(* Title: HOL/Analysis/Change_Of_Vars.thy
Authors: LC Paulson, based on material from HOL Light
*)
section\<open>Change of Variables Theorems\<close>
theory Change_Of_Vars
imports Vitali_Covering_Theorem Determinants
begin
subsection \<open>Measurable Shear and Stretch\<close>
proposition
fixes a :: "real^'n"
assumes "m \ n" and ab_ne: "cbox a b \ {}" and an: "0 \ a$n"
shows measurable_shear_interval: "(\x. \ i. if i = m then x$m + x$n else x$i) ` (cbox a b) \ lmeasurable"
(is "?f ` _ \ _")
and measure_shear_interval: "measure lebesgue ((\x. \ i. if i = m then x$m + x$n else x$i) ` cbox a b)
= measure lebesgue (cbox a b)" (is "?Q")
proof -
have lin: "linear ?f"
by (rule linearI) (auto simp: plus_vec_def scaleR_vec_def algebra_simps)
show fab: "?f ` cbox a b \ lmeasurable"
by (simp add: lin measurable_linear_image_interval)
let ?c = "\ i. if i = m then b$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 \<inter> {x. ?mn \<bullet> x \<le> a$m})
+ measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$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 \<open>m \<noteq> n\<close> 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 \<open>m \<noteq> n\<close> 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$m}"
by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
moreover have "(?f ` cbox a b) \ (cbox a ?c \ {x. ?mn \ x \ b$m}) \ {x. ?mn \ x = b$m}"
using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
ultimately show "negligible (?f ` cbox a b \ (cbox a ?c \ {x. ?mn \ x \ b$m}))"
by (rule negligible_subset)
have "negligible {x. ?mn \ x = b$m}"
by (metis \<open>m \<noteq> n\<close> 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 \<open>m \<noteq> n\<close> ab_ne
apply (auto simp: algebra_simps mem_box_cart inner_axis')
apply (drule_tac x=m in spec)+
apply simp
done
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 (auto simp: algebra_simps image_iff inner_axis' lambda_add_Galois [OF \m \ n\])
apply (auto simp: mem_box_cart split: if_split_asm)
done
qed
qed (fact fab)
let ?d = "\ 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 (\<chi> 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}"
"(\ i. if i = m then a$m - b$m else 0)" "cbox a (\ 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\
\<Longrightarrow> x \<in> (+) (\<chi> i. if i = m then a $ m - b $ m else 0) ` {x. x $ n + b $ m \<le> x $ m}"
using \<open>m \<noteq> n\<close>
by (rule_tac x="x - (\ 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 \<open>m \<noteq> n\<close> by (auto simp: mem_box_cart inner_axis' algebra_simps)
have "\x. \0 \ a $ n; x $ n + a $ m \ x $ m;
\<forall>i. i \<noteq> m \<longrightarrow> a $ i \<le> x $ i \<and> x $ i \<le> b $ i\<rbrakk>
\<Longrightarrow> a $ m \<le> x $ m"
using \<open>m \<noteq> n\<close> by force
then have "(+) ?d ` (cbox a ?c \ {x. b $ m \ ?mn \ x})
= cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i) \<inter> {x. a $ m \<le> ?mn \<bullet> 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 \<inter> {x. b $ m \<le> ?mn \<bullet> x}) =
cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i)" (is "?lhs = ?rhs")
using an \<open>m \<noteq> n\<close>
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 \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
moreover have "(cbox a ?c \ {x. ?mn \ x \ a $ m} \
(+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x})) \<subseteq> {x. ?mn \<bullet> x = a$m}"
using \<open>m \<noteq> n\<close> 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 \<inter> {x. b $ m \<le> ?mn \<bullet> x}))"
by (rule negligible_subset)
qed
have ac_ne: "cbox a ?c \ {}"
using ab_ne an
by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
have ax_ne: "cbox a (\ i. if i = m then a $ m + b $ n else b $ i) \ {}"
using ab_ne an
by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
have eq3: "measure lebesgue (cbox a ?c) = measure lebesgue (cbox a (\ i. if i = m then a$m + b$n else b$i)) + measure lebesgue (cbox a b)"
by (simp add: content_cbox_if_cart ab_ne ac_ne ax_ne algebra_simps prod.delta_remove
if_distrib [of "\u. u - z" for z] prod.remove)
show ?Q
using eq1 eq2 eq3
by (simp add: algebra_simps)
qed
proposition
fixes S :: "(real^'n) set"
assumes "S \ lmeasurable"
shows measurable_stretch: "((\x. \ k. m k * x$k) ` S) \ lmeasurable" (is "?f ` S \ _")
and measure_stretch: "measure lebesgue ((\x. \ 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 \
\<exists>z. (indicat_real S has_integral z) (cbox a b) \<and>
\<bar>z - measure lebesgue S\<bar> < e / \<bar>prod m UNIV\<bar>"
by (simp add: has_integral_alt [where i=UNIV]) (metis (full_types) divide_pos_pos m0 m0 \<open>e > 0\<close>)
show "\B>0. \a b. ball 0 B \ cbox a b \
(\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox a b) \<and>
\<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)"
proof (intro exI conjI allI)
let ?C = "Max (range (\k. \m k\)) * B"
show "?C > 0"
using True \<open>B > 0\<close> by (simp add: Max_gr_iff)
show "ball 0 ?C \ cbox u v \
(\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox u v) \<and>
\<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)" for u v
proof
assume uv: "ball 0 ?C \ cbox u v"
with \<open>?C > 0\<close> have cbox_ne: "cbox u v \<noteq> {}"
using centre_in_ball by blast
let ?\<alpha> = "\<lambda>k. u$k / m k"
let ?\<beta> = "\<lambda>k. v$k / m k"
have invm0: "\k. inverse (m k) \ 0"
using True by auto
have "ball 0 B \ (\x. \ k. x $ k / m k) ` ball 0 ?C"
proof clarsimp
fix x :: "real^'n"
assume x: "norm x < B"
have [simp]: "\Max (range (\k. \m k\))\ = Max (range (\k. \m k\))"
by (meson Max_ge abs_ge_zero abs_of_nonneg finite finite_imageI order_trans rangeI)
have "norm (\ k. m k * x $ k) \ norm (Max (range (\k. \m k\)) *\<^sub>R x)"
by (rule norm_le_componentwise_cart) (auto simp: abs_mult intro: mult_right_mono)
also have "\ < ?C"
using x \<open>0 < (MAX k. \<bar>m k\<bar>) * B\<close> \<open>0 < B\<close> zero_less_mult_pos2 by fastforce
finally have "norm (\ k. m k * x $ k) < ?C" .
then show "x \ (\x. \ 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 (\ k. min (?\ k) (?\ k)) (\ k. max (?\ k) (?\ 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 (\ k. min (?\ k) (?\ k)) (\ k. max (?\ k) (?\ k)))"
and zless: "\z - measure lebesgue S\ < e / \prod m UNIV\"
using B [OF Bsub] by blast
have ind: "indicat_real (?f ` S) = (\x. indicator S (\ 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) \
\<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e"
proof (simp add: ind, intro conjI exI)
have "((\x. indicat_real S (\ k. x $ k/ m k)) has_integral z *\<^sub>R \prod m UNIV\)
((\<lambda>x. \<chi> k. x $ k * m k) ` cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k)))"
using True has_integral_stretch_cart [OF zint, of "inverse \ m"]
by (simp add: field_simps prod_dividef)
moreover have "((\x. \ k. x $ k * m k) ` cbox (\ k. min (?\ k) (?\ k)) (\ k. max (?\ k) (?\ k))) = cbox u v"
using True image_stretch_interval_cart [of "inverse \ m" u v, symmetric]
image_stretch_interval_cart [of "\k. 1" u v, symmetric] \cbox u v \ {}\
by (simp add: field_simps image_comp o_def)
ultimately show "((\x. indicat_real S (\ k. x $ k/ m k)) has_integral z *\<^sub>R \prod m UNIV\) (cbox u v)"
by simp
have "\z *\<^sub>R \prod m UNIV\ - \prod m UNIV\ * measure lebesgue S\
= \<bar>prod m UNIV\<bar> * \<bar>z - measure lebesgue S\<bar>"
by (metis (no_types, hide_lams) 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 *\<^sub>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 \<open>m k = 0\<close> in auto)
then have "(?f ` S) \ lmeasurable"
by (simp add: negligible_iff_measure)
with nfS show ?thesis
by (simp add: prm negligible_iff_measure0)
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 \<open>linear f\<close>]; 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 \<open>linear g\<close> \<open>linear f\<close>]
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 \<open>\<not> inj f\<close> det_nz_iff_inj[OF \<open>linear f\<close>] by blast
show "f ` S \ lmeasurable \ ?Q f S"
proof
show "f ` S \ lmeasurable"
using lmeasurable_iff_indicator_has_integral \<open>linear f\<close> \<open>\<not> inj f\<close> negligible_UNIV negligible_linear_singular_image by blast
have "measure lebesgue (f ` S) = 0"
by (meson \<open>\<not> inj f\<close> \<open>linear f\<close> 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 "(\a. \ i. c i * a $ i) ` S \ lmeasurable \ ?Q (\a. \ i. c i * a $ i) S"
proof
show "(\a. \ i. c i * a $ i) ` S \ lmeasurable"
by (simp add: \<open>S \<in> lmeasurable\<close> measurable_stretch)
show "?Q (\a. \ i. c i * a $ i) S"
by (simp add: measure_stretch [OF \<open>S \<in> lmeasurable\<close>, 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 = "\v::(real, 'n) vec. \ i. v $ Fun.swap m n id i"
have lin: "linear ?h"
by (rule linearI) (simp_all add: plus_vec_def scaleR_vec_def)
have meq: "measure lebesgue ((\v::(real, 'n) vec. \ i. v $ Fun.swap m n id 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 swap_id_eq)+
show ?thesis
using him prod.permute [OF permutes_swap_id, where S=UNIV and g="\i. (b - a)$i", symmetric]
by (simp add: eq content_cbox_cart False)
qed
have "(\ i j. if Fun.swap m n id i = j then 1 else 0) = (\ i j. if j = Fun.swap m n id i then 1 else (0::real))"
by (auto intro!: Cart_lambda_cong)
then have "matrix ?h = transpose(\ i j. mat 1 $ i $ Fun.swap m n id 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"
proof
show "?h ` S \ lmeasurable" "?Q ?h S"
using measure_linear_sufficient [OF lin \<open>S \<in> lmeasurable\<close>] meq 1 by force+
qed
next
fix m n :: "'n" and S :: "(real, 'n) vec set"
assume "m \ n" and "S \ lmeasurable"
let ?h = "\v::(real, 'n) vec. \ 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 \<open>m \<noteq> n\<close> 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 = (\ n. if n = j then 1 else (0::real))"
using axis_def by blast
then have "(\ p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
using \<open>j < i\<close> axis_def \<open>m < n\<close> by auto
with \<open>m < n\<close> show ?thesis
by (auto simp: matrix_def axis_def cong: if_cong)
qed
show ?thesis
using \<open>m \<noteq> n\<close> 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 = (\ n. if n = j then 1 else (0::real))"
using axis_def by blast
then have "(\ p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
using \<open>j > i\<close> axis_def \<open>m > n\<close> by auto
with \<open>m > n\<close> show ?thesis
by (auto simp: matrix_def axis_def cong: if_cong)
qed
show ?thesis
using \<open>m \<noteq> n\<close>
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: "(+) (\ i. if i = n then - a $ n else 0) ` cbox a b \ {}"
by auto
let ?v = "\ i. if i = n then - a $ n else 0"
have "?h ` cbox a b
= (+) (\<chi> i. if i = m \<or> i = n then a $ n else 0) ` ?h ` (+) ?v ` (cbox a b)"
using \<open>m \<noteq> n\<close> unfolding image_comp o_def by (force simp: vec_eq_iff)
then have "measure lebesgue (?h ` (cbox a b))
= measure lebesgue ((\<lambda>v. \<chi> 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 ((\v. \ 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 ((+) (\ i. if i = n then - a $ n else 0) ` cbox a b)"
apply (subst measure_shear_interval)
using \<open>m \<noteq> n\<close> 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 \<open>S \<in> lmeasurable\<close>] 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 \<open>linear f\<close> 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\
\<Longrightarrow> measure lebesgue T < measure lebesgue S + e"
proof (cases "S = {}")
case True
with that \<open>e > 0\<close> show ?thesis by force
next
case False
then have frS: "frontier S \ {}"
using \<open>bounded S\<close> frontier_eq_empty not_bounded_UNIV by blast
have "S \ lmeasurable"
by (simp add: \<open>bounded S\<close> 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 \<open>e > 0\<close> 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: \<open>U \<in> lmeasurable\<close> 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 \<open>U \<in> lmeasurable\<close> by auto
then have "- U \ {}"
by blast
with \<open>open U\<close> \<open>frontier S \<subseteq> U\<close> show "setdist (frontier S) (- U) > 0"
by (auto simp: \<open>bounded S\<close> 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: \<open>S \<in> lmeasurable\<close> 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 \<open>x \<notin> S\<close> \<open>y \<in> S\<close> 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 \<open>z \<in> frontier S\<close> that] by auto
then show "x \ U"
by blast
qed
then show ?thesis
by (simp add: \<open>S \<in> lmeasurable\<close> \<open>T \<in> lmeasurable\<close> \<open>U \<in> lmeasurable\<close> 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: "(\x. \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 ?\<mu> = "measure lebesgue"
have f_diff: "f differentiable_on S"
using deriv by (auto simp: differentiable_on_def differentiable_def)
have eps: "f ` S \ lmeasurable" "?\ (f ` S) \ (B+e) * ?\ S" (is "?ME")
if "e > 0" for e
proof -
have eps_d: "f ` S \ lmeasurable" "?\ (f ` S) \ (B+e) * (?\ 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 \<open>d > 0\<close> sets_lebesgue_outer_open by blast
then have "?\ (T-S) < d"
by (metis emeasure_eq_measure2 ennreal_leI not_less)
with S T TS have "T \ lmeasurable" and Tless: "?\ T < ?\ 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 \
?\<mu> (f ` (S \<inter> ball x r)) \<le> (B + e) * ?\<mu> (ball x r)"
if "x \ S" "d > 0" for x d
proof -
have lin: "linear (f' x)"
and lim0: "((\y. (f y - (f x + f' x (y - x))) /\<^sub>R norm(y - x)) \ 0) (at x within S)"
using deriv \<open>x \<in> S\<close> 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 \<open>x \<in> S\<close>
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 \<open>e > 0\<close> 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\
\<Longrightarrow> ?\<mu> U < ?\<mu> (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 \<open>x \<in> S\<close> \<open>open T\<close> \<open>S \<subseteq> T\<close> openE by blast
obtain \<zeta> where "0 < \<zeta>"
and \<zeta>: "\<And>y. \<lbrakk>y \<in> S; y \<noteq> x; dist y x < \<zeta>\<rbrakk>
\<Longrightarrow> norm (f y - (f x + f' x (y - x))) / norm (y - x) < k"
using lim0 \<open>k > 0\<close> by (simp add: Lim_within) (auto simp add: field_simps)
define r where "r \ min (min l (\/2)) (min 1 (d/2))"
show ?thesis
proof (intro exI conjI)
show "r > 0" "r < d"
using \<open>l > 0\<close> \<open>\<zeta> > 0\<close> \<open>d > 0\<close> 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) /\<^sub>R r) < k"
if "y \ S" and "dist x y < r" for y
proof (cases "y = x")
case True
with lin linear_0 \<open>k > 0\<close> 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) /\<^sub>R r" in bexI)
have "f' x ((y - x) /\<^sub>R r) = f' x (y - x) /\<^sub>R r"
by (simp add: lin linear_scale)
then have "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) = norm (f' x (y - x) /\<^sub>R r - (f y - f x) /\<^sub>R r)"
by (simp add: dist_norm)
also have "\ = norm (f' x (y - x) - (f y - f x)) / r"
using \<open>r > 0\<close> 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 \<open>r > 0\<close> False by (simp add: field_split_simps dist_norm norm_minus_commute mult_right_mono)
also have "\ < k"
using that \<open>0 < \<zeta>\<close> by (simp add: dist_commute r_def \<zeta> [OF \<open>y \<in> S\<close> False])
finally show "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) < k" .
show "(y - x) /\<^sub>R r \ ball 0 1"
using that \<open>r > 0\<close> by (simp add: dist_norm divide_simps norm_minus_commute)
qed
qed
let ?rfs = "(\x. x /\<^sub>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 = "(\(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, hide_lams) 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 "(\x. r *\<^sub>R x) ` ?rfs \ lmeasurable"
by (simp add: measurable_linear_image)
with \<open>r > 0\<close> have "(+) (- f x) ` f ` (S \<inter> ball x r) \<in> 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 "?\ (f ` (S \ ball x r)) = ?\ (?rfs) * r ^ CARD('n)"
using \<open>r > 0\<close> 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 "?\ (?rfs) < ?\ (f' x ` ball 0 1) + e * ?unit_vol"
using rfs_mble by (force intro: k dest!: ex_lessK)
then have "?\ (?rfs) < \det (matrix (f' x))\ * ?unit_vol + e * ?unit_vol"
by (simp add: lin measure_linear_image [of "f' x"])
with \<open>r > 0\<close> show ?thesis
by auto
qed
also have "\ \ (B + e) * ?\ (ball x r)"
using bounded [OF \<open>x \<in> S\<close>] \<open>r > 0\<close>
by (simp add: algebra_simps content_ball_conv_unit_ball[of r] content_ball_pos)
finally show "?\ (f ` (S \ ball x r)) \ (B + e) * ?\ (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 \<inter> ball x (r x d))) \<in> lmeasurable \<and>
?\<mu> (f ` (S \<inter> ball x (r x d))) \<le> (B + e) * ?\<mu> (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 (\i 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: "?\ (\(x,s) \ K. f ` (S \ ball x s)) \ (B + e) * (?\ 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 \<open>K \<subseteq> C\<close> r0d by auto
have inj: "inj_on (\(x, y). ball x y) K"
by (force simp: inj_on_def ball_eq_ball_iff dest: gt0)
have disjnt: "disjoint ((\(x, y). ball x y) ` K)"
using pwC that
apply (clarsimp simp: pairwise_def case_prod_unfold ball_eq_ball_iff)
by (metis subsetD fst_conv snd_conv)
have "?l \ (\i\K. ?\ (case i of (x, s) \ f ` (S \ ball x s)))"
proof (rule measure_UNION_le [OF \<open>finite K\<close>], clarify)
fix x r
assume "(x,r) \ K"
then have "x \ S"
using Csub \<open>K \<subseteq> C\<close> 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) * ?\ (ball x s))"
apply (rule sum_mono)
using Csub r \<open>K \<subseteq> C\<close> by auto
also have "\ = (B + e) * (\(x,s) \ K. ?\ (ball x s))"
by (simp add: prod.case_distrib sum_distrib_left)
also have "\ = (B + e) * sum ?\ ((\(x, y). ball x y) ` K)"
using \<open>B \<ge> 0\<close> \<open>e > 0\<close> by (simp add: inj sum.reindex prod.case_distrib)
also have "\ = (B + e) * ?\ (\(x,s) \ K. ball x s)"
using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that
by (subst measure_Union') (auto simp: disjnt measure_Union')
also have "\ \ (B + e) * ?\ T"
using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that apply simp
apply (rule measure_mono_fmeasurable [OF _ _ \<open>T \<in> lmeasurable\<close>])
using Csub rT by force+
also have "\ \ (B + e) * (?\ S + d)"
using \<open>B \<ge> 0\<close> \<open>e > 0\<close> Tless by simp
finally show ?thesis .
qed
have fSUB_mble: "(f ` (S \ ?UB)) \ lmeasurable"
unfolding eq using Csub r False \<open>e > 0\<close> that
by (auto simp: intro!: fmeasurable_UN_bound [OF \<open>countable C\<close> _ mle])
have fSUB_meas: "?\ (f ` (S \ ?UB)) \ (B + e) * (?\ S + d)" (is "?MUB")
unfolding eq using Csub r False \<open>e > 0\<close> that
by (auto simp: intro!: measure_UN_bound [OF \<open>countable C\<close> _ 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 \<delta> :: real
assume "0 < \"
then show "?\ (f ` S) \ (B + e) * ?\ S + \"
using eps_d [of "\ / (B+e)"] \e > 0\ \B \ 0\ by (auto simp: divide_simps mult_ac)
qed
qed
show ?thesis
proof (cases "?\ S = 0")
case True
with eps have "?\ (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 "?\ (f ` S) \ B * ?\ S"
proof (rule field_le_epsilon)
fix e :: real
assume "e > 0"
then show "?\ (f ` S) \ B * ?\ S + e"
using eps [of "e / ?\ 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: "(\x. \det (matrix (f' x))\) integrable_on S"
shows "f ` S \ lmeasurable \ measure lebesgue (f ` S) \ integral S (\x. \det (matrix (f' x))\)"
proof -
let ?\<mu> = "measure lebesgue"
have aint_S: "(\x. \det (matrix (f' x))\) absolutely_integrable_on S"
using int unfolding absolutely_integrable_on_def by auto
define m where "m \ integral S (\x. \det (matrix (f' x))\)"
have *: "f ` S \ lmeasurable" "?\ (f ` S) \ m + e * ?\ S"
if "e > 0" for e
proof -
define T where "T \ \n. {x \ S. n * e \ \det (matrix (f' x))\ \
\<bar>det (matrix (f' x))\<bar> < (Suc n) * e}"
have meas_t: "T n \ lmeasurable" for n
proof -
have *: "(\x. \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. (\x. \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(floor(abs(det(matrix(f' x))) / e))" in exI)
using that apply auto
using of_int_floor_le pos_le_divide_eq apply blast
by (metis add.commute pos_divide_less_eq real_of_int_floor_add_one_gt)
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) \<open>x \<in> T n\<close> deriv has_derivative_subset mem_Collect_eq subsetI T_def)
show "\det (matrix (f' x))\ \ (Suc n) * e"
using \<open>x \<in> T n\<close> T_def by auto
next
show "(\x. \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 \<open>e > 0\<close> 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, hide_lams) less_le_trans mult.commute not_le mult_le_cancel_iff2)
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 \<open>e > 0\<close> \<open>m < n\<close> that
apply (auto simp: T_def mult.commute intro: less_le_trans dest!: subsetD)
by (metis add.commute less_le_trans nat_less_real_le not_le mult_le_cancel_iff2)
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 * ?\ S"
have "(\k\n. ?\ (f ` T k)) \ ?B" for n
proof -
have "(\k\n. ?\ (f ` T k)) \ (\k\n. ((k+1) * e) * ?\(T k))"
proof (rule sum_mono [OF measure_bounded_differentiable_image])
show "(f has_derivative f' x) (at x within T k)" if "x \ T k" for k x
using that unfolding T_def by (blast intro: deriv has_derivative_subset)
show "(\x. \det (matrix (f' x))\) 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) * ?\(T k)) + (\k\n. e * ?\(T k))"
by (simp add: algebra_simps sum.distrib)
also have "\ \ ?B"
proof (rule add_mono)
have "(\k\n. real k * e * ?\ (T k)) = (\k\n. integral (T k) (\x. k * e))"
by (simp add: lmeasure_integral [OF meas_t]
flip: integral_mult_right integral_mult_left)
also have "\ \ (\k\n. integral (T k) (\x. (abs (det (matrix (f' x))))))"
proof (rule sum_mono)
fix k
assume "k \ {..n}"
show "integral (T k) (\x. k * e) \ integral (T k) (\x. \det (matrix (f' x))\)"
proof (rule integral_le [OF integrable_on_const [OF meas_t]])
show "(\x. \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 \<open>x \<in> T k\<close> T_def by blast
qed
qed
also have "\ = sum (\T. integral T (\x. \det (matrix (f' x))\)) (T ` {..n})"
by (auto intro: sum_eq_Tim)
also have "\ = integral (\k\n. T k) (\x. \det (matrix (f' x))\)"
proof (rule integral_unique [OF has_integral_Union, symmetric])
fix S assume "S \ T ` {..n}"
then show "((\x. \det (matrix (f' x))\) has_integral integral S (\x. \det (matrix (f' x))\)) S"
using absolutely_integrable_on_def aint_T by blast
next
show "pairwise (\S 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 "(\x. \det (matrix (f' x))\) absolutely_integrable_on (\k\n. T k)"
apply (rule set_integrable_subset [OF aint_S])
apply (intro measurable meas_t fmeasurableD)
apply (force simp: Seq)
done
then show "(\x. \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 * ?\ (T k)) \ m" .
next
have "(\k\n. ?\ (T k)) = sum ?\ (T ` {..n})"
by (auto intro: sum_eq_Tim)
also have "\ = ?\ (\k\n. T k)"
using S disT by (auto simp: pairwise_def meas_t intro: measure_Union' [symmetric])
also have "\ \ ?\ S"
using S by (auto simp: Seq intro: meas_t fmeasurableD measure_mono_fmeasurable)
finally have "(\k\n. ?\ (T k)) \ ?\ S" .
then show "(\k\n. e * ?\ (T k)) \ e * ?\ S"
by (metis less_eq_real_def ordered_comm_semiring_class.comm_mult_left_mono sum_distrib_left that)
qed
finally show "(\k\n. ?\ (f ` T k)) \ ?B" .
qed
moreover have "measure lebesgue (\k\n. f ` T k) \ (\k\n. ?\ (f ` T k))" for n
by (simp add: fmeasurableD meas_ft measure_UNION_le)
ultimately have B_ge_m: "?\ (\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 "?\ (\n. f ` T n) \ m + e * ?\ S"
by (rule measure_countable_Union_le [OF meas_ft B_ge_m])
ultimately show "f ` S \ lmeasurable" "?\ (f ` S) \ m + e * ?\ 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 - ?\ (f ` S)"
have False if "?\ (f ` S) > integral S (\x. \det (matrix (f' x))\)"
proof -
have ml: "m < ?\ (f ` S)"
using m_def that by blast
then have "?\ S \ 0"
using "*"(2) bgauge_existence_lemma by fastforce
with ml have 0: "0 < - (m - ?\ (f ` S))/2 / ?\ 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 "?\ (f ` S) \ integral S (\x. \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: "(\x. \det (matrix (f' x))\) integrable_on S"
shows measurable_differentiable_image: "f ` S \ lmeasurable"
and measure_differentiable_image:
"measure lebesgue (f ` S) \ integral S (\x. \det (matrix (f' x))\)" (is "?M")
proof -
let ?I = "\n::nat. cbox (vec (-n)) (vec n) \ S"
let ?\<mu> = "measure lebesgue"
have "x \ cbox (vec (- real (nat \norm x\))) (vec (real (nat \norm x\)))" for x :: "real^'n::_"
apply (auto simp: mem_box_cart)
apply (metis abs_le_iff component_le_norm_cart minus_le_iff of_nat_ceiling order.trans)
by (meson abs_le_D1 norm_bound_component_le_cart real_nat_ceiling_ge)
then have Seq: "S = (\n. ?I n)"
by auto
have fIn: "f ` ?I n \ lmeasurable"
and mfIn: "?\ (f ` ?I n) \ integral S (\x. \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: "(\x. \det (matrix (f' x))\) integrable_on ?I n"
proof -
have "(\x. \det (matrix (f' x))\) absolutely_integrable_on S"
using int absolutely_integrable_integrable_bound by force
then have "(\x. \det (matrix (f' x))\) absolutely_integrable_on ?I n"
by (metis (no_types) Int_lower1 In fmeasurableD inf_commute set_integrable_subset)
then show ?thesis
using absolutely_integrable_on_def by blast
qed
ultimately have "f ` ?I n \ lmeasurable" "?\ (f ` ?I n) \ integral (?I n) (\x. \det (matrix (f' x))\)"
using m_diff_image_weak by metis+
moreover have "integral (?I n) (\x. \det (matrix (f' x))\) \ integral S (\x. \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 \<open>auto simp: subset_interval_imp_cart\<close>)
then have "(\k\n. f ` ?I k) = f ` ?I n" for n
by (fastforce simp add:)
with mfIn have "?\ (\k\n. f ` ?I k) \ integral S (\x. \det (matrix (f' x))\)" for n
by simp
then have "(\n. f ` ?I n) \ lmeasurable" "?\ (\n. f ` ?I n) \ integral S (\x. \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)) \
(\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> (g(Suc n) x)) \<and>
(\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>
(\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> 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. (\n. 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 "\na\N. g n x - f x \ dist (g na x) (f x)" for N
apply (rule_tac x="N+n" in exI)
using g [of N] by (auto simp: dist_norm)
with that show ?thesis
using diff_gt_0_iff_gt by blast
qed
with lim show ?thesis
apply (auto simp: lim_sequentially)
by (meson less_le_not_le not_le_imp_less)
qed
moreover
let ?\<Omega> = "\<lambda>n k. indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n}"
let ?g = "\n x. (\k::real | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * ?\ n k x)"
have "\g. (\n x. 0 \ g n x \ g n x \ (g(Suc n) x)) \
(\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>(\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> 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 \ \" and k: "\k\ \ 2 ^ (2*n)"
show "0 \ k/2^n * ?\ n k x"
using f \<open>k \<in> \<int>\<close> apply (auto simp: indicator_def field_split_simps Ints_def)
apply (drule spec [where x=x])
using zero_le_power [of "2::real" n] mult_nonneg_nonneg [of "f x" "2^n"]
by linarith
qed
show "?g n x \ ?g (Suc n) x" for n x
proof -
have "?g n x =
(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
k/2^n * (indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x +
indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x))"
by (rule sum.cong [OF refl]) (simp add: indicator_def field_split_simps)
also have "\ = (\k | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * indicator {y. k/2^n \ f y \ f y < (k+1/2)/2^n} x) +
(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x)"
by (simp add: comm_monoid_add_class.sum.distrib algebra_simps)
also have "\ = (\k | k \ \ \ \k\ \ 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k)/2 ^ Suc n \ f y \ f y < (2 * k+1)/2 ^ Suc n} x) +
(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2 * k+1) + 1)/2 ^ Suc n} x)"
by (force simp: field_simps indicator_def intro: sum.cong)
also have "\ \ (\k | k \ \ \ \k\ \ 2 ^ (2 * Suc n). k/2 ^ Suc n * (indicator {y. k/2 ^ Suc n \ f y \ f y < (k+1)/2 ^ Suc n} x))"
(is "?a + _ \ ?b")
proof -
have *: "\sum f I \ sum h I; a + sum h I \ b\ \ a + sum f I \ b" for I a b f and h :: "real\real"
by linarith
let ?h = "\k. (2*k+1)/2 ^ Suc n *
(indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2*k+1) + 1)/2 ^ Suc n} x)"
show ?thesis
proof (rule *)
show "(\k | k \ \ \ \k\ \ 2 ^ (2*n).
2 * k/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1 + 1)/2 ^ Suc n} x)
\<le> sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
by (rule sum_mono) (simp add: indicator_def field_split_simps)
next
have \<alpha>: "?a = (\<Sum>k \<in> (*)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
have \<beta>: "sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}
= (\<Sum>k \<in> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
have 0: "(*) 2 ` {k \ \. P k} \ (\x. 2 * x + 1) ` {k \ \. P k} = {}" for P :: "real \ bool"
proof -
have "2 * i \ 2 * j + 1" for i j :: int by arith
thus ?thesis
unfolding Ints_def by auto (use of_int_eq_iff in fastforce)
qed
have "?a + sum ?h {k \ \. \k\ \ 2 ^ (2*n)}
= (\<Sum>k \<in> (*)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
unfolding \<alpha> \<beta>
using finite_abs_int_segment [of "2 ^ (2*n)"]
by (subst sum_Un) (auto simp: 0)
also have "\ \ ?b"
proof (rule sum_mono2)
show "finite {k::real. k \ \ \ \k\ \ 2 ^ (2 * Suc n)}"
by (rule finite_abs_int_segment)
show "(*) 2 ` {k::real. k \ \ \ \k\ \ 2^(2*n)} \ (\x. 2*x + 1) ` {k \ \. \k\ \ 2^(2*n)} \ {k \ \. \k\ \ 2 ^ (2 * Suc n)}"
apply auto
using one_le_power [of "2::real" "2*n"] by linarith
have *: "\x \ (S \ T) - U; \x. x \ S \ x \ U; \x. x \ T \ x \ U\ \ P x" for S T U P
by blast
have "0 \ b" if "b \ \" "f x * (2 * 2^n) < b + 1" for b
proof -
have "0 \ f x * (2 * 2^n)"
by (simp add: f)
also have "\ < b+1"
by (simp add: that)
finally show "0 \ b"
using \<open>b \<in> \<int>\<close> by (auto simp: elim!: Ints_cases)
qed
then show "0 \ b/2 ^ Suc n * indicator {y. b/2 ^ Suc n \ f y \ f y < (b + 1)/2 ^ Suc n} x"
if "b \ {k \ \. \k\ \ 2 ^ (2 * Suc n)} -
((*) 2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})" for b
using that by (simp add: indicator_def divide_simps)
qed
finally show "?a + sum ?h {k \ \. \k\ \ 2 ^ (2*n)} \ ?b" .
qed
qed
finally show ?thesis .
qed
show "?g n \ borel_measurable lebesgue" for n
apply (intro borel_measurable_indicator borel_measurable_times borel_measurable_sum)
using leb_f sets_restrict_UNIV by auto
show "finite (range (?g n))" for n
proof -
have "(\k | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * ?\ n k x)
\<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}" for x
proof (cases "\k. k \ \ \ \k\ \ 2 ^ (2*n) \ k/2^n \ f x \ f x < (k+1)/2^n")
case True
then show ?thesis
by (blast intro: indicator_sum_eq)
next
case False
then have "(\k | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * ?\ n k x) = 0"
by auto
then show ?thesis by force
qed
then have "range (?g n) \ ((\k. (k/2^n)) ` {k. k \ \ \ \k\ \ 2 ^ (2*n)})"
by auto
moreover have "finite ((\k::real. (k/2^n)) ` {k \ \. \k\ \ 2 ^ (2*n)})"
by (intro finite_imageI finite_abs_int_segment)
ultimately show ?thesis
by (rule finite_subset)
qed
show "(\n. ?g n x) \ f x" for x
proof (clarsimp simp add: lim_sequentially)
fix e::real
assume "e > 0"
obtain N1 where N1: "2 ^ N1 > abs(f x)"
using real_arch_pow by fastforce
obtain N2 where N2: "(1/2) ^ N2 < e"
using real_arch_pow_inv \<open>e > 0\<close> by fastforce
have "dist (\k | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * ?\ n k x) (f x) < e" if "N1 + N2 \ n" for n
proof -
let ?m = "real_of_int \2^n * f x\"
have "\?m\ \ 2^n * 2^N1"
using N1 apply (simp add: f)
by (meson floor_mono le_floor_iff less_le_not_le mult_le_cancel_left_pos zero_less_numeral zero_less_power)
also have "\ \ 2 ^ (2*n)"
by (metis that add_leD1 add_le_cancel_left mult.commute mult_2_right one_less_numeral_iff
power_add power_increasing_iff semiring_norm(76))
finally have m_le: "\?m\ \ 2 ^ (2*n)" .
have "?m/2^n \ f x" "f x < (?m + 1)/2^n"
by (auto simp: mult.commute pos_divide_le_eq mult_imp_less_div_pos)
then have eq: "dist (\k | k \ \ \ \k\ \ 2 ^ (2*n). k/2^n * ?\ n k x) (f x)
= dist (?m/2^n) (f x)"
--> --------------------
--> maximum size reached
--> --------------------
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