Theory Lebesgue_Measure

```(*  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
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 ‹Measures defined by monotonous functions›

text ‹
Every right-continuous and nondecreasing function gives rise to a measure on the reals:
›

definition✐‹tag important› 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✐‹tag important› 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 (simp add: subset_eq Ball_def Bex_def)
by (metis order_le_less_trans order_less_le_trans order_less_not_sym)
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 *: "?S1 ∩ ?S2 = {}"
using j by (fastforce simp add: disjoint_iff)
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) by (simp add: * sum.union_disjoint disjoint_iff_not_equal)
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"]
then have "∃d>0. F (r i + d) - F (r i) < epsilon / 2 ^ (i + 2)"
thus "∃d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
by force
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"
using right_cont_F [of a]
by (metis continuous_at_right_real_increasing egt0 half_gt_zero less_add_same_cancel1 mono_F)
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})"
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
obtain n where Sbound: "⋀i. i ∈ S ⟹ i ≤ n"
using Max_ge finS by blast
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"
proof -
have "F b - F a' ≤ (∑i∈S. F (r i + delta i) - F (l i))"
also have "... ≤ (∑i ∈ S. F(r i) - F(l i) + epsilon / 2^(i+2))"
by (smt (verit) sum_mono deltai_prop)
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)"
using egt0 Sbound by (intro add_left_mono mult_left_mono sum_mono2) force+
also have "... ≤ ?t + (epsilon / 2)"
finally have "F b - F a' ≤ (∑i∈S. F (r i) - F (l i)) + epsilon / 2"
by simp
then show ?thesis
by linarith
qed
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" and "⋀x y. x ≤ y ⟹ F x ≤ F y" and "⋀a. continuous (at_right a) F"
shows "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def

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✐‹tag important› 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 image_def split: prod.split)
by (metis greaterThanAtMost_empty nle_le)

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 X: "⋀n. X n < a" "incseq X" "X ⇢ a"
then have "emeasure (interval_measure F) {X n<..b} ≠ ∞" for n
by (smt (verit) "*" ‹a ≤ b› ennreal_neq_top infinity_ennreal_def)
with X have "(λn. emeasure ?F {X n<..b}) ⇢ emeasure ?F (⋂n. {X n <..b})"
by (intro Lim_emeasure_decseq; force simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
also have "(⋂n. {X n <..b}) = {a..b}"
apply auto
apply (rule LIMSEQ_le_const2[OF ‹X ⇢ a›])
using less_eq_real_def apply presburger
using X(1) order_less_le_trans by blast
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)
qed (rule trivial_limit_at_left_real)

lemma✐‹tag important› 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✐‹tag important› 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✐‹tag important› lebesgue :: "'a::euclidean_space measure"
where "lebesgue ≡ completion lborel"

abbreviation✐‹tag important› lebesgue_on :: "'a set ⇒ 'a::euclidean_space measure"
where "lebesgue_on Ω ≡ restrict_space (completion lborel) Ω"

lemma lebesgue_on_mono:
assumes major: "AE x in lebesgue_on S. P x" and minor: "⋀x.⟦P x; x ∈ S⟧ ⟹ Q x"
shows "AE x in lebesgue_on S. Q x"
proof -
have "AE a in lebesgue_on S. P a ⟶ Q a"
using minor space_restrict_space by fastforce
then show ?thesis
using major by auto
qed

lemma integral_eq_zero_null_sets:
assumes "S ∈ null_sets lebesgue"
shows "integral⇧L (lebesgue_on S) f = 0"
proof (rule integral_eq_zero_AE)
show "AE x in lebesgue_on S. f x = 0"
by (metis (no_types, lifting) assms AE_not_in lebesgue_on_mono null_setsD2 null_sets_restrict_space order_refl)
qed

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 space_lebesgue_on [simp]: "space (lebesgue_on S) = S"

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_simp space_restrict_space)

lemma lebesgue_on_UNIV_eq: "lebesgue_on UNIV = lebesgue"

lemma integral_restrict_Int:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "S ∈ sets lebesgue" "T ∈ sets lebesgue"
shows "integral⇧L (lebesgue_on T) (λx. if x ∈ S then f x else 0) = integral⇧L (lebesgue_on (S ∩ T)) f"
proof -
have "(λx. indicat_real T x *⇩R (if x ∈ S then f x else 0)) = (λx. indicat_real (S ∩ T) x *⇩R f x)"
by (force simp: indicator_def)
then show ?thesis
by (simp add: assms sets.Int Bochner_Integration.integral_restrict_space)
qed

lemma integral_restrict:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "S ⊆ T" "S ∈ sets lebesgue" "T ∈ sets lebesgue"
shows "integral⇧L (lebesgue_on T) (λx. if x ∈ S then f x else 0) = integral⇧L (lebesgue_on S) f"
using integral_restrict_Int [of S T f] assms

lemma integral_restrict_UNIV:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "S ∈ sets lebesgue"
shows "integral⇧L lebesgue (λx. if x ∈ S then f x else 0) = integral⇧L (lebesgue_on S) f"
using integral_restrict_Int [of S UNIV f] assms

lemma integrable_lebesgue_on_empty [iff]:
fixes f :: "'a::euclidean_space ⇒ 'b::{second_countable_topology,banach}"
shows "integrable (lebesgue_on {}) f"

lemma integral_lebesgue_on_empty [simp]:
fixes f :: "'a::euclidean_space ⇒ 'b::{second_countable_topology,banach}"
shows "integral⇧L (lebesgue_on {}) f = 0"
lemma has_bochner_integral_restrict_space:
fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}"
assumes Ω: "Ω ∩ space M ∈ sets M"
shows "has_bochner_integral (restrict_space M Ω) f i
⟷ has_bochner_integral M (λx. indicator Ω x *⇩R f x) i"
by (simp add: integrable_restrict_space [OF assms] integral_restrict_space [OF assms] has_bochner_integral_iff)

lemma integrable_restrict_UNIV:
fixes f :: "'a::euclidean_space ⇒ 'b::{banach, second_countable_topology}"
assumes S: "S ∈ sets lebesgue"
shows  "integrable lebesgue (λx. if x ∈ S then f x else 0) ⟷ integrable (lebesgue_on S) f"
using has_bochner_integral_restrict_space [of S lebesgue f] assms

lemma integral_mono_lebesgue_on_AE:
fixes f::"_ ⇒ real"
assumes f: "integrable (lebesgue_on T) f"
and gf: "AE x in (lebesgue_on S). g x ≤ f x"
and f0: "AE x in (lebesgue_on T). 0 ≤ f x"
and "S ⊆ T" and S: "S ∈ sets lebesgue" and T: "T ∈ sets lebesgue"
shows "(∫x. g x ∂(lebesgue_on S)) ≤ (∫x. f x ∂(lebesgue_on T))"
proof -
have "(∫x. g x ∂(lebesgue_on S)) = (∫x. (if x ∈ S then g x else 0) ∂lebesgue)"
also have "… ≤ (∫x. (if x ∈ T then f x else 0) ∂lebesgue)"
proof (rule Bochner_Integration.integral_mono_AE')
show "integrable lebesgue (λx. if x ∈ T then f x else 0)"
by (simp add: integrable_restrict_UNIV T f)
show "AE x in lebesgue. (if x ∈ S then g x else 0) ≤ (if x ∈ T then f x else 0)"
using assms by (auto simp: AE_restrict_space_iff)
show "AE x in lebesgue. 0 ≤ (if x ∈ T then f x else 0)"
using f0 by (simp add: AE_restrict_space_iff T)
qed
also have "… = (∫x. f x ∂(lebesgue_on T))"
using Lebesgue_Measure.integral_restrict_UNIV T by blast
finally show ?thesis .
qed

subsection ‹Borel measurability›

lemma borel_measurable_if_I:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "f ∈ borel_measurable (lebesgue_on S)" and S: "S ∈ sets lebesgue"
shows "(λx. if x ∈ S then f x else 0) ∈ borel_measurable lebesgue"
proof -
have eq: "{x. x ∉ S} ∪ {x. f x ∈ Y} = {x. x ∉ S} ∪ {x. f x ∈ Y} ∩ S" for Y
by blast
show ?thesis
using f S
apply (simp add: vimage_def in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (simp only: Collect_conj_eq Collect_disj_eq imp_conv_disj eq)
apply (auto simp: Compl_eq [symmetric] Compl_in_sets_lebesgue sets_restrict_space_iff)
done
qed

lemma borel_measurable_if_D:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "(λx. if x ∈ S then f x else 0) ∈ borel_measurable lebesgue"
shows "f ∈ borel_measurable (lebesgue_on S)"
using assms by (smt (verit) measurable_lebesgue_cong measurable_restrict_space1)

lemma borel_measurable_if:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "S ∈ sets lebesgue"
shows "(λx. if x ∈ S then f x else 0) ∈ borel_measurable lebesgue ⟷ f ∈ borel_measurable (lebesgue_on S)"
using assms borel_measurable_if_D borel_measurable_if_I by blast

lemma borel_measurable_if_lebesgue_on:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "S ∈ sets lebesgue" "T ∈ sets lebesgue" "S ⊆ T"
shows "(λx. if x ∈ S then f x else 0) ∈ borel_measurable (lebesgue_on T) ⟷ f ∈ borel_measurable (lebesgue_on S)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using measurable_restrict_mono [OF _ ‹S ⊆ T›]
by (subst measurable_lebesgue_cong [where g = "(λx. if x ∈ S then f x else 0)"]) auto
next
assume ?rhs then show ?lhs
by (simp add: ‹S ∈ sets lebesgue› borel_measurable_if_I measurable_restrict_space1)
qed

lemma borel_measurable_vimage_halfspace_component_lt:
"f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a i. i ∈ Basis ⟶ {x ∈ S. f x ∙ i < a} ∈ sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_less])

lemma borel_measurable_vimage_halfspace_component_ge:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a i. i ∈ Basis ⟶ {x ∈ S. f x ∙ i ≥ a} ∈ sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_ge])

lemma borel_measurable_vimage_halfspace_component_gt:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a i. i ∈ Basis ⟶ {x ∈ S. f x ∙ i > a} ∈ sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_greater])

lemma borel_measurable_vimage_halfspace_component_le:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a i. i ∈ Basis ⟶ {x ∈ S. f x ∙ i ≤ a} ∈ sets (lebesgue_on S))"
by (force simp add: space_restrict_space trans [OF borel_measurable_iff_halfspace_le])

lemma
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows borel_measurable_vimage_open_interval:
"f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a b. {x ∈ S. f x ∈ box a b} ∈ sets (lebesgue_on S))" (is ?thesis1)
and borel_measurable_vimage_open:
"f ∈ borel_measurable (lebesgue_on S) ⟷
(∀T. open T ⟶ {x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S))" (is ?thesis2)
proof -
have "{x ∈ S. f x ∈ box a b} ∈ sets (lebesgue_on S)" if "f ∈ borel_measurable (lebesgue_on S)" for a b
proof -
have "S = S ∩ space lebesgue"
by simp
then have "S ∩ (f -` box a b) ∈ sets (lebesgue_on S)"
by (metis (no_types) box_borel in_borel_measurable_borel inf_sup_aci(1) space_restrict_space that)
then show ?thesis
qed
moreover
have "{x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S)"
if T: "⋀a b. {x ∈ S. f x ∈ box a b} ∈ sets (lebesgue_on S)" "open T" for T
proof -
obtain 𝒟 where "countable 𝒟" and 𝒟: "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = box a b" "⋃𝒟 = T"
using open_countable_Union_open_box that ‹open T› by metis
then have eq: "{x ∈ S. f x ∈ T} = (⋃U ∈ 𝒟. {x ∈ S. f x ∈ U})"
by blast
have "{x ∈ S. f x ∈ U} ∈ sets (lebesgue_on S)" if "U ∈ 𝒟" for U
using that T 𝒟 by blast
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF ‹countable 𝒟›])
qed
moreover
have eq: "{x ∈ S. f x ∙ i < a} = {x ∈ S. f x ∈ {y. y ∙ i < a}}" for i a
by auto
have "f ∈ borel_measurable (lebesgue_on S)"
if "⋀T. open T ⟹ {x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S)"
by (metis (no_types) eq borel_measurable_vimage_halfspace_component_lt open_halfspace_component_lt that)
ultimately show "?thesis1" "?thesis2"
by blast+
qed

lemma borel_measurable_vimage_closed:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀T. closed T ⟶ {x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S))"
proof -
have eq: "{x ∈ S. f x ∈ T} = S - (S ∩ f -` (- T))" for T
by auto
show ?thesis
unfolding borel_measurable_vimage_open eq
by (metis Diff_Diff_Int closed_Compl diff_eq open_Compl sets.Diff sets_lebesgue_on_refl vimage_Compl)
qed

lemma borel_measurable_vimage_closed_interval:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀a b. {x ∈ S. f x ∈ cbox a b} ∈ sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using borel_measurable_vimage_closed by blast
next
assume RHS: ?rhs
have "{x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S)" if "open T" for T
proof -
obtain 𝒟 where "countable 𝒟" and 𝒟: "𝒟 ⊆ Pow T" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = cbox a b" "⋃𝒟 = T"
using open_countable_Union_open_cbox that ‹open T› by metis
then have eq: "{x ∈ S. f x ∈ T} = (⋃U ∈ 𝒟. {x ∈ S. f x ∈ U})"
by blast
have "{x ∈ S. f x ∈ U} ∈ sets (lebesgue_on S)" if "U ∈ 𝒟" for U
using that 𝒟 by (metis RHS)
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF ‹countable 𝒟›])
qed
then show ?lhs
qed

lemma borel_measurable_vimage_borel:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable (lebesgue_on S) ⟷
(∀T. T ∈ sets borel ⟶ {x ∈ S. f x ∈ T} ∈ sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume f: ?lhs
then show ?rhs
using measurable_sets [OF f]
by (simp add: Collect_conj_eq inf_sup_aci(1) space_restrict_space vimage_def)

lemma lebesgue_measurable_vimage_borel:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "f ∈ borel_measurable lebesgue" "T ∈ sets borel"
shows "{x. f x ∈ T} ∈ sets lebesgue"
using assms borel_measurable_vimage_borel [of f UNIV] by auto

lemma borel_measurable_lebesgue_preimage_borel:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f ∈ borel_measurable lebesgue ⟷
(∀T. T ∈ sets borel ⟶ {x. f x ∈ T} ∈ sets lebesgue)"
by (smt (verit, best) Collect_cong UNIV_I borel_measurable_vimage_borel lebesgue_on_UNIV_eq)

subsection ✐‹tag unimportant› ‹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)"
by (metis borel_measurable_continuous_on_restrict borel_measurable_subalgebra f
lebesgue_on_UNIV_eq mono_restrict_space sets_completionI_sets sets_lborel space_borel
space_lebesgue_on space_restrict_space subsetI)

lemma id_borel_measurable_lebesgue [iff]: "id ∈ borel_measurable lebesgue"

lemma id_borel_measurable_lebesgue_on [iff]: "id ∈ borel_measurable (lebesgue_on S)"

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"

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

lemma✐‹tag important› 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)"

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)

lemma emeasure_lborel_cbox_finite: "emeasure lborel (cbox a b) < ∞"
by (auto simp: emeasure_lborel_cbox_eq)

lemma emeasure_lborel_box_finite: "emeasure lborel (box a b) < ∞"
by (auto simp: emeasure_lborel_box_eq)

lemma emeasure_lborel_ball_finite: "emeasure lborel (ball c r) < ∞"
by (metis bounded_ball bounded_subset_cbox_symmetric cbox_borel emeasure_lborel_cbox_finite
emeasure_mono order_le_less_trans sets_lborel)

lemma emeasure_lborel_cball_finite: "emeasure lborel (cball c r) < ∞"
by (metis bounded_cball bounded_subset_cbox_symmetric cbox_borel emeasure_lborel_cbox_finite
emeasure_mono order_le_less_trans sets_lborel)

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)
apply (simp)
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
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
by simp
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 insert_null_sets_iff [simp]: "insert a N ∈ null_sets lebesgue ⟷ N ∈ null_sets lebesgue"
by (meson completion.complete2 finite.simps finite_imp_null_set_lborel null_sets.insert_in_sets
null_sets_completionI subset_insertI)

lemma insert_null_sets_lebesgue_on_iff [simp]:
assumes "a ∈ S" "S ∈ sets lebesgue"
shows "insert a N ∈ null_sets (lebesgue_on S) ⟷ N ∈ null_sets (lebesgue_on S)"

lemma lborel_neq_count_space[simp]:
fixes A :: "('a::ordered_euclidean_space) set"
shows "lborel ≠ count_space A"
by (metis finite.simps finite_imp_null_set_lborel insert_not_empty null_sets_count_space singleton_iff)

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✐‹tag important› 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
show "emeasure lborel (?A i) ≠ ∞" for i by auto
show "emeasure lborel X = emeasure M X" if "X ∈ ?E" for X
using that box_eq_empty(1) by (fastforce simp: emeasure_eq emeasure_lborel_box_eq)
qed

lemma✐‹tag important› 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_split_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✐‹tag important› 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 (simp add: completion.measurable_completion2 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

corollary lebesgue_real_affine:
"c ≠ 0 ⟹ lebesgue = density (distr lebesgue lebesgue (λx. t + c * x)) (λ_. ennreal (abs c))"
using lebesgue_affine_euclidean [where c= "λx::real. c"] by simp

lemma nn_integral_real_affine_lebesgue:
fixes c :: real assumes f[measurable]: "f ∈ borel_measurable lebesgue" and c: "c ≠ 0"
shows "(∫⇧+x. f x ∂lebesgue) = ennreal¦c¦ * (∫⇧+x. f(t + c * x) ∂lebesgue)"
proof -
have "(∫⇧+x. f x ∂lebesgue) = (∫⇧+x. f x ∂density (distr lebesgue lebesgue (λx. t + c * x)) (λx. ennreal ¦c¦))"
using lebesgue_real_affine c by auto
also have "… = ∫⇧+ x. ennreal ¦c¦ * f x ∂distr lebesgue lebesgue (λx. t + c * x)"
by (subst nn_integral_density) auto
also have "… = ennreal ¦c¦ * integral⇧N (distr lebesgue lebesgue (λx. t + c * x)) f"
using f measurable_distr_eq1 nn_integral_cmult by blast
also have "… = ¦c¦ * (∫⇧+x. f(t + c * x) ∂lebesgue)"
using lebesgue_affine_measurable[where c= "λx::real. c"]
by (subst nn_integral_distr) (force+)
finally show ?thesis .
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)
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:
fixes c :: "'a::euclidean_space"
shows "distr lborel borel ((+) c) = lborel"
by (subst lborel_affine[of 1 c], auto simp: density_1)

interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)

interpretation lborel_pair: pair_sigma_finite lborel lborel ..

lemma✐‹tag important› 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```