuous_at_imp_continuous_on)
qed simp
then show ?thesis
by (auto simp: mult.commute)
qed

subsection‹Lebesgue measurable sets›

abbreviation lmeasurable :: "'a::euclidean_space set set"
where
"lmeasurable ≡ fmeasurable lebesgue"

lemma not_measurable_UNIV [simp]: "UNIV ∉ lmeasurable"

lemma lmeasurable_iff_integrable:
"S ∈ lmeasurable ⟷ integrable lebesgue (indicator S :: 'a::euclidean_space ⇒ real)"
by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)

lemma lmeasurable_cbox [iff]: "cbox a b ∈ lmeasurable"
and lmeasurable_box [iff]: "box a b ∈ lmeasurable"
by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)

lemma fmeasurable_compact: "compact S ⟹ S ∈ fmeasurable lborel"
using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_compact)

lemma lmeasurable_compact: "compact S ⟹ S ∈ lmeasurable"
using fmeasurable_compact by (force simp: fmeasurable_def)

lemma measure_frontier:
"bounded S ⟹ measure lebesgue (frontier S) = measure lebesgue (closure S) - measure lebesgue (interior S)"
using closure_subset interior_subset
by (auto simp: frontier_def fmeasurable_compact intro!: measurable_measure_Diff)

lemma lmeasurable_closure:
"bounded S ⟹ closure S ∈ lmeasurable"

lemma lmeasurable_frontier:
"bounded S ⟹ frontier S ∈ lmeasurable"

lemma lmeasurable_open: "bounded S ⟹ open S ⟹ S ∈ lmeasurable"
using emeasure_bounded_finite[of S] by (intro fmeasurableI) (auto simp: borel_open)

lemma lmeasurable_ball [simp]: "ball a r ∈ lmeasurable"

lemma lmeasurable_cball [simp]: "cball a r ∈ lmeasurable"

lemma lmeasurable_interior: "bounded S ⟹ interior S ∈ lmeasurable"

lemma null_sets_cbox_Diff_box: "cbox a b - box a b ∈ null_sets lborel"
proof -
have "emeasure lborel (cbox a b - box a b) = 0"
by (subst emeasure_Diff) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq box_subset_cbox)
then have "cbox a b - box a b ∈ null_sets lborel"
by (auto simp: null_sets_def)
then show ?thesis
by (auto dest!: AE_not_in)
qed

lemma bounded_set_imp_lmeasurable:
assumes "bounded S" "S ∈ sets lebesgue" shows "S ∈ lmeasurable"
by (metis assms bounded_Un emeasure_bounded_finite emeasure_completion fmeasurableI main_part_null_part_Un)

subsection‹Translation preserves Lebesgue measure›

lemma sigma_sets_image:
assumes S: "S ∈ sigma_sets Ω M" and "M ⊆ Pow Ω" "f ` Ω = Ω" "inj_on f Ω"
and M: "⋀y. y ∈ M ⟹ f ` y ∈ M"
shows "(f ` S) ∈ sigma_sets Ω M"
using S
proof (induct S rule: sigma_sets.induct)
case (Basic a) then show ?case
next
case Empty then show ?case
next
case (Compl a)
then have "Ω - a ⊆ Ω" "a ⊆ Ω"
by (auto simp: sigma_sets_into_sp [OF ‹M ⊆ Pow Ω›])
then show ?case
by (auto simp: inj_on_image_set_diff [OF ‹inj_on f Ω›] assms intro: Compl sigma_sets.Compl)
next
case (Union a) then show ?case
by (metis image_UN sigma_sets.simps)
qed

lemma null_sets_translation:
assumes "N ∈ null_sets lborel" shows "{x. x - a ∈ N} ∈ null_sets lborel"
proof -
have [simp]: "(λx. x + a) ` N = {x. x - a ∈ N}"
by force
show ?thesis
using assms emeasure_lebesgue_affine [of 1 a N] by (auto simp: null_sets_def)
qed

lemma lebesgue_sets_translation:
fixes f :: "'a ⇒ 'a::euclidean_space"
assumes S: "S ∈ sets lebesgue"
shows "((λx. a + x) ` S) ∈ sets lebesgue"
proof -
have im_eq: "(+) a ` A = {x. x - a ∈ A}" for A
by force
have "((λx. a + x) ` S) = ((λx. -a + x) -` S) ∩ (space lebesgue)"
using image_iff by fastforce
also have "… ∈ sets lebesgue"
proof (rule measurable_sets [OF measurableI assms])
fix A :: "'b set"
assume A: "A ∈ sets lebesgue"
have vim_eq: "(λx. x - a) -` A = (+) a ` A" for A
by force
have "∃s n N'. (+) a ` (S ∪ N) = s ∪ n ∧ s ∈ sets borel ∧ N' ∈ null_sets lborel ∧ n ⊆ N'"
if "S ∈ sets borel" and "N' ∈ null_sets lborel" and "N ⊆ N'" for S N N'
proof (intro exI conjI)
show "(+) a ` (S ∪ N) = (λx. a + x) ` S ∪ (λx. a + x) ` N"
by auto
show "(λx. a + x) ` N' ∈ null_sets lborel"
using that by (auto simp: null_sets_translation im_eq)
qed (use that im_eq in auto)
with A have "(λx. x - a) -` A ∈ sets lebesgue"
by (force simp: vim_eq completion_def intro!: sigma_sets_image)
then show "(+) (- a) -` A ∩ space lebesgue ∈ sets lebesgue"
by (auto simp: vimage_def im_eq)
qed auto
finally show ?thesis .
qed

lemma measurable_translation:
"S ∈ lmeasurable ⟹ ((λx. a + x) ` S) ∈ lmeasurable"
unfolding fmeasurable_def
apply (auto intro: lebesgue_sets_translation)
using  emeasure_lebesgue_affine [of 1 a S]
by (auto simp: add.commute [of _ a])

lemma measure_translation:
"measure lebesgue ((λx. a + x) ` S) = measure lebesgue S"
using measure_lebesgue_affine [of 1 a S]
by (auto simp: add.commute [of _ a])

subsection ‹A nice lemma for negligibility proofs›

lemma summable_iff_suminf_neq_top: "(⋀n. f n ≥ 0) ⟹ ¬ summable f ⟹ (∑i. ennreal (f i)) = top"
by (metis summable_suminf_not_top)

proposition starlike_negligible_bounded_gmeasurable:
fixes S :: "'a :: euclidean_space set"
assumes S: "S ∈ sets lebesgue" and "bounded S"
and eq1: "⋀c x. ⟦(c *⇩R x) ∈ S; 0 ≤ c; x ∈ S⟧ ⟹ c = 1"
shows "S ∈ null_sets lebesgue"
proof -
obtain M where "0 < M" "S ⊆ ball 0 M"
using ‹bounded S› by (auto dest: bounded_subset_ballD)

let ?f = "λn. root DIM('a) (Suc n)"

have vimage_eq_image: "( *⇩R) (?f n) -` S = ( *⇩R) (1 / ?f n) ` S" for n
apply safe
subgoal for x by (rule image_eqI[of _ _ "?f n *⇩R x"]) auto
subgoal by auto
done

have eq: "(1 / ?f n) ^ DIM('a) = 1 / Suc n" for n

{ fix n x assume x: "root DIM('a) (1 + real n) *⇩R x ∈ S"
have "1 * norm x ≤ root DIM('a) (1 + real n) * norm x"
by (rule mult_mono) auto
also have "… < M"
using x ‹S ⊆ ball 0 M› by auto
finally have "norm x < M" by simp }
note less_M = this

have "(∑n. ennreal (1 / Suc n)) = top"
using not_summable_harmonic[where 'a=real] summable_Suc_iff[where f="λn. 1 / (real n)"]
by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
then have "top * emeasure lebesgue S = (∑n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
unfolding ennreal_suminf_multc eq by simp
also have "… = (∑n. emeasure lebesgue (( *⇩R) (?f n) -` S))"
unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
also have "… = emeasure lebesgue (⋃n. ( *⇩R) (?f n) -` S)"
proof (intro suminf_emeasure)
show "disjoint_family (λn. ( *⇩R) (?f n) -` S)"
unfolding disjoint_family_on_def
proof safe
fix m n :: nat and x assume "m ≠ n" "?f m *⇩R x ∈ S" "?f n *⇩R x ∈ S"
with eq1[of "?f m / ?f n" "?f n *⇩R x"] show "x ∈ {}"
by auto
qed
have "( *⇩R) (?f i) -` S ∈ sets lebesgue" for i
using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
then show "range (λi. ( *⇩R) (?f i) -` S) ⊆ sets lebesgue"
by auto
qed
also have "… ≤ emeasure lebesgue (ball 0 M :: 'a set)"
using less_M by (intro emeasure_mono) auto
also have "… < top"
using lmeasurable_ball by (auto simp: fmeasurable_def)
finally have "emeasure lebesgue S = 0"
by (simp add: ennreal_top_mult split: if_split_asm)
then show "S ∈ null_sets lebesgue"
unfolding null_sets_def using ‹S ∈ sets lebesgue› by auto
qed

corollary starlike_negligible_compact:
"compact S ⟹ (⋀c x. ⟦(c *⇩R x) ∈ S; 0 ≤ c; x ∈ S⟧ ⟹ c = 1) ⟹ S ∈ null_sets lebesgue"
using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)

proposition outer_regular_lborel_le:
assumes B[measurable]: "B ∈ sets borel" and "0 < (e::real)"
obtains U where "open U" "B ⊆ U" and "emeasure lborel (U - B) ≤ e"
proof -
let ?μ = "emeasure lborel"
let ?B = "λn::nat. ball 0 n :: 'a set"
let ?e = "λn. e*((1/2)^Suc n)"
have "∀n. ∃U. open U ∧ ?B n ∩ B ⊆ U ∧ ?μ (U - B) < ?e n"
proof
fix n :: nat
let ?A = "density lborel (indicator (?B n))"
have emeasure_A: "X ∈ sets borel ⟹ emeasure ?A X = ?μ (?B n ∩ X)" for X
by (auto simp: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])

have finite_A: "emeasure ?A (space ?A) ≠ ∞"
using emeasure_bounded_finite[of "?B n"] by (auto simp: emeasure_A)
interpret A: finite_measure ?A
by rule fact
have "emeasure ?A B + ?e n > (INF U:{U. B ⊆ U ∧ open U}. emeasure ?A U)"
using ‹0<e› by (auto simp: outer_regular[OF _ finite_A B, symmetric])
then obtain U where U: "B ⊆ U" "open U" and muU: "?μ (?B n ∩ B) + ?e n > ?μ (?B n ∩ U)"
unfolding INF_less_iff by (auto simp: emeasure_A)
moreover
{ have "?μ ((?B n ∩ U) - B) = ?μ ((?B n ∩ U) - (?B n ∩ B))"
using U by (intro arg_cong[where f="?μ"]) auto
also have "… = ?μ (?B n ∩ U) - ?μ (?B n ∩ B)"
using U A.emeasure_finite[of B]
by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
also have "… < ?e n"
using U muU A.emeasure_finite[of B]
by (subst minus_less_iff_ennreal)
(auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
finally have "?μ ((?B n ∩ U) - B) < ?e n" . }
ultimately show "∃U. open U ∧ ?B n ∩ B ⊆ U ∧ ?μ (U - B) < ?e n"
by (intro exI[of _ "?B n ∩ U"]) auto
qed
then obtain U
where U: "⋀n. open (U n)" "⋀n. ?B n ∩ B ⊆ U n" "⋀n. ?μ (U n - B) < ?e n"
by metis
show ?thesis
proof
{ fix x assume "x ∈ B"
moreover
obtain n where "norm x < real n"
using reals_Archimedean2 by blast
ultimately have "x ∈ (⋃n. U n)"
using U(2)[of n] by auto }
note * = this
then show "open (⋃n. U n)" "B ⊆ (⋃n. U n)"
using U by auto
have "?μ (⋃n. U n - B) ≤ (∑n. ?μ (U n - B))"
using U(1) by (intro emeasure_subadditive_countably) auto
also have "… ≤ (∑n. ennreal (?e n))"
using U(3) by (intro suminf_le) (auto intro: less_imp_le)
also have "… = ennreal (e * 1)"
using ‹0<e› by (intro suminf_ennreal_eq sums_mult power_half_series) auto
finally show "emeasure lborel ((⋃n. U n) - B) ≤ ennreal e"
by simp
qed
qed

lemma outer_regular_lborel:
assumes B: "B ∈ sets borel" and "0 < (e::real)"
obtains U where "open U" "B ⊆ U" "emeasure lborel (U - B) < e"
proof -
obtain U where U: "open U" "B ⊆ U" and "emeasure lborel (U-B) ≤ e/2"
using outer_regular_lborel_le [OF B, of "e/2"] ‹e > 0›
by force
moreover have "ennreal (e/2) < ennreal e"
using ‹e > 0› by (simp add: ennreal_lessI)
ultimately have "emeasure lborel (U-B) < e"
by auto
with U show ?thesis
using that by auto
qed

lemma completion_upper:
assumes A: "A ∈ sets (completion M)"
obtains A' where "A ⊆ A'" "A' ∈ sets M" "A' - A ∈ null_sets (completion M)"
"emeasure (completion M) A = emeasure M A'"
proof -
from AE_notin_null_part[OF A] obtain N where N: "N ∈ null_sets M" "null_part M A ⊆ N"
unfolding eventually_ae_filter using null_part_null_sets[OF A, THEN null_setsD2, THEN sets.sets_into_space] by auto
let ?A' = "main_part M A ∪ N"
show ?thesis
proof
show "A ⊆ ?A'"
using ‹null_part M A ⊆ N› by (subst main_part_null_part_Un[symmetric, OF A]) auto
have "main_part M A ⊆ A"
using assms main_part_null_part_Un by auto
then have "?A' - A ⊆ N"
by blast
with N show "?A' - A ∈ null_sets (completion M)"
by (blast intro: null_sets_completionI completion.complete_measure_axioms complete_measure.complete2)
show "emeasure (completion M) A = emeasure M (main_part M A ∪ N)"
using A ‹N ∈ null_sets M› by (simp add: emeasure_Un_null_set)
qed (use A N in auto)
qed

lemma lmeasurable_outer_open:
assumes S: "S ∈ sets lebesgue" and "e > 0"
obtains T where "open T" "S ⊆ T" "(T - S) ∈ lmeasurable" "measure lebesgue (T - S) < e"
proof -
obtain S' where S': "S ⊆ S'" "S' ∈ sets borel"
and null: "S' - S ∈ null_sets lebesgue"
and em: "emeasure lebesgue S = emeasure lborel S'"
using completion_upper[of S lborel] S by auto
then have f_S': "S' ∈ sets borel"
using S by (auto simp: fmeasurable_def)
with outer_regular_lborel[OF _ ‹0<e›]
obtain U where U: "open U" "S' ⊆ U" "emeasure lborel (U - S') < e"
by blast
show thesis
proof
show "open U" "S ⊆ U"
using f_S' U S' by auto
have "(U - S) = (U - S') ∪ (S' - S)"
using S' U by auto
then have eq: "emeasure lebesgue (U - S) = emeasure lborel (U - S')"
using null  by (simp add: U(1) emeasure_Un_null_set f_S' sets.Diff)
have "(U - S) ∈ sets lebesgue"
by (simp add: S U(1) sets.Diff)
then show "(U - S) ∈ lmeasurable"
unfolding fmeasurable_def using U(3) eq less_le_trans by fastforce
with eq U show "measure lebesgue (U - S) < e"
by (metis ‹U - S ∈ lmeasurable› emeasure_eq_measure2 ennreal_leI not_le)
qed
qed

lemma sets_lebesgue_inner_closed:
assumes "S ∈ sets lebesgue" "e > 0"
obtains T where "closed T" "T ⊆ S" "S-T ∈ lmeasurable" "measure lebesgue (S - T) < e"
proof -
have "-S ∈ sets lebesgue"
using assms by (simp add: Compl_in_sets_lebesgue)
then obtain T where "open T" "-S ⊆ T"
and T: "(T - -S) ∈ lmeasurable" "measure lebesgue (T - -S) < e"
using lmeasurable_outer_open assms  by blast
show thesis
proof
show "closed (-T)"
using ‹open T› by blast
show "-T ⊆ S"
using ‹- S ⊆ T› by auto
show "S - ( -T) ∈ lmeasurable" "measure lebesgue (S - (- T)) < e"
using T by (auto simp: Int_commute)
qed
qed

lemma lebesgue_openin:
"⟦openin (subtopology euclidean S) T; S ∈ sets lebesgue⟧ ⟹ T ∈ sets lebesgue"
by (metis borel_open openin_open sets.Int sets_completionI_sets sets_lborel)

lemma lebesgue_closedin:
"⟦closedin (subtopology euclidean S) T; S ∈ sets lebesgue⟧ ⟹ T ∈ sets lebesgue"
by (metis borel_closed closedin_closed sets.Int sets_completionI_sets sets_lborel)

end
```