(* Title: HOL/Analysis/Change_Of_Vars.thy Authors: LC Paulson, based on material from HOL Light *) section‹Change of Variables Theorems› theory Change_Of_Vars imports Vitali_Covering_Theorem Determinants begin subsection ‹Measurable Shear and Stretch› proposition fixes a :: "real^'n" assumes "m ≠ n" and ab_ne: "cbox a b ≠ {}" and an: "0 ≤ a$n" shows measurable_shear_interval: "(λx. χ i. if i = m then x$m + x$n else x$i) ` (cbox a b) ∈ lmeasurable" (is "?f ` _ ∈ _") and measure_shear_interval: "measure lebesgue ((λx. χ i. if i = m then x$m + x$n else x$i) ` cbox a b) = measure lebesgue (cbox a b)" (is "?Q") proof - have lin: "linear ?f" by (rule linearI) (auto simp: plus_vec_def scaleR_vec_def algebra_simps) show fab: "?f ` cbox a b ∈ lmeasurable" by (simp add: lin measurable_linear_image_interval) let ?c = "χ i. if i = m then b$m + b$n else b$i" let ?mn = "axis m 1 - axis n (1::real)" have eq1: "measure lebesgue (cbox a ?c) = measure lebesgue (?f ` cbox a b) + measure lebesgue (cbox a ?c ∩ {x. ?mn ∙ x ≤ a$m}) + measure lebesgue (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m})" proof (rule measure_Un3_negligible) show "cbox a ?c ∩ {x. ?mn ∙ x ≤ a$m} ∈ lmeasurable" "cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m} ∈ lmeasurable" by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex) have "negligible {x. ?mn ∙ x = a$m}" by (metis ‹m ≠ n› axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane) moreover have "?f ` cbox a b ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m}) ⊆ {x. ?mn ∙ x = a$m}" using ‹m ≠ n› antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis') ultimately show "negligible ((?f ` cbox a b) ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m}))" by (rule negligible_subset) have "negligible {x. ?mn ∙ x = b$m}" by (metis ‹m ≠ n› axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane) moreover have "(?f ` cbox a b) ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m}) ⊆ {x. ?mn ∙ x = b$m}" using ‹m ≠ n› antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis') ultimately show "negligible (?f ` cbox a b ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m}))" by (rule negligible_subset) have "negligible {x. ?mn ∙ x = b$m}" by (metis ‹m ≠ n› axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane) moreover have "(cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m})) ⊆ {x. ?mn ∙ x = b$m}" using ‹m ≠ n› ab_ne apply (clarsimp simp: algebra_simps mem_box_cart inner_axis') by (smt (verit, ccfv_SIG) interval_ne_empty_cart(1)) ultimately show "negligible (cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∩ (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m}))" by (rule negligible_subset) show "?f ` cbox a b ∪ cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∪ cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m} = cbox a ?c" (is "?lhs = _") proof show "?lhs ⊆ cbox a ?c" by (auto simp: mem_box_cart add_mono) (meson add_increasing2 an order_trans) show "cbox a ?c ⊆ ?lhs" apply (clarsimp simp: algebra_simps image_iff inner_axis' lambda_add_Galois [OF ‹m ≠ n›]) by (smt (verit, del_insts) mem_box_cart(2) vec_lambda_beta) qed qed (fact fab) let ?d = "χ i. if i = m then a $ m - b $ m else 0" have eq2: "measure lebesgue (cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m}) + measure lebesgue (cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m}) = measure lebesgue (cbox a (χ i. if i = m then a $ m + b $ n else b $ i))" proof (rule measure_translate_add[of "cbox a ?c ∩ {x. ?mn ∙ x ≤ a$m}" "cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m}" "(χ i. if i = m then a$m - b$m else 0)" "cbox a (χ i. if i = m then a$m + b$n else b$i)"]) show "(cbox a ?c ∩ {x. ?mn ∙ x ≤ a$m}) ∈ lmeasurable" "cbox a ?c ∩ {x. ?mn ∙ x ≥ b$m} ∈ lmeasurable" by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex) have "⋀x. ⟦x $ n + a $ m ≤ x $ m⟧ ⟹ x ∈ (+) (χ i. if i = m then a $ m - b $ m else 0) ` {x. x $ n + b $ m ≤ x $ m}" using ‹m ≠ n› by (rule_tac x="x - (χ i. if i = m then a$m - b$m else 0)" in image_eqI) (simp_all add: mem_box_cart) then have imeq: "(+) ?d ` {x. b $ m ≤ ?mn ∙ x} = {x. a $ m ≤ ?mn ∙ x}" using ‹m ≠ n› by (auto simp: mem_box_cart inner_axis' algebra_simps) have "⋀x. ⟦0 ≤ a $ n; x $ n + a $ m ≤ x $ m; ∀i. i ≠ m ⟶ a $ i ≤ x $ i ∧ x $ i ≤ b $ i⟧ ⟹ a $ m ≤ x $ m" using ‹m ≠ n› by force then have "(+) ?d ` (cbox a ?c ∩ {x. b $ m ≤ ?mn ∙ x}) = cbox a (χ i. if i = m then a $ m + b $ n else b $ i) ∩ {x. a $ m ≤ ?mn ∙ x}" using an ab_ne apply (simp add: cbox_translation [symmetric] translation_Int interval_ne_empty_cart imeq) apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib) by (metis (full_types) add_mono mult_2_right) then show "cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∪ (+) ?d ` (cbox a ?c ∩ {x. b $ m ≤ ?mn ∙ x}) = cbox a (χ i. if i = m then a $ m + b $ n else b $ i)" (is "?lhs = ?rhs") using an ‹m ≠ n› apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib, force) apply (drule_tac x=n in spec)+ by (meson ab_ne add_mono_thms_linordered_semiring(3) dual_order.trans interval_ne_empty_cart(1)) have "negligible{x. ?mn ∙ x = a$m}" by (metis ‹m ≠ n› axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane) moreover have "(cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∩ (+) ?d ` (cbox a ?c ∩ {x. b $ m ≤ ?mn ∙ x})) ⊆ {x. ?mn ∙ x = a$m}" using ‹m ≠ n› antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis') ultimately show "negligible (cbox a ?c ∩ {x. ?mn ∙ x ≤ a $ m} ∩ (+) ?d ` (cbox a ?c ∩ {x. b $ m ≤ ?mn ∙ x}))" by (rule negligible_subset) qed have ac_ne: "cbox a ?c ≠ {}" by (smt (verit, del_insts) ab_ne an interval_ne_empty_cart(1) vec_lambda_beta) have ax_ne: "cbox a (χ i. if i = m then a $ m + b $ n else b $ i) ≠ {}" using ab_ne an by (smt (verit, ccfv_threshold) interval_ne_empty_cart(1) vec_lambda_beta) have eq3: "measure lebesgue (cbox a ?c) = measure lebesgue (cbox a (χ i. if i = m then a$m + b$n else b$i)) + measure lebesgue (cbox a b)" by (simp add: content_cbox_if_cart ab_ne ac_ne ax_ne algebra_simps prod.delta_remove if_distrib [of "λu. u - z" for z] prod.remove) show ?Q using eq1 eq2 eq3 by (simp add: algebra_simps) qed proposition fixes S :: "(real^'n) set" assumes "S ∈ lmeasurable" shows measurable_stretch: "((λx. χ k. m k * x$k) ` S) ∈ lmeasurable" (is "?f ` S ∈ _") and measure_stretch: "measure lebesgue ((λx. χ k. m k * x$k) ` S) = ¦prod m UNIV¦ * measure lebesgue S" (is "?MEQ") proof - have "(?f ` S) ∈ lmeasurable ∧ ?MEQ" proof (cases "∀k. m k ≠ 0") case True have m0: "0 < ¦prod m UNIV¦" using True by simp have "(indicat_real (?f ` S) has_integral ¦prod m UNIV¦ * measure lebesgue S) UNIV" proof (clarsimp simp add: has_integral_alt [where i=UNIV]) fix e :: "real" assume "e > 0" have "(indicat_real S has_integral (measure lebesgue S)) UNIV" using assms lmeasurable_iff_has_integral by blast then obtain B where "B>0" and B: "⋀a b. ball 0 B ⊆ cbox a b ⟹ ∃z. (indicat_real S has_integral z) (cbox a b) ∧ ¦z - measure lebesgue S¦ < e / ¦prod m UNIV¦" by (simp add: has_integral_alt [where i=UNIV]) (metis (full_types) divide_pos_pos m0 m0 ‹e > 0›) show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶ (∃z. (indicat_real (?f ` S) has_integral z) (cbox a b) ∧ ¦z - ¦prod m UNIV¦ * measure lebesgue S¦ < e)" proof (intro exI conjI allI) let ?C = "Max (range (λk. ¦m k¦)) * B" show "?C > 0" using True ‹B > 0› by (simp add: Max_gr_iff) show "ball 0 ?C ⊆ cbox u v ⟶ (∃z. (indicat_real (?f ` S) has_integral z) (cbox u v) ∧ ¦z - ¦prod m UNIV¦ * measure lebesgue S¦ < e)" for u v proof assume uv: "ball 0 ?C ⊆ cbox u v" with ‹?C > 0› have cbox_ne: "cbox u v ≠ {}" using centre_in_ball by blast let ?α = "λk. u$k / m k" let ?β = "λk. v$k / m k" have invm0: "⋀k. inverse (m k) ≠ 0" using True by auto have "ball 0 B ⊆ (λx. χ k. x $ k / m k) ` ball 0 ?C" proof clarsimp fix x :: "real^'n" assume x: "norm x < B" have [simp]: "¦Max (range (λk. ¦m k¦))¦ = Max (range (λk. ¦m k¦))" by (meson Max_ge abs_ge_zero abs_of_nonneg finite finite_imageI order_trans rangeI) have "norm (χ k. m k * x $ k) ≤ norm (Max (range (λk. ¦m k¦)) *⇩_{R}x)" by (rule norm_le_componentwise_cart) (auto simp: abs_mult intro: mult_right_mono) also have "… < ?C" using x ‹0 < (MAX k. ¦m k¦) * B› ‹0 < B› zero_less_mult_pos2 by fastforce finally have "norm (χ k. m k * x $ k) < ?C" . then show "x ∈ (λx. χ k. x $ k / m k) ` ball 0 ?C" using stretch_Galois [of "inverse ∘ m"] True by (auto simp: image_iff field_simps) qed then have Bsub: "ball 0 B ⊆ cbox (χ k. min (?α k) (?β k)) (χ k. max (?α k) (?β k))" using cbox_ne uv image_stretch_interval_cart [of "inverse ∘ m" u v, symmetric] by (force simp: field_simps) obtain z where zint: "(indicat_real S has_integral z) (cbox (χ k. min (?α k) (?β k)) (χ k. max (?α k) (?β k)))" and zless: "¦z - measure lebesgue S¦ < e / ¦prod m UNIV¦" using B [OF Bsub] by blast have ind: "indicat_real (?f ` S) = (λx. indicator S (χ k. x$k / m k))" using True stretch_Galois [of m] by (force simp: indicator_def) show "∃z. (indicat_real (?f ` S) has_integral z) (cbox u v) ∧ ¦z - ¦prod m UNIV¦ * measure lebesgue S¦ < e" proof (simp add: ind, intro conjI exI) have "((λx. indicat_real S (χ k. x $ k/ m k)) has_integral z *⇩_{R}¦prod m UNIV¦) ((λx. χ k. x $ k * m k) ` cbox (χ k. min (?α k) (?β k)) (χ k. max (?α k) (?β k)))" using True has_integral_stretch_cart [OF zint, of "inverse ∘ m"] by (simp add: field_simps prod_dividef) moreover have "((λx. χ k. x $ k * m k) ` cbox (χ k. min (?α k) (?β k)) (χ k. max (?α k) (?β k))) = cbox u v" using True image_stretch_interval_cart [of "inverse ∘ m" u v, symmetric] image_stretch_interval_cart [of "λk. 1" u v, symmetric] ‹cbox u v ≠ {}› by (simp add: field_simps image_comp o_def) ultimately show "((λx. indicat_real S (χ k. x $ k/ m k)) has_integral z *⇩_{R}¦prod m UNIV¦) (cbox u v)" by simp have "¦z *⇩_{R}¦prod m UNIV¦ - ¦prod m UNIV¦ * measure lebesgue S¦ = ¦prod m UNIV¦ * ¦z - measure lebesgue S¦" by (metis (no_types, opaque_lifting) abs_abs abs_scaleR mult.commute real_scaleR_def right_diff_distrib') also have "… < e" using zless True by (simp add: field_simps) finally show "¦z *⇩_{R}¦prod m UNIV¦ - ¦prod m UNIV¦ * measure lebesgue S¦ < e" . qed qed qed qed then show ?thesis by (auto simp: has_integral_integrable integral_unique lmeasure_integral_UNIV measurable_integrable) next case False then obtain k where "m k = 0" and prm: "prod m UNIV = 0" by auto have nfS: "negligible (?f ` S)" by (rule negligible_subset [OF negligible_standard_hyperplane_cart]) (use ‹m k = 0› in auto) then show ?thesis by (simp add: negligible_iff_measure prm) qed then show "(?f ` S) ∈ lmeasurable" ?MEQ by metis+ qed proposition fixes f :: "real^'n::{finite,wellorder} ⇒ real^'n::_" assumes "linear f" "S ∈ lmeasurable" shows measurable_linear_image: "(f ` S) ∈ lmeasurable" and measure_linear_image: "measure lebesgue (f ` S) = ¦det (matrix f)¦ * measure lebesgue S" (is "?Q f S") proof - have "∀S ∈ lmeasurable. (f ` S) ∈ lmeasurable ∧ ?Q f S" proof (rule induct_linear_elementary [OF ‹linear f›]; intro ballI) fix f g and S :: "(real,'n) vec set" assume "linear f" and "linear g" and f [rule_format]: "∀S ∈ lmeasurable. f ` S ∈ lmeasurable ∧ ?Q f S" and g [rule_format]: "∀S ∈ lmeasurable. g ` S ∈ lmeasurable ∧ ?Q g S" and S: "S ∈ lmeasurable" then have gS: "g ` S ∈ lmeasurable" by blast show "(f ∘ g) ` S ∈ lmeasurable ∧ ?Q (f ∘ g) S" using f [OF gS] g [OF S] matrix_compose [OF ‹linear g› ‹linear f›] by (simp add: o_def image_comp abs_mult det_mul) next fix f :: "real^'n::_ ⇒ real^'n::_" and i and S :: "(real^'n::_) set" assume "linear f" and 0: "⋀x. f x $ i = 0" and "S ∈ lmeasurable" then have "¬ inj f" by (metis (full_types) linear_injective_imp_surjective one_neq_zero surjE vec_component) have detf: "det (matrix f) = 0" using ‹¬ inj f› det_nz_iff_inj[OF ‹linear f›] by blast show "f ` S ∈ lmeasurable ∧ ?Q f S" proof show "f ` S ∈ lmeasurable" using lmeasurable_iff_indicator_has_integral ‹linear f› ‹¬ inj f› negligible_UNIV negligible_linear_singular_image by blast have "measure lebesgue (f ` S) = 0" by (meson ‹¬ inj f› ‹linear f› negligible_imp_measure0 negligible_linear_singular_image) also have "… = ¦det (matrix f)¦ * measure lebesgue S" by (simp add: detf) finally show "?Q f S" . qed next fix c and S :: "(real^'n::_) set" assume "S ∈ lmeasurable" show "(λa. χ i. c i * a $ i) ` S ∈ lmeasurable ∧ ?Q (λa. χ i. c i * a $ i) S" proof show "(λa. χ i. c i * a $ i) ` S ∈ lmeasurable" by (simp add: ‹S ∈ lmeasurable› measurable_stretch) show "?Q (λa. χ i. c i * a $ i) S" by (simp add: measure_stretch [OF ‹S ∈ lmeasurable›, of c] axis_def matrix_def det_diagonal) qed next fix m :: "'n" and n :: "'n" and S :: "(real, 'n) vec set" assume "m ≠ n" and "S ∈ lmeasurable" let ?h = "λv::(real, 'n) vec. χ i. v $ Transposition.transpose m n i" have lin: "linear ?h" by (rule linearI) (simp_all add: plus_vec_def scaleR_vec_def) have meq: "measure lebesgue ((λv::(real, 'n) vec. χ i. v $ Transposition.transpose m n i) ` cbox a b) = measure lebesgue (cbox a b)" for a b proof (cases "cbox a b = {}") case True then show ?thesis by simp next case False then have him: "?h ` (cbox a b) ≠ {}" by blast have eq: "?h ` (cbox a b) = cbox (?h a) (?h b)" by (auto simp: image_iff lambda_swap_Galois mem_box_cart) (metis transpose_involutory)+ show ?thesis using him prod.permute [OF permutes_swap_id, where S=UNIV and g="λi. (b - a)$i", symmetric] by (simp add: eq content_cbox_cart False) qed have "(χ i j. if Transposition.transpose m n i = j then 1 else 0) = (χ i j. if j = Transposition.transpose m n i then 1 else (0::real))" by (auto intro!: Cart_lambda_cong) then have "matrix ?h = transpose(χ i j. mat 1 $ i $ Transposition.transpose m n j)" by (auto simp: matrix_eq transpose_def axis_def mat_def matrix_def) then have 1: "¦det (matrix ?h)¦ = 1" by (simp add: det_permute_columns permutes_swap_id sign_swap_id abs_mult) show "?h ` S ∈ lmeasurable ∧ ?Q ?h S" using measure_linear_sufficient [OF lin ‹S ∈ lmeasurable›] meq 1 by force next fix m n :: "'n" and S :: "(real, 'n) vec set" assume "m ≠ n" and "S ∈ lmeasurable" let ?h = "λv::(real, 'n) vec. χ i. if i = m then v $ m + v $ n else v $ i" have lin: "linear ?h" by (rule linearI) (auto simp: algebra_simps plus_vec_def scaleR_vec_def vec_eq_iff) consider "m < n" | " n < m" using ‹m ≠ n› less_linear by blast then have 1: "det(matrix ?h) = 1" proof cases assume "m < n" have *: "matrix ?h $ i $ j = (0::real)" if "j < i" for i j :: 'n proof - have "axis j 1 = (χ n. if n = j then 1 else (0::real))" using axis_def by blast then have "(χ p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)" using ‹j < i› axis_def ‹m < n› by auto with ‹m < n› show ?thesis by (auto simp: matrix_def axis_def cong: if_cong) qed show ?thesis using ‹m ≠ n› by (subst det_upperdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong) next assume "n < m" have *: "matrix ?h $ i $ j = (0::real)" if "j > i" for i j :: 'n proof - have "axis j 1 = (χ n. if n = j then 1 else (0::real))" using axis_def by blast then have "(χ p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)" using ‹j > i› axis_def ‹m > n› by auto with ‹m > n› show ?thesis by (auto simp: matrix_def axis_def cong: if_cong) qed show ?thesis using ‹m ≠ n› by (subst det_lowerdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong) qed have meq: "measure lebesgue (?h ` (cbox a b)) = measure lebesgue (cbox a b)" for a b proof (cases "cbox a b = {}") case True then show ?thesis by simp next case False then have ne: "(+) (χ i. if i = n then - a $ n else 0) ` cbox a b ≠ {}" by auto let ?v = "χ i. if i = n then - a $ n else 0" have "?h ` cbox a b = (+) (χ i. if i = m ∨ i = n then a $ n else 0) ` ?h ` (+) ?v ` (cbox a b)" using ‹m ≠ n› unfolding image_comp o_def by (force simp: vec_eq_iff) then have "measure lebesgue (?h ` (cbox a b)) = measure lebesgue ((λv. χ i. if i = m then v $ m + v $ n else v $ i) ` (+) ?v ` cbox a b)" by (rule ssubst) (rule measure_translation) also have "… = measure lebesgue ((λv. χ i. if i = m then v $ m + v $ n else v $ i) ` cbox (?v +a) (?v + b))" by (metis (no_types, lifting) cbox_translation) also have "… = measure lebesgue ((+) ?v ` cbox a b)" apply (subst measure_shear_interval) using ‹m ≠ n› ne apply auto apply (simp add: cbox_translation) by (metis cbox_borel cbox_translation measure_completion sets_lborel) also have "… = measure lebesgue (cbox a b)" by (rule measure_translation) finally show ?thesis . qed show "?h ` S ∈ lmeasurable ∧ ?Q ?h S" using measure_linear_sufficient [OF lin ‹S ∈ lmeasurable›] meq 1 by force qed with assms show "(f ` S) ∈ lmeasurable" "?Q f S" by metis+ qed lemma fixes f :: "real^'n::{finite,wellorder} ⇒ real^'n::_" assumes f: "orthogonal_transformation f" and S: "S ∈ lmeasurable" shows measurable_orthogonal_image: "f ` S ∈ lmeasurable" and measure_orthogonal_image: "measure lebesgue (f ` S) = measure lebesgue S" proof - have "linear f" by (simp add: f orthogonal_transformation_linear) then show "f ` S ∈ lmeasurable" by (metis S measurable_linear_image) show "measure lebesgue (f ` S) = measure lebesgue S" by (simp add: measure_linear_image ‹linear f› S f) qed proposition measure_semicontinuous_with_hausdist_explicit: assumes "bounded S" and neg: "negligible(frontier S)" and "e > 0" obtains d where "d > 0" "⋀T. ⟦T ∈ lmeasurable; ⋀y. y ∈ T ⟹ ∃x. x ∈ S ∧ dist x y < d⟧ ⟹ measure lebesgue T < measure lebesgue S + e" proof (cases "S = {}") case True with that ‹e > 0› show ?thesis by force next case False then have frS: "frontier S ≠ {}" using ‹bounded S› frontier_eq_empty not_bounded_UNIV by blast have "S ∈ lmeasurable" by (simp add: ‹bounded S› measurable_Jordan neg) have null: "(frontier S) ∈ null_sets lebesgue" by (metis neg negligible_iff_null_sets) have "frontier S ∈ lmeasurable" and mS0: "measure lebesgue (frontier S) = 0" using neg negligible_imp_measurable negligible_iff_measure by blast+ with ‹e > 0› sets_lebesgue_outer_open obtain U where "open U" and U: "frontier S ⊆ U" "U - frontier S ∈ lmeasurable" "emeasure lebesgue (U - frontier S) < e" by (metis fmeasurableD) with null have "U ∈ lmeasurable" by (metis borel_open measurable_Diff_null_set sets_completionI_sets sets_lborel) have "measure lebesgue (U - frontier S) = measure lebesgue U" using mS0 by (simp add: ‹U ∈ lmeasurable› fmeasurableD measure_Diff_null_set null) with U have mU: "measure lebesgue U < e" by (simp add: emeasure_eq_measure2 ennreal_less_iff) show ?thesis proof have "U ≠ UNIV" using ‹U ∈ lmeasurable› by auto then have "- U ≠ {}" by blast with ‹open U› ‹frontier S ⊆ U› show "setdist (frontier S) (- U) > 0" by (auto simp: ‹bounded S› open_closed compact_frontier_bounded setdist_gt_0_compact_closed frS) fix T assume "T ∈ lmeasurable" and T: "⋀t. t ∈ T ⟹ ∃y. y ∈ S ∧ dist y t < setdist (frontier S) (- U)" then have "measure lebesgue T - measure lebesgue S ≤ measure lebesgue (T - S)" by (simp add: ‹S ∈ lmeasurable› measure_diff_le_measure_setdiff) also have "… ≤ measure lebesgue U" proof - have "T - S ⊆ U" proof clarify fix x assume "x ∈ T" and "x ∉ S" then obtain y where "y ∈ S" and y: "dist y x < setdist (frontier S) (- U)" using T by blast have "closed_segment x y ∩ frontier S ≠ {}" using connected_Int_frontier ‹x ∉ S› ‹y ∈ S› by blast then obtain z where z: "z ∈ closed_segment x y" "z ∈ frontier S" by auto with y have "dist z x < setdist(frontier S) (- U)" by (auto simp: dist_commute dest!: dist_in_closed_segment) with z have False if "x ∈ -U" using setdist_le_dist [OF ‹z ∈ frontier S› that] by auto then show "x ∈ U" by blast qed then show ?thesis by (simp add: ‹S ∈ lmeasurable› ‹T ∈ lmeasurable› ‹U ∈ lmeasurable› fmeasurableD measure_mono_fmeasurable sets.Diff) qed finally have "measure lebesgue T - measure lebesgue S ≤ measure lebesgue U" . with mU show "measure lebesgue T < measure lebesgue S + e" by linarith qed qed proposition fixes f :: "real^'n::{finite,wellorder} ⇒ real^'n::_" assumes S: "S ∈ lmeasurable" and deriv: "⋀x. x ∈ S ⟹ (f has_derivative f' x) (at x within S)" and int: "(λx. ¦det (matrix (f' x))¦) integrable_on S" and bounded: "⋀x. x ∈ S ⟹ ¦det (matrix (f' x))¦ ≤ B" shows measurable_bounded_differentiable_image: "f ` S ∈ lmeasurable" and measure_bounded_differentiable_image: "measure lebesgue (f ` S) ≤ B * measure lebesgue S" (is "?M") proof - have "f ` S ∈ lmeasurable ∧ measure lebesgue (f ` S) ≤ B * measure lebesgue S" proof (cases "B < 0") case True then have "S = {}" by (meson abs_ge_zero bounded empty_iff equalityI less_le_trans linorder_not_less subsetI) then show ?thesis by auto next case False then have "B ≥ 0" by arith let ?μ = "measure lebesgue" have f_diff: "f differentiable_on S" using deriv by (auto simp: differentiable_on_def differentiable_def) have eps: "f ` S ∈ lmeasurable" "?μ (f ` S) ≤ (B+e) * ?μ S" (is "?ME") if "e > 0" for e proof - have eps_d: "f ` S ∈ lmeasurable" "?μ (f ` S) ≤ (B+e) * (?μ S + d)" (is "?MD") if "d > 0" for d proof - obtain T where T: "open T" "S ⊆ T" and TS: "(T-S) ∈ lmeasurable" and "emeasure lebesgue (T-S) < ennreal d" using S ‹d > 0› sets_lebesgue_outer_open by blast then have "?μ (T-S) < d" by (metis emeasure_eq_measure2 ennreal_leI not_less) with S T TS have "T ∈ lmeasurable" and Tless: "?μ T < ?μ S + d" by (auto simp: measurable_measure_Diff dest!: fmeasurable_Diff_D) have "∃r. 0 < r ∧ r < d ∧ ball x r ⊆ T ∧ f ` (S ∩ ball x r) ∈ lmeasurable ∧ ?μ (f ` (S ∩ ball x r)) ≤ (B + e) * ?μ (ball x r)" if "x ∈ S" "d > 0" for x d proof - have lin: "linear (f' x)" and lim0: "((λy. (f y - (f x + f' x (y - x))) /⇩_{R}norm(y - x)) ⤏ 0) (at x within S)" using deriv ‹x ∈ S› by (auto simp: has_derivative_within bounded_linear.linear field_simps) have bo: "bounded (f' x ` ball 0 1)" by (simp add: bounded_linear_image linear_linear lin) have neg: "negligible (frontier (f' x ` ball 0 1))" using deriv has_derivative_linear ‹x ∈ S› by (auto intro!: negligible_convex_frontier [OF convex_linear_image]) let ?unit_vol = "content (ball (0 :: real ^ 'n :: {finite, wellorder}) 1)" have 0: "0 < e * ?unit_vol" using ‹e > 0› by (simp add: content_ball_pos) obtain k where "k > 0" and k: "⋀U. ⟦U ∈ lmeasurable; ⋀y. y ∈ U ⟹ ∃z. z ∈ f' x ` ball 0 1 ∧ dist z y < k⟧ ⟹ ?μ U < ?μ (f' x ` ball 0 1) + e * ?unit_vol" using measure_semicontinuous_with_hausdist_explicit [OF bo neg 0] by blast obtain l where "l > 0" and l: "ball x l ⊆ T" using ‹x ∈ S› ‹open T› ‹S ⊆ T› openE by blast obtain ζ where "0 < ζ" and ζ: "⋀y. ⟦y ∈ S; y ≠ x; dist y x < ζ⟧ ⟹ norm (f y - (f x + f' x (y - x))) / norm (y - x) < k" using lim0 ‹k > 0› by (simp add: Lim_within) (auto simp add: field_simps) define r where "r ≡ min (min l (ζ/2)) (min 1 (d/2))" show ?thesis proof (intro exI conjI) show "r > 0" "r < d" using ‹l > 0› ‹ζ > 0› ‹d > 0› by (auto simp: r_def) have "r ≤ l" by (auto simp: r_def) with l show "ball x r ⊆ T" by auto have ex_lessK: "∃x' ∈ ball 0 1. dist (f' x x') ((f y - f x) /⇩_{R}r) < k" if "y ∈ S" and "dist x y < r" for y proof (cases "y = x") case True with lin linear_0 ‹k > 0› that show ?thesis by (rule_tac x=0 in bexI) (auto simp: linear_0) next case False then show ?thesis proof (rule_tac x="(y - x) /⇩_{R}r" in bexI) have "f' x ((y - x) /⇩_{R}r) = f' x (y - x) /⇩_{R}r" by (simp add: lin linear_scale) then have "dist (f' x ((y - x) /⇩_{R}r)) ((f y - f x) /⇩_{R}r) = norm (f' x (y - x) /⇩_{R}r - (f y - f x) /⇩_{R}r)" by (simp add: dist_norm) also have "… = norm (f' x (y - x) - (f y - f x)) / r" using ‹r > 0› by (simp add: divide_simps scale_right_diff_distrib [symmetric]) also have "… ≤ norm (f y - (f x + f' x (y - x))) / norm (y - x)" using that ‹r > 0› False by (simp add: field_split_simps dist_norm norm_minus_commute mult_right_mono) also have "… < k" using that ‹0 < ζ› by (simp add: dist_commute r_def ζ [OF ‹y ∈ S› False]) finally show "dist (f' x ((y - x) /⇩_{R}r)) ((f y - f x) /⇩_{R}r) < k" . show "(y - x) /⇩_{R}r ∈ ball 0 1" using that ‹r > 0› by (simp add: dist_norm divide_simps norm_minus_commute) qed qed let ?rfs = "(λx. x /⇩_{R}r) ` (+) (- f x) ` f ` (S ∩ ball x r)" have rfs_mble: "?rfs ∈ lmeasurable" proof (rule bounded_set_imp_lmeasurable) have "f differentiable_on S ∩ ball x r" using f_diff by (auto simp: fmeasurableD differentiable_on_subset) with S show "?rfs ∈ sets lebesgue" by (auto simp: sets.Int intro!: lebesgue_sets_translation differentiable_image_in_sets_lebesgue) let ?B = "(λ(x, y). x + y) ` (f' x ` ball 0 1 × ball 0 k)" have "bounded ?B" by (simp add: bounded_plus [OF bo]) moreover have "?rfs ⊆ ?B" apply (auto simp: dist_norm image_iff dest!: ex_lessK) by (metis (no_types, opaque_lifting) add.commute diff_add_cancel dist_0_norm dist_commute dist_norm mem_ball) ultimately show "bounded (?rfs)" by (rule bounded_subset) qed then have "(λx. r *⇩_{R}x) ` ?rfs ∈ lmeasurable" by (simp add: measurable_linear_image) with ‹r > 0› have "(+) (- f x) ` f ` (S ∩ ball x r) ∈ lmeasurable" by (simp add: image_comp o_def) then have "(+) (f x) ` (+) (- f x) ` f ` (S ∩ ball x r) ∈ lmeasurable" using measurable_translation by blast then show fsb: "f ` (S ∩ ball x r) ∈ lmeasurable" by (simp add: image_comp o_def) have "?μ (f ` (S ∩ ball x r)) = ?μ (?rfs) * r ^ CARD('n)" using ‹r > 0› fsb by (simp add: measure_linear_image measure_translation_subtract measurable_translation_subtract field_simps cong: image_cong_simp) also have "… ≤ (¦det (matrix (f' x))¦ * ?unit_vol + e * ?unit_vol) * r ^ CARD('n)" proof - have "?μ (?rfs) < ?μ (f' x ` ball 0 1) + e * ?unit_vol" using rfs_mble by (force intro: k dest!: ex_lessK) then have "?μ (?rfs) < ¦det (matrix (f' x))¦ * ?unit_vol + e * ?unit_vol" by (simp add: lin measure_linear_image [of "f' x"]) with ‹r > 0› show ?thesis by auto qed also have "… ≤ (B + e) * ?μ (ball x r)" using bounded [OF ‹x ∈ S›] ‹r > 0› by (simp add: algebra_simps content_ball_conv_unit_ball[of r] content_ball_pos) finally show "?μ (f ` (S ∩ ball x r)) ≤ (B + e) * ?μ (ball x r)" . qed qed then obtain r where r0d: "⋀x d. ⟦x ∈ S; d > 0⟧ ⟹ 0 < r x d ∧ r x d < d" and rT: "⋀x d. ⟦x ∈ S; d > 0⟧ ⟹ ball x (r x d) ⊆ T" and r: "⋀x d. ⟦x ∈ S; d > 0⟧ ⟹ (f ` (S ∩ ball x (r x d))) ∈ lmeasurable ∧ ?μ (f ` (S ∩ ball x (r x d))) ≤ (B + e) * ?μ (ball x (r x d))" by metis obtain C where "countable C" and Csub: "C ⊆ {(x,r x t) |x t. x ∈ S ∧ 0 < t}" and pwC: "pairwise (λi j. disjnt (ball (fst i) (snd i)) (ball (fst j) (snd j))) C" and negC: "negligible(S - (⋃i ∈ C. ball (fst i) (snd i)))" apply (rule Vitali_covering_theorem_balls [of S "{(x,r x t) |x t. x ∈ S ∧ 0 < t}" fst snd]) apply auto by (metis dist_eq_0_iff r0d) let ?UB = "(⋃(x,s) ∈ C. ball x s)" have eq: "f ` (S ∩ ?UB) = (⋃(x,s) ∈ C. f ` (S ∩ ball x s))" by auto have mle: "?μ (⋃(x,s) ∈ K. f ` (S ∩ ball x s)) ≤ (B + e) * (?μ S + d)" (is "?l ≤ ?r") if "K ⊆ C" and "finite K" for K proof - have gt0: "b > 0" if "(a, b) ∈ K" for a b using Csub that ‹K ⊆ C› r0d by auto have inj: "inj_on (λ(x, y). ball x y) K" by (force simp: inj_on_def ball_eq_ball_iff dest: gt0) have disjnt: "disjoint ((λ(x, y). ball x y) ` K)" using pwC that pairwise_image pairwise_mono by fastforce have "?l ≤ (∑i∈K. ?μ (case i of (x, s) ⇒ f ` (S ∩ ball x s)))" proof (rule measure_UNION_le [OF ‹finite K›], clarify) fix x r assume "(x,r) ∈ K" then have "x ∈ S" using Csub ‹K ⊆ C› by auto show "f ` (S ∩ ball x r) ∈ sets lebesgue" by (meson Int_lower1 S differentiable_on_subset f_diff fmeasurableD lmeasurable_ball order_refl sets.Int differentiable_image_in_sets_lebesgue) qed also have "… ≤ (∑(x,s) ∈ K. (B + e) * ?μ (ball x s))" using Csub r ‹K ⊆ C› by (intro sum_mono) auto also have "… = (B + e) * (∑(x,s) ∈ K. ?μ (ball x s))" by (simp add: prod.case_distrib sum_distrib_left) also have "… = (B + e) * sum ?μ ((λ(x, y). ball x y) ` K)" using ‹B ≥ 0› ‹e > 0› by (simp add: inj sum.reindex prod.case_distrib) also have "… = (B + e) * ?μ (⋃(x,s) ∈ K. ball x s)" using ‹B ≥ 0› ‹e > 0› that by (subst measure_Union') (auto simp: disjnt measure_Union') also have "… ≤ (B + e) * ?μ T" using ‹B ≥ 0› ‹e > 0› that apply simp using measure_mono_fmeasurable [OF _ _ ‹T ∈ lmeasurable›] Csub rT by (smt (verit) SUP_least measure_nonneg measure_notin_sets mem_Collect_eq old.prod.case subset_iff) also have "… ≤ (B + e) * (?μ S + d)" using ‹B ≥ 0› ‹e > 0› Tless by simp finally show ?thesis . qed have fSUB_mble: "(f ` (S ∩ ?UB)) ∈ lmeasurable" unfolding eq using Csub r False ‹e > 0› that by (auto simp: intro!: fmeasurable_UN_bound [OF ‹countable C› _ mle]) have fSUB_meas: "?μ (f ` (S ∩ ?UB)) ≤ (B + e) * (?μ S + d)" (is "?MUB") unfolding eq using Csub r False ‹e > 0› that by (auto simp: intro!: measure_UN_bound [OF ‹countable C› _ mle]) have neg: "negligible ((f ` (S ∩ ?UB) - f ` S) ∪ (f ` S - f ` (S ∩ ?UB)))" proof (rule negligible_subset [OF negligible_differentiable_image_negligible [OF order_refl negC, where f=f]]) show "f differentiable_on S - (⋃i∈C. ball (fst i) (snd i))" by (meson DiffE differentiable_on_subset subsetI f_diff) qed force show "f ` S ∈ lmeasurable" by (rule lmeasurable_negligible_symdiff [OF fSUB_mble neg]) show ?MD using fSUB_meas measure_negligible_symdiff [OF fSUB_mble neg] by simp qed show "f ` S ∈ lmeasurable" using eps_d [of 1] by simp show ?ME proof (rule field_le_epsilon) fix δ :: real assume "0 < δ" then show "?μ (f ` S) ≤ (B + e) * ?μ S + δ" using eps_d [of "δ / (B+e)"] ‹e > 0› ‹B ≥ 0› by (auto simp: divide_simps mult_ac) qed qed show ?thesis proof (cases "?μ S = 0") case True with eps have "?μ (f ` S) = 0" by (metis mult_zero_right not_le zero_less_measure_iff) then show ?thesis using eps [of 1] by (simp add: True) next case False have "?μ (f ` S) ≤ B * ?μ S" proof (rule field_le_epsilon) fix e :: real assume "e > 0" then show "?μ (f ` S) ≤ B * ?μ S + e" using eps [of "e / ?μ S"] False by (auto simp: algebra_simps zero_less_measure_iff) qed with eps [of 1] show ?thesis by auto qed qed then show "f ` S ∈ lmeasurable" ?M by blast+ qed lemma m_diff_image_weak: fixes f :: "real^'n::{finite,wellorder} ⇒ real^'n::_" assumes S: "S ∈ lmeasurable" and deriv: "⋀x. x ∈ S ⟹ (f has_derivative f' x) (at x within S)" and int: "(λx. ¦det (matrix (f' x))¦) integrable_on S" shows "f ` S ∈ lmeasurable ∧ measure lebesgue (f ` S) ≤ integral S (λx. ¦det (matrix (f' x))¦)" proof - let ?μ = "measure lebesgue" have aint_S: "(λx. ¦det (matrix (f' x))¦) absolutely_integrable_on S" using int unfolding absolutely_integrable_on_def by auto define m where "m ≡ integral S (λx. ¦det (matrix (f' x))¦)" have *: "f ` S ∈ lmeasurable" "?μ (f ` S) ≤ m + e * ?μ S" if "e > 0" for e proof - define T where "T ≡ λn. {x ∈ S. n * e ≤ ¦det (matrix (f' x))¦ ∧ ¦det (matrix (f' x))¦ < (Suc n) * e}" have meas_t: "T n ∈ lmeasurable" for n proof - have *: "(λx. ¦det (matrix (f' x))¦) ∈ borel_measurable (lebesgue_on S)" using aint_S by (simp add: S borel_measurable_restrict_space_iff fmeasurableD set_integrable_def) have [intro]: "x ∈ sets (lebesgue_on S) ⟹ x ∈ sets lebesgue" for x using S sets_restrict_space_subset by blast have "{x ∈ S. real n * e ≤ ¦det (matrix (f' x))¦} ∈ sets lebesgue" using * by (auto simp: borel_measurable_iff_halfspace_ge space_restrict_space) then have 1: "{x ∈ S. real n * e ≤ ¦det (matrix (f' x))¦} ∈ lmeasurable" using S by (simp add: fmeasurableI2) have "{x ∈ S. ¦det (matrix (f' x))¦ < (1 + real n) * e} ∈ sets lebesgue" using * by (auto simp: borel_measurable_iff_halfspace_less space_restrict_space) then have 2: "{x ∈ S. ¦det (matrix (f' x))¦ < (1 + real n) * e} ∈ lmeasurable" using S by (simp add: fmeasurableI2) show ?thesis using fmeasurable.Int [OF 1 2] by (simp add: T_def Int_def cong: conj_cong) qed have aint_T: "⋀k. (λx. ¦det (matrix (f' x))¦) absolutely_integrable_on T k" using set_integrable_subset [OF aint_S] meas_t T_def by blast have Seq: "S = (⋃n. T n)" apply (auto simp: T_def) apply (rule_tac x = "nat ⌊¦det (matrix (f' x))¦ / e⌋" in exI) by (smt (verit, del_insts) divide_nonneg_nonneg floor_eq_iff of_nat_nat pos_divide_less_eq that zero_le_floor) have meas_ft: "f ` T n ∈ lmeasurable" for n proof (rule measurable_bounded_differentiable_image) show "T n ∈ lmeasurable" by (simp add: meas_t) next fix x :: "(real,'n) vec" assume "x ∈ T n" show "(f has_derivative f' x) (at x within T n)" by (metis (no_types, lifting) ‹x ∈ T n› deriv has_derivative_subset mem_Collect_eq subsetI T_def) show "¦det (matrix (f' x))¦ ≤ (Suc n) * e" using ‹x ∈ T n› T_def by auto next show "(λx. ¦det (matrix (f' x))¦) integrable_on T n" using aint_T absolutely_integrable_on_def by blast qed have disT: "disjoint (range T)" unfolding disjoint_def proof clarsimp show "T m ∩ T n = {}" if "T m ≠ T n" for m n using that proof (induction m n rule: linorder_less_wlog) case (less m n) with ‹e > 0› show ?case unfolding T_def proof (clarsimp simp add: Collect_conj_eq [symmetric]) fix x assume "e > 0" "m < n" "n * e ≤ ¦det (matrix (f' x))¦" "¦det (matrix (f' x))¦ < (1 + real m) * e" then have "n < 1 + real m" by (metis (no_types, opaque_lifting) less_le_trans mult.commute not_le mult_le_cancel_left_pos) then show "False" using less.hyps by linarith qed qed auto qed have injT: "inj_on T ({n. T n ≠ {}})" unfolding inj_on_def proof clarsimp show "m = n" if "T m = T n" "T n ≠ {}" for m n using that proof (induction m n rule: linorder_less_wlog) case (less m n) have False if "T n ⊆ T m" "x ∈ T n" for x using ‹e > 0› ‹m < n› that apply (auto simp: T_def mult.commute intro: less_le_trans dest!: subsetD) by (smt (verit, best) mult_less_cancel_left_disj nat_less_real_le) then show ?case using less.prems by blast qed auto qed have sum_eq_Tim: "(∑k≤n. f (T k)) = sum f (T ` {..n})" if "f {} = 0" for f :: "_ ⇒ real" and n proof (subst sum.reindex_nontrivial) fix i j assume "i ∈ {..n}" "j ∈ {..n}" "i ≠ j" "T i = T j" with that injT [unfolded inj_on_def] show "f (T i) = 0" by simp metis qed (use atMost_atLeast0 in auto) let ?B = "m + e * ?μ S" have "(∑k≤n. ?μ (f ` T k)) ≤ ?B" for n proof - have "(∑k≤n. ?μ (f ` T k)) ≤ (∑k≤n. ((k+1) * e) * ?μ(T k))" proof (rule sum_mono [OF measure_bounded_differentiable_image]) show "(f has_derivative f' x) (at x within T k)" if "x ∈ T k" for k x using that unfolding T_def by (blast intro: deriv has_derivative_subset) show "(λx. ¦det (matrix (f' x))¦) integrable_on T k" for k using absolutely_integrable_on_def aint_T by blast show "¦det (matrix (f' x))¦ ≤ real (k + 1) * e" if "x ∈ T k" for k x using T_def that by auto qed (use meas_t in auto) also have "… ≤ (∑k≤n. (k * e) * ?μ(T k)) + (∑k≤n. e * ?μ(T k))" by (simp add: algebra_simps sum.distrib) also have "… ≤ ?B" proof (rule add_mono) have "(∑k≤n. real k * e * ?μ (T k)) = (∑k≤n. integral (T k) (λx. k * e))" by (simp add: lmeasure_integral [OF meas_t] flip: integral_mult_right integral_mult_left) also have "… ≤ (∑k≤n. integral (T k) (λx. (abs (det (matrix (f' x))))))" proof (rule sum_mono) fix k assume "k ∈ {..n}" show "integral (T k) (λx. k * e) ≤ integral (T k) (λx. ¦det (matrix (f' x))¦)" proof (rule integral_le [OF integrable_on_const [OF meas_t]]) show "(λx. ¦det (matrix (f' x))¦) integrable_on T k" using absolutely_integrable_on_def aint_T by blast next fix x assume "x ∈ T k" show "k * e ≤ ¦det (matrix (f' x))¦" using ‹x ∈ T k› T_def by blast qed qed also have "… = sum (λT. integral T (λx. ¦det (matrix (f' x))¦)) (T ` {..n})" by (auto intro: sum_eq_Tim) also have "… = integral (⋃k≤n. T k) (λx. ¦det (matrix (f' x))¦)" proof (rule integral_unique [OF has_integral_Union, symmetric]) fix S assume "S ∈ T ` {..n}" then show "((λx. ¦det (matrix (f' x))¦) has_integral integral S (λx. ¦det (matrix (f' x))¦)) S" using absolutely_integrable_on_def aint_T by blast next show "pairwise (λS S'. negligible (S ∩ S')) (T ` {..n})" using disT unfolding disjnt_iff by (auto simp: pairwise_def intro!: empty_imp_negligible) qed auto also have "… ≤ m" unfolding m_def proof (rule integral_subset_le) have "(λx. ¦det (matrix (f' x))¦) absolutely_integrable_on (⋃k≤n. T k)" proof (rule set_integrable_subset [OF aint_S]) show "⋃ (T ` {..n}) ∈ sets lebesgue" by (intro measurable meas_t fmeasurableD) qed (force simp: Seq) then show "(λx. ¦det (matrix (f' x))¦) integrable_on (⋃k≤n. T k)" using absolutely_integrable_on_def by blast qed (use Seq int in auto) finally show "(∑k≤n. real k * e * ?μ (T k)) ≤ m" . next have "(∑k≤n. ?μ (T k)) = sum ?μ (T ` {..n})" by (auto intro: sum_eq_Tim) also have "… = ?μ (⋃k≤n. T k)" using S disT by (auto simp: pairwise_def meas_t intro: measure_Union' [symmetric]) also have "… ≤ ?μ S" using S by (auto simp: Seq intro: meas_t fmeasurableD measure_mono_fmeasurable) finally have "(∑k≤n. ?μ (T k)) ≤ ?μ S" . then show "(∑k≤n. e * ?μ (T k)) ≤ e * ?μ S" by (metis less_eq_real_def ordered_comm_semiring_class.comm_mult_left_mono sum_distrib_left that) qed finally show "(∑k≤n. ?μ (f ` T k)) ≤ ?B" . qed moreover have "measure lebesgue (⋃k≤n. f ` T k) ≤ (∑k≤n. ?μ (f ` T k))" for n by (simp add: fmeasurableD meas_ft measure_UNION_le) ultimately have B_ge_m: "?μ (⋃k≤n. (f ` T k)) ≤ ?B" for n by (meson order_trans) have "(⋃n. f ` T n) ∈ lmeasurable" by (rule fmeasurable_countable_Union [OF meas_ft B_ge_m]) moreover have "?μ (⋃n. f ` T n) ≤ m + e * ?μ S" by (rule measure_countable_Union_le [OF meas_ft B_ge_m]) ultimately show "f ` S ∈ lmeasurable" "?μ (f ` S) ≤ m + e * ?μ S" by (auto simp: Seq image_Union) qed show ?thesis proof show "f ` S ∈ lmeasurable" using * linordered_field_no_ub by blast let ?x = "m - ?μ (f ` S)" have False if "?μ (f ` S) > integral S (λx. ¦det (matrix (f' x))¦)" proof - have ml: "m < ?μ (f ` S)" using m_def that by blast then have "?μ S ≠ 0" using "*"(2) bgauge_existence_lemma by fastforce with ml have 0: "0 < - (m - ?μ (f ` S))/2 / ?μ S" using that zero_less_measure_iff by force then show ?thesis using * [OF 0] that by (auto simp: field_split_simps m_def split: if_split_asm) qed then show "?μ (f ` S) ≤ integral S (λx. ¦det (matrix (f' x))¦)" by fastforce qed qed theorem fixes f :: "real^'n::{finite,wellorder} ⇒ real^'n::_" assumes S: "S ∈ sets lebesgue" and deriv: "⋀x. x ∈ S ⟹ (f has_derivative f' x) (at x within S)" and int: "(λx. ¦det (matrix (f' x))¦) integrable_on S" shows measurable_differentiable_image: "f ` S ∈ lmeasurable" and measure_differentiable_image: "measure lebesgue (f ` S) ≤ integral S (λx. ¦det (matrix (f' x))¦)" (is "?M") proof - let ?I = "λn::nat. cbox (vec (-n)) (vec n) ∩ S" let ?μ = "measure lebesgue" have "x ∈ cbox (vec (- real (nat ⌈norm x⌉))) (vec (real (nat ⌈norm x⌉)))" for x :: "real^'n::_" apply (simp add: mem_box_cart) by (smt (verit, best) component_le_norm_cart le_of_int_ceiling) then have Seq: "S = (⋃n. ?I n)" by blast have fIn: "f ` ?I n ∈ lmeasurable" and mfIn: "?μ (f ` ?I n) ≤ integral S (λx. ¦det (matrix (f' x))¦)" (is ?MN) for n proof - have In: "?I n ∈ lmeasurable" by (simp add: S bounded_Int bounded_set_imp_lmeasurable sets.Int) moreover have "⋀x. x ∈ ?I n ⟹ (f has_derivative f' x) (at x within ?I n)" by (meson Int_iff deriv has_derivative_subset subsetI) moreover have int_In: "(λx. ¦det (matrix (f' x))¦) integrable_on ?I n" by (metis (mono_tags) Int_commute int integrable_altD(1) integrable_restrict_Int) ultimately have "f ` ?I n ∈ lmeasurable" "?μ (f ` ?I n) ≤ integral (?I n) (λx. ¦det (matrix (f' x))¦)" using m_diff_image_weak by metis+ moreover have "integral (?I n) (λx. ¦det (matrix (f' x))¦) ≤ integral S (λx. ¦det (matrix (f' x))¦)" by (simp add: int_In int integral_subset_le) ultimately show "f ` ?I n ∈ lmeasurable" ?MN by auto qed have "?I k ⊆ ?I n" if "k ≤ n" for k n by (rule Int_mono) (use that in ‹auto simp: subset_interval_imp_cart›) then have "(⋃k≤n. f ` ?I k) = f ` ?I n" for n by (fastforce simp add:) with mfIn have "?μ (⋃k≤n. f ` ?I k) ≤ integral S (λx. ¦det (matrix (f' x))¦)" for n by simp then have "(⋃n. f ` ?I n) ∈ lmeasurable" "?μ (⋃n. f ` ?I n) ≤ integral S (λx. ¦det (matrix (f' x))¦)" by (rule fmeasurable_countable_Union [OF fIn] measure_countable_Union_le [OF fIn])+ then show "f ` S ∈ lmeasurable" ?M by (metis Seq image_UN)+ qed lemma borel_measurable_simple_function_limit_increasing: fixes f :: "'a::euclidean_space ⇒ real" shows "(f ∈ borel_measurable lebesgue ∧ (∀x. 0 ≤ f x)) ⟷ (∃g. (∀n x. 0 ≤ g n x ∧ g n x ≤ f x) ∧ (∀n x. g n x ≤ (g(Suc n) x)) ∧ (∀n. g n ∈ borel_measurable lebesgue) ∧ (∀n. finite(range (g n))) ∧ (∀x. (λn. g n x) ⇢ f x))" (is "?lhs = ?rhs") proof assume f: ?lhs have leb_f: "{x. a ≤ f x ∧ f x < b} ∈ sets lebesgue" for a b proof - have "{x. a ≤ f x ∧ f x < b} = {x. f x < b} - {x. f x < a}" by auto also have "… ∈ sets lebesgue" using borel_measurable_vimage_halfspace_component_lt [of f UNIV] f by auto finally show ?thesis . qed have "g n x ≤ f x" if inc_g: "⋀n x. 0 ≤ g n x ∧ g n x ≤ g (Suc n) x" and meas_g: "⋀n. g n ∈ borel_measurable lebesgue" and fin: "⋀n. finite(range (g n))" and lim: "⋀x. (λn. g n x) ⇢ f x" for g n x proof - have "∃r>0. ∀N. ∃n≥N. dist (g n x) (f x) ≥ r" if "g n x > f x" proof - have g: "g n x ≤ g (N + n) x" for N by (rule transitive_stepwise_le) (use inc_g in auto) have "∃m≥N. g n x - f x ≤ dist (g m x) (f x)" for N proof show "N ≤ N + n ∧ g n x - f x ≤ dist (g (N + n) x) (f x)" using g [of N] by (auto simp: dist_norm) qed with that show ?thesis using diff_gt_0_iff_gt by blast qed with lim show ?thesis unfolding lim_sequentially by (meson less_le_not_le not_le_imp_less) qed moreover let ?Ω = "λn k. indicator {y. k/2^n ≤ f y ∧ f y < (k+1)/2^n}" let ?g = "λn x. (∑k::real | k ∈ ℤ ∧ ¦k¦ ≤ 2 ^ (2*n). k/2^n * ?Ω n k x)" have "∃g. (∀n x. 0 ≤ g n x ∧ g n x ≤ (g(Suc n) x)) ∧ (∀n. g n ∈ borel_measurable lebesgue) ∧ (∀n. finite(range (g n))) ∧(∀x. (λn. g n x) ⇢ f x)" proof (intro exI allI conjI) show "0 ≤ ?g n x" for n x proof (clarify intro!: ordered_comm_monoid_add_class.sum_nonneg) fix k::real assume "k ∈ ℤ" and k: "¦k¦ ≤ 2 ^ (2*n)" show "0 ≤ k/2^n * ?Ω n k x" using f ‹k ∈ ℤ› apply (clarsimp simp: indicator_def field_split_simps Ints_def) by (smt (verit) int_less_real_le mult_nonneg_nonneg of_int_0 zero_le_power) qed show "?g n x ≤ ?g (Suc n) x" for n x proof - have "?g n x = (∑k | k ∈ ℤ ∧ ¦k¦ ≤ 2 ^ (2*n). k/2^n * (indicator {y. k/2^n ≤ f y ∧ f y < (k+1/2)/2^n} x + indicator {y. (k+1/2)/2^n ≤ f y ∧ f y < (k+1)/2^n} x))" by (rule sum.cong [OF refl]) (simp add: indicator_def field_split_simps) also have "… = (∑k | k ∈ ℤ ∧ ¦k¦ ≤ 2 ^ (2*n). k/2^n * indicator {y. k/2^n ≤ f y ∧ f y < (k+1/2)/2^n} x) + (∑k | k ∈ ℤ ∧ ¦k¦ ≤ 2 ^ (2*n). k/2^n * indicator {y. (k+1/2)/2^n ≤ f y ∧ f y < (k+1)/2^n} x)" by (simp add: comm_monoid_add_class.sum.distrib algebra_simps) also have "… = (∑k | k ∈ ℤ ∧ ¦k¦ ≤ 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k)/2 ^ Suc n ≤ f y ∧ f y < (2 * k<