(* Title: HOL/Probability/Lebesgue_Measure.thy Author: Johannes Hölzl, TU München Author: Robert Himmelmann, TU München *) header {* Lebsegue measure *} theory Lebesgue_Measure imports Finite_Product_Measure begin lemma absolutely_integrable_on_indicator[simp]: fixes A :: "'a::ordered_euclidean_space set" shows "((indicator A :: _ \ real) absolutely_integrable_on X) \ (indicator A :: _ \ real) integrable_on X" unfolding absolutely_integrable_on_def by simp lemma has_integral_indicator_UNIV: fixes s A :: "'a::ordered_euclidean_space set" and x :: real shows "((indicator (s \ A) :: 'a\real) has_integral x) UNIV = ((indicator s :: _\real) has_integral x) A" proof - have "(\x. if x \ A then indicator s x else 0) = (indicator (s \ A) :: _\real)" by (auto simp: fun_eq_iff indicator_def) then show ?thesis unfolding has_integral_restrict_univ[where s=A, symmetric] by simp qed lemma fixes s a :: "'a::ordered_euclidean_space set" shows integral_indicator_UNIV: "integral UNIV (indicator (s \ A) :: 'a\real) = integral A (indicator s :: _\real)" and integrable_indicator_UNIV: "(indicator (s \ A) :: 'a\real) integrable_on UNIV \ (indicator s :: 'a\real) integrable_on A" unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto subsection {* Standard Cubes *} definition cube :: "nat \ 'a::ordered_euclidean_space set" where "cube n \ {\i\Basis. - n *\<^sub>R i .. \i\Basis. n *\<^sub>R i}" lemma borel_cube[intro]: "cube n \ sets borel" unfolding cube_def by auto lemma cube_closed[intro]: "closed (cube n)" unfolding cube_def by auto lemma cube_subset[intro]: "n \ N \ cube n \ cube N" by (fastforce simp: eucl_le[where 'a='a] cube_def setsum_negf) lemma cube_subset_iff: "cube n \ cube N \ n \ N" unfolding cube_def subset_interval by (simp add: setsum_negf ex_in_conv) lemma ball_subset_cube: "ball (0::'a::ordered_euclidean_space) (real n) \ cube n" apply (simp add: cube_def subset_eq mem_interval setsum_negf eucl_le[where 'a='a]) proof safe fix x i :: 'a assume x: "x \ ball 0 (real n)" and i: "i \ Basis" thus "- real n \ x \ i" "real n \ x \ i" using Basis_le_norm[OF i, of x] by(auto simp: dist_norm) qed lemma mem_big_cube: obtains n where "x \ cube n" proof - from reals_Archimedean2[of "norm x"] guess n .. with ball_subset_cube[unfolded subset_eq, of n] show ?thesis by (intro that[where n=n]) (auto simp add: dist_norm) qed lemma cube_subset_Suc[intro]: "cube n \ cube (Suc n)" unfolding cube_def subset_interval by (simp add: setsum_negf) lemma has_integral_interval_cube: fixes a b :: "'a::ordered_euclidean_space" shows "(indicator {a .. b} has_integral content ({a .. b} \ cube n)) (cube n)" (is "(?I has_integral content ?R) (cube n)") proof - have [simp]: "(\x. if x \ cube n then ?I x else 0) = indicator ?R" by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a]) have "(?I has_integral content ?R) (cube n) \ (indicator ?R has_integral content ?R) UNIV" unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp also have "\ \ ((\x. 1::real) has_integral content ?R *\<^sub>R 1) ?R" unfolding indicator_def [abs_def] has_integral_restrict_univ real_scaleR_def mult_1_right .. also have "((\x. 1) has_integral content ?R *\<^sub>R 1) ?R" unfolding cube_def inter_interval by (rule has_integral_const) finally show ?thesis . qed subsection {* Lebesgue measure *} definition lebesgue :: "'a::ordered_euclidean_space measure" where "lebesgue = measure_of UNIV {A. \n. (indicator A :: 'a \ real) integrable_on cube n} (\A. SUP n. ereal (integral (cube n) (indicator A)))" lemma space_lebesgue[simp]: "space lebesgue = UNIV" unfolding lebesgue_def by simp lemma lebesgueI: "(\n. (indicator A :: _ \ real) integrable_on cube n) \ A \ sets lebesgue" unfolding lebesgue_def by simp lemma sigma_algebra_lebesgue: defines "leb \ {A. \n. (indicator A :: 'a::ordered_euclidean_space \ real) integrable_on cube n}" shows "sigma_algebra UNIV leb" proof (safe intro!: sigma_algebra_iff2[THEN iffD2]) fix A assume A: "A \ leb" moreover have "indicator (UNIV - A) = (\x. 1 - indicator A x :: real)" by (auto simp: fun_eq_iff indicator_def) ultimately show "UNIV - A \ leb" using A by (auto intro!: integrable_sub simp: cube_def leb_def) next fix n show "{} \ leb" by (auto simp: cube_def indicator_def[abs_def] leb_def) next fix A :: "nat \ _" assume A: "range A \ leb" have "\n. (indicator (\i. A i) :: _ \ real) integrable_on cube n" (is "\n. ?g integrable_on _") proof (intro dominated_convergence[where g="?g"] ballI allI) fix k n show "(indicator (\i real) integrable_on cube n" proof (induct k) case (Suc k) have *: "(\ i i A k" unfolding lessThan_Suc UN_insert by auto have *: "(\x. max (indicator (\ i ix. max (?f x) (?g x)) = _") by (auto simp: fun_eq_iff * indicator_def) show ?case using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: * leb_def subset_eq) qed auto qed (auto intro: LIMSEQ_indicator_UN simp: cube_def) then show "(\i. A i) \ leb" by (auto simp: leb_def) qed simp lemma sets_lebesgue: "sets lebesgue = {A. \n. (indicator A :: _ \ real) integrable_on cube n}" unfolding lebesgue_def sigma_algebra.sets_measure_of_eq[OF sigma_algebra_lebesgue] .. lemma lebesgueD: "A \ sets lebesgue \ (indicator A :: _ \ real) integrable_on cube n" unfolding sets_lebesgue by simp lemma emeasure_lebesgue: assumes "A \ sets lebesgue" shows "emeasure lebesgue A = (SUP n. ereal (integral (cube n) (indicator A)))" (is "_ = ?\ A") proof (rule emeasure_measure_of[OF lebesgue_def]) have *: "indicator {} = (\x. 0 :: real)" by (simp add: fun_eq_iff) show "positive (sets lebesgue) ?\" proof (unfold positive_def, intro conjI ballI) show "?\ {} = 0" by (simp add: integral_0 *) fix A :: "'a set" assume "A \ sets lebesgue" then show "0 \ ?\ A" by (auto intro!: SUP_upper2 Integration.integral_nonneg simp: sets_lebesgue) qed next show "countably_additive (sets lebesgue) ?\" proof (intro countably_additive_def[THEN iffD2] allI impI) fix A :: "nat \ 'a set" assume rA: "range A \ sets lebesgue" "disjoint_family A" then have A[simp, intro]: "\i n. (indicator (A i) :: _ \ real) integrable_on cube n" by (auto dest: lebesgueD) let ?m = "\n i. integral (cube n) (indicator (A i) :: _\real)" let ?M = "\n I. integral (cube n) (indicator (\i\I. A i) :: _\real)" have nn[simp, intro]: "\i n. 0 \ ?m n i" by (auto intro!: Integration.integral_nonneg) assume "(\i. A i) \ sets lebesgue" then have UN_A[simp, intro]: "\i n. (indicator (\i. A i) :: _ \ real) integrable_on cube n" by (auto simp: sets_lebesgue) show "(\n. ?\ (A n)) = ?\ (\i. A i)" proof (subst suminf_SUP_eq, safe intro!: incseq_SucI) fix i n show "ereal (?m n i) \ ereal (?m (Suc n) i)" using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI) next fix i n show "0 \ ereal (?m n i)" using rA unfolding lebesgue_def by (auto intro!: SUP_upper2 integral_nonneg) next show "(SUP n. \i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))" proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2]) fix n have "\m. (UNION {.. sets lebesgue" using rA by auto from lebesgueD[OF this] have "(\m. ?M n {..< m}) ----> ?M n UNIV" (is "(\m. integral _ (?A m)) ----> ?I") by (intro dominated_convergence(2)[where f="?A" and h="\x. 1::real"]) (auto intro: LIMSEQ_indicator_UN simp: cube_def) moreover { fix m have *: "(\xi sets lebesgue" using rA by auto then have "(indicator (\ireal) integrable_on (cube n)" by (auto dest!: lebesgueD) moreover have "(\i A m = {}" using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m] by auto then have "\x. indicator (\iim. \x = 0.. ?M n UNIV" by (simp add: atLeast0LessThan) qed qed qed qed (auto, fact) lemma lebesgueI_borel[intro, simp]: fixes s::"'a::ordered_euclidean_space set" assumes "s \ sets borel" shows "s \ sets lebesgue" proof - have "s \ sigma_sets (space lebesgue) (range (\(a, b). {a .. (b :: 'a\ordered_euclidean_space)}))" using assms by (simp add: borel_eq_atLeastAtMost) also have "\ \ sets lebesgue" proof (safe intro!: sets.sigma_sets_subset lebesgueI) fix n :: nat and a b :: 'a show "(indicator {a..b} :: 'a\real) integrable_on cube n" unfolding integrable_on_def using has_integral_interval_cube[of a b] by auto qed finally show ?thesis . qed lemma borel_measurable_lebesgueI: "f \ borel_measurable borel \ f \ borel_measurable lebesgue" unfolding measurable_def by simp lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set" assumes "negligible s" shows "s \ sets lebesgue" using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI) lemma lmeasure_eq_0: fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "emeasure lebesgue S = 0" proof - have "\n. integral (cube n) (indicator S :: 'a\real) = 0" unfolding lebesgue_integral_def using assms by (intro integral_unique some1_equality ex_ex1I) (auto simp: cube_def negligible_def) then show ?thesis using assms by (simp add: emeasure_lebesgue lebesgueI_negligible) qed lemma lmeasure_iff_LIMSEQ: assumes A: "A \ sets lebesgue" and "0 \ m" shows "emeasure lebesgue A = ereal m \ (\n. integral (cube n) (indicator A :: _ \ real)) ----> m" proof (subst emeasure_lebesgue[OF A], intro SUP_eq_LIMSEQ) show "mono (\n. integral (cube n) (indicator A::_=>real))" using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) qed lemma lmeasure_finite_has_integral: fixes s :: "'a::ordered_euclidean_space set" assumes "s \ sets lebesgue" "emeasure lebesgue s = ereal m" shows "(indicator s has_integral m) UNIV" proof - let ?I = "indicator :: 'a set \ 'a \ real" have "0 \ m" using emeasure_nonneg[of lebesgue s] `emeasure lebesgue s = ereal m` by simp have **: "(?I s) integrable_on UNIV \ (\k. integral UNIV (?I (s \ cube k))) ----> integral UNIV (?I s)" proof (intro monotone_convergence_increasing allI ballI) have LIMSEQ: "(\n. integral (cube n) (?I s)) ----> m" using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \ m`] . { fix n have "integral (cube n) (?I s) \ m" using cube_subset assms by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI) (auto dest!: lebesgueD) } moreover { fix n have "0 \ integral (cube n) (?I s)" using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) } ultimately show "bounded {integral UNIV (?I (s \ cube k)) |k. True}" unfolding bounded_def apply (rule_tac exI[of _ 0]) apply (rule_tac exI[of _ m]) by (auto simp: dist_real_def integral_indicator_UNIV) fix k show "?I (s \ cube k) integrable_on UNIV" unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) fix x show "?I (s \ cube k) x \ ?I (s \ cube (Suc k)) x" using cube_subset[of k "Suc k"] by (auto simp: indicator_def) next fix x :: 'a from mem_big_cube obtain k where k: "x \ cube k" . { fix n have "?I (s \ cube (n + k)) x = ?I s x" using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } note * = this show "(\k. ?I (s \ cube k) x) ----> ?I s x" by (rule LIMSEQ_offset[where k=k]) (auto simp: *) qed note ** = conjunctD2[OF this] have m: "m = integral UNIV (?I s)" apply (intro LIMSEQ_unique[OF _ **(2)]) using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \ m`] integral_indicator_UNIV . show ?thesis unfolding m by (intro integrable_integral **) qed lemma lmeasure_finite_integrable: assumes s: "s \ sets lebesgue" and "emeasure lebesgue s \ \" shows "(indicator s :: _ \ real) integrable_on UNIV" proof (cases "emeasure lebesgue s") case (real m) with lmeasure_finite_has_integral[OF `s \ sets lebesgue` this] emeasure_nonneg[of lebesgue s] show ?thesis unfolding integrable_on_def by auto qed (insert assms emeasure_nonneg[of lebesgue s], auto) lemma has_integral_lebesgue: assumes "((indicator s :: _\real) has_integral m) UNIV" shows "s \ sets lebesgue" proof (intro lebesgueI) let ?I = "indicator :: 'a set \ 'a \ real" fix n show "(?I s) integrable_on cube n" unfolding cube_def proof (intro integrable_on_subinterval) show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto qed auto qed lemma has_integral_lmeasure: assumes "((indicator s :: _\real) has_integral m) UNIV" shows "emeasure lebesgue s = ereal m" proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) let ?I = "indicator :: 'a set \ 'a \ real" show "s \ sets lebesgue" using has_integral_lebesgue[OF assms] . show "0 \ m" using assms by (rule has_integral_nonneg) auto have "(\n. integral UNIV (?I (s \ cube n))) ----> integral UNIV (?I s)" proof (intro dominated_convergence(2) ballI) show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto fix n show "?I (s \ cube n) integrable_on UNIV" unfolding integrable_indicator_UNIV using `s \ sets lebesgue` by (auto dest: lebesgueD) fix x show "norm (?I (s \ cube n) x) \ ?I s x" by (auto simp: indicator_def) next fix x :: 'a from mem_big_cube obtain k where k: "x \ cube k" . { fix n have "?I (s \ cube (n + k)) x = ?I s x" using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } note * = this show "(\k. ?I (s \ cube k) x) ----> ?I s x" by (rule LIMSEQ_offset[where k=k]) (auto simp: *) qed then show "(\n. integral (cube n) (?I s)) ----> m" unfolding integral_unique[OF assms] integral_indicator_UNIV by simp qed lemma has_integral_iff_lmeasure: "(indicator A has_integral m) UNIV \ (A \ sets lebesgue \ emeasure lebesgue A = ereal m)" proof assume "(indicator A has_integral m) UNIV" with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] show "A \ sets lebesgue \ emeasure lebesgue A = ereal m" by (auto intro: has_integral_nonneg) next assume "A \ sets lebesgue \ emeasure lebesgue A = ereal m" then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto qed lemma lmeasure_eq_integral: assumes "(indicator s::_\real) integrable_on UNIV" shows "emeasure lebesgue s = ereal (integral UNIV (indicator s))" using assms unfolding integrable_on_def proof safe fix y :: real assume "(indicator s has_integral y) UNIV" from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] show "emeasure lebesgue s = ereal (integral UNIV (indicator s))" by simp qed lemma lebesgue_simple_function_indicator: fixes f::"'a::ordered_euclidean_space \ ereal" assumes f:"simple_function lebesgue f" shows "f = (\x. (\y \ f ` UNIV. y * indicator (f -` {y}) x))" by (rule, subst simple_function_indicator_representation[OF f]) auto lemma integral_eq_lmeasure: "(indicator s::_\real) integrable_on UNIV \ integral UNIV (indicator s) = real (emeasure lebesgue s)" by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) lemma lmeasure_finite: assumes "(indicator s::_\real) integrable_on UNIV" shows "emeasure lebesgue s \ \" using lmeasure_eq_integral[OF assms] by auto lemma negligible_iff_lebesgue_null_sets: "negligible A \ A \ null_sets lebesgue" proof assume "negligible A" from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] show "A \ null_sets lebesgue" by auto next assume A: "A \ null_sets lebesgue" then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by (auto simp: null_sets_def) show "negligible A" unfolding negligible_def proof (intro allI) fix a b :: 'a have integrable: "(indicator A :: _\real) integrable_on {a..b}" by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) then have "integral {a..b} (indicator A) \ (integral UNIV (indicator A) :: real)" using * by (auto intro!: integral_subset_le) moreover have "(0::real) \ integral {a..b} (indicator A)" using integrable by (auto intro!: integral_nonneg) ultimately have "integral {a..b} (indicator A) = (0::real)" using integral_unique[OF *] by auto then show "(indicator A has_integral (0::real)) {a..b}" using integrable_integral[OF integrable] by simp qed qed lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \" proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI) fix n :: nat have "indicator UNIV = (\x::'a. 1 :: real)" by auto moreover { have "real n \ (2 * real n) ^ DIM('a)" proof (cases n) case 0 then show ?thesis by auto next case (Suc n') have "real n \ (2 * real n)^1" by auto also have "(2 * real n)^1 \ (2 * real n) ^ DIM('a)" using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc simp del: DIM_positive) finally show ?thesis . qed } ultimately show "ereal (real n) \ ereal (integral (cube n) (indicator UNIV::'a\real))" using integral_const DIM_positive[where 'a='a] by (auto simp: cube_def content_closed_interval_cases setprod_constant setsum_negf) qed simp lemma lmeasure_complete: "A \ B \ B \ null_sets lebesgue \ A \ null_sets lebesgue" unfolding negligible_iff_lebesgue_null_sets[symmetric] by (auto simp: negligible_subset) lemma fixes a b ::"'a::ordered_euclidean_space" shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {a..b} = ereal (content {a..b})" proof - have "(indicator (UNIV \ {a..b})::_\real) integrable_on UNIV" unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def]) from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV by (simp add: indicator_def [abs_def]) qed lemma lmeasure_singleton[simp]: fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0" using lmeasure_atLeastAtMost[of a a] by simp lemma AE_lebesgue_singleton: fixes a :: "'a::ordered_euclidean_space" shows "AE x in lebesgue. x \ a" by (rule AE_I[where N="{a}"]) auto declare content_real[simp] lemma fixes a b :: real shows lmeasure_real_greaterThanAtMost[simp]: "emeasure lebesgue {a <.. b} = ereal (if a \ b then b - a else 0)" proof - have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b}" using AE_lebesgue_singleton[of a] by (intro emeasure_eq_AE) auto then show ?thesis by auto qed lemma fixes a b :: real shows lmeasure_real_atLeastLessThan[simp]: "emeasure lebesgue {a ..< b} = ereal (if a \ b then b - a else 0)" proof - have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b}" using AE_lebesgue_singleton[of b] by (intro emeasure_eq_AE) auto then show ?thesis by auto qed lemma fixes a b :: real shows lmeasure_real_greaterThanLessThan[simp]: "emeasure lebesgue {a <..< b} = ereal (if a \ b then b - a else 0)" proof - have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a .. b}" using AE_lebesgue_singleton[of a] AE_lebesgue_singleton[of b] by (intro emeasure_eq_AE) auto then show ?thesis by auto qed subsection {* Lebesgue-Borel measure *} definition "lborel = measure_of UNIV (sets borel) (emeasure lebesgue)" lemma shows space_lborel[simp]: "space lborel = UNIV" and sets_lborel[simp]: "sets lborel = sets borel" and measurable_lborel1[simp]: "measurable lborel = measurable borel" and measurable_lborel2[simp]: "measurable A lborel = measurable A borel" using sets.sigma_sets_eq[of borel] by (auto simp add: lborel_def measurable_def[abs_def]) lemma emeasure_lborel[simp]: "A \ sets borel \ emeasure lborel A = emeasure lebesgue A" by (rule emeasure_measure_of[OF lborel_def]) (auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure) interpretation lborel: sigma_finite_measure lborel proof (default, intro conjI exI[of _ "\n. cube n"]) show "range cube \ sets lborel" by (auto intro: borel_closed) { fix x :: 'a have "\n. x\cube n" using mem_big_cube by auto } then show "(\i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto show "\i. emeasure lborel (cube i) \ \" by (simp add: cube_def) qed interpretation lebesgue: sigma_finite_measure lebesgue proof from lborel.sigma_finite guess A :: "nat \ 'a set" .. then show "\A::nat \ 'a set. range A \ sets lebesgue \ (\i. A i) = space lebesgue \ (\i. emeasure lebesgue (A i) \ \)" by (intro exI[of _ A]) (auto simp: subset_eq) qed lemma Int_stable_atLeastAtMost: fixes x::"'a::ordered_euclidean_space" shows "Int_stable (range (\(a, b::'a). {a..b}))" by (auto simp: inter_interval Int_stable_def) lemma lborel_eqI: fixes M :: "'a::ordered_euclidean_space measure" assumes emeasure_eq: "\a b. emeasure M {a .. b} = content {a .. b}" assumes sets_eq: "sets M = sets borel" shows "lborel = M" proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost]) let ?P = "\\<^sub>M i\{.. Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E" by (simp_all add: borel_eq_atLeastAtMost sets_eq) show "range cube \ ?E" unfolding cube_def [abs_def] by auto { fix x :: 'a have "\n. x \ cube n" using mem_big_cube[of x] by fastforce } then show "(\i. cube i :: 'a set) = UNIV" by auto { fix i show "emeasure lborel (cube i) \ \" unfolding cube_def by auto } { fix X assume "X \ ?E" then show "emeasure lborel X = emeasure M X" by (auto simp: emeasure_eq) } qed lemma lebesgue_real_affine: fixes c :: real assumes "c \ 0" shows "lborel = density (distr lborel borel (\x. t + c * x)) (\_. \c\)" (is "_ = ?D") proof (rule lborel_eqI) fix a b show "emeasure ?D {a..b} = content {a .. b}" proof cases assume "0 < c" then have "(\x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}" by (auto simp: field_simps) with `0 < c` show ?thesis by (cases "a \ b") (auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult borel_measurable_indicator' emeasure_distr) next assume "\ 0 < c" with `c \ 0` have "c < 0" by auto then have *: "(\x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}" by (auto simp: field_simps) with `c < 0` show ?thesis by (cases "a \ b") (auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult borel_measurable_indicator' emeasure_distr) qed qed simp lemma lebesgue_integral_real_affine: fixes c :: real assumes c: "c \ 0" and f: "f \ borel_measurable borel" shows "(\ x. f x \ lborel) = \c\ * (\ x. f (t + c * x) \lborel)" by (subst lebesgue_real_affine[OF c, of t]) (simp add: f integral_density integral_distr lebesgue_integral_cmult) subsection {* Lebesgue integrable implies Gauge integrable *} lemma simple_function_has_integral: fixes f::"'a::ordered_euclidean_space \ ereal" assumes f:"simple_function lebesgue f" and f':"range f \ {0..<\}" and om:"\x. x \ range f \ emeasure lebesgue (f -` {x} \ UNIV) = \ \ x = 0" shows "((\x. real (f x)) has_integral (real (integral\<^sup>S lebesgue f))) UNIV" unfolding simple_integral_def space_lebesgue proof (subst lebesgue_simple_function_indicator) let ?M = "\x. emeasure lebesgue (f -` {x} \ UNIV)" let ?F = "\x. indicator (f -` {x})" { fix x y assume "y \ range f" from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)" by (cases rule: ereal2_cases[of y "?F y x"]) (auto simp: indicator_def one_ereal_def split: split_if_asm) } moreover { fix x assume x: "x\range f" have "x * ?M x = real x * real (?M x)" proof cases assume "x \ 0" with om[OF x] have "?M x \ \" by auto with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis by (cases rule: ereal2_cases[of x "?M x"]) auto qed simp } ultimately have "((\x. real (\y\range f. y * ?F y x)) has_integral real (\x\range f. x * ?M x)) UNIV \ ((\x. \y\range f. real y * ?F y x) has_integral (\x\range f. real x * real (?M x))) UNIV" by simp also have \ proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral real_of_ereal_pos emeasure_nonneg ballI) show *: "finite (range f)" "\y. f -` {y} \ sets lebesgue" using simple_functionD[OF f] by auto fix y assume "real y \ 0" "y \ range f" with * om[OF this(2)] show "emeasure lebesgue (f -` {y}) = ereal (real (?M y))" by (auto simp: ereal_real) qed finally show "((\x. real (\y\range f. y * ?F y x)) has_integral real (\x\range f. x * ?M x)) UNIV" . qed fact lemma simple_function_has_integral': fixes f::"'a::ordered_euclidean_space \ ereal" assumes f: "simple_function lebesgue f" "\x. 0 \ f x" and i: "integral\<^sup>S lebesgue f \ \" shows "((\x. real (f x)) has_integral (real (integral\<^sup>S lebesgue f))) UNIV" proof - let ?f = "\x. if x \ f -` {\} then 0 else f x" note f(1)[THEN simple_functionD(2)] then have [simp, intro]: "\X. f -` X \ sets lebesgue" by auto have f': "simple_function lebesgue ?f" using f by (intro simple_function_If_set) auto have rng: "range ?f \ {0..<\}" using f by auto have "AE x in lebesgue. f x = ?f x" using simple_integral_PInf[OF f i] by (intro AE_I[where N="f -` {\} \ space lebesgue"]) auto from f(1) f' this have eq: "integral\<^sup>S lebesgue f = integral\<^sup>S lebesgue ?f" by (rule simple_integral_cong_AE) have real_eq: "\x. real (f x) = real (?f x)" by auto show ?thesis unfolding eq real_eq proof (rule simple_function_has_integral[OF f' rng]) fix x assume x: "x \ range ?f" and inf: "emeasure lebesgue (?f -` {x} \ UNIV) = \" have "x * emeasure lebesgue (?f -` {x} \ UNIV) = (\\<^sup>S y. x * indicator (?f -` {x}) y \lebesgue)" using f'[THEN simple_functionD(2)] by (simp add: simple_integral_cmult_indicator) also have "\ \ integral\<^sup>S lebesgue f" using f'[THEN simple_functionD(2)] f by (intro simple_integral_mono simple_function_mult simple_function_indicator) (auto split: split_indicator) finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm) qed qed lemma positive_integral_has_integral: fixes f :: "'a::ordered_euclidean_space \ ereal" assumes f: "f \ borel_measurable lebesgue" "range f \ {0..<\}" "integral\<^sup>P lebesgue f \ \" shows "((\x. real (f x)) has_integral (real (integral\<^sup>P lebesgue f))) UNIV" proof - from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u . note u = this have SUP_eq: "\x. (SUP i. u i x) = f x" using u(4) f(2)[THEN subsetD] by (auto split: split_max) let ?u = "\i x. real (u i x)" note u_eq = positive_integral_eq_simple_integral[OF u(1,5), symmetric] { fix i note u_eq also have "integral\<^sup>P lebesgue (u i) \ (\\<^sup>+x. max 0 (f x) \lebesgue)" by (intro positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric]) finally have "integral\<^sup>S lebesgue (u i) \ \" unfolding positive_integral_max_0 using f by auto } note u_fin = this then have u_int: "\i. (?u i has_integral real (integral\<^sup>S lebesgue (u i))) UNIV" by (rule simple_function_has_integral'[OF u(1,5)]) have "\x. \r\0. f x = ereal r" proof fix x from f(2) have "0 \ f x" "f x \ \" by (auto simp: subset_eq) then show "\r\0. f x = ereal r" by (cases "f x") auto qed from choice[OF this] obtain f' where f': "f = (\x. ereal (f' x))" "\x. 0 \ f' x" by auto have "\i. \r. \x. 0 \ r x \ u i x = ereal (r x)" proof fix i show "\r. \x. 0 \ r x \ u i x = ereal (r x)" proof (intro choice allI) fix i x have "u i x \ \" using u(3)[of i] by (auto simp: image_iff) metis then show "\r\0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto qed qed from choice[OF this] obtain u' where u': "u = (\i x. ereal (u' i x))" "\i x. 0 \ u' i x" by (auto simp: fun_eq_iff) have convergent: "f' integrable_on UNIV \ (\k. integral UNIV (u' k)) ----> integral UNIV f'" proof (intro monotone_convergence_increasing allI ballI) show int: "\k. (u' k) integrable_on UNIV" using u_int unfolding integrable_on_def u' by auto show "\k x. u' k x \ u' (Suc k) x" using u(2,3,5) by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono) show "\x. (\k. u' k x) ----> f' x" using SUP_eq u(2) by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_Suc_iff le_fun_def) show "bounded {integral UNIV (u' k)|k. True}" proof (safe intro!: bounded_realI) fix k have "\integral UNIV (u' k)\ = integral UNIV (u' k)" by (intro abs_of_nonneg integral_nonneg int ballI u') also have "\ = real (integral\<^sup>S lebesgue (u k))" using u_int[THEN integral_unique] by (simp add: u') also have "\ = real (integral\<^sup>P lebesgue (u k))" using positive_integral_eq_simple_integral[OF u(1,5)] by simp also have "\ \ real (integral\<^sup>P lebesgue f)" using f by (auto intro!: real_of_ereal_positive_mono positive_integral_positive positive_integral_mono SUP_upper simp: SUP_eq[symmetric]) finally show "\integral UNIV (u' k)\ \ real (integral\<^sup>P lebesgue f)" . qed qed have "integral\<^sup>P lebesgue f = ereal (integral UNIV f')" proof (rule tendsto_unique[OF trivial_limit_sequentially]) have "(\i. integral\<^sup>S lebesgue (u i)) ----> (SUP i. integral\<^sup>P lebesgue (u i))" unfolding u_eq by (intro LIMSEQ_SUP incseq_positive_integral u) also note positive_integral_monotone_convergence_SUP [OF u(2) borel_measurable_simple_function[OF u(1)] u(5), symmetric] finally show "(\k. integral\<^sup>S lebesgue (u k)) ----> integral\<^sup>P lebesgue f" unfolding SUP_eq . { fix k have "0 \ integral\<^sup>S lebesgue (u k)" using u by (auto intro!: simple_integral_positive) then have "integral\<^sup>S lebesgue (u k) = ereal (real (integral\<^sup>S lebesgue (u k)))" using u_fin by (auto simp: ereal_real) } note * = this show "(\k. integral\<^sup>S lebesgue (u k)) ----> ereal (integral UNIV f')" using convergent using u_int[THEN integral_unique, symmetric] by (subst *) (simp add: u') qed then show ?thesis using convergent by (simp add: f' integrable_integral) qed lemma lebesgue_integral_has_integral: fixes f :: "'a::ordered_euclidean_space \ real" assumes f: "integrable lebesgue f" shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV" proof - let ?n = "\x. real (ereal (max 0 (- f x)))" and ?p = "\x. real (ereal (max 0 (f x)))" have *: "f = (\x. ?p x - ?n x)" by (auto simp del: ereal_max) { fix f :: "'a \ real" have "(\\<^sup>+ x. ereal (f x) \lebesgue) = (\\<^sup>+ x. ereal (max 0 (f x)) \lebesgue)" by (intro positive_integral_cong_pos) (auto split: split_max) } note eq = this show ?thesis unfolding lebesgue_integral_def apply (subst *) apply (rule has_integral_sub) unfolding eq[of f] eq[of "\x. - f x"] apply (safe intro!: positive_integral_has_integral) using integrableD[OF f] by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max intro!: measurable_If) qed lemma lebesgue_simple_integral_eq_borel: assumes f: "f \ borel_measurable borel" shows "integral\<^sup>S lebesgue f = integral\<^sup>S lborel f" using f[THEN measurable_sets] by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_lborel[symmetric] simp: simple_integral_def) lemma lebesgue_positive_integral_eq_borel: assumes f: "f \ borel_measurable borel" shows "integral\<^sup>P lebesgue f = integral\<^sup>P lborel f" proof - from f have "integral\<^sup>P lebesgue (\x. max 0 (f x)) = integral\<^sup>P lborel (\x. max 0 (f x))" by (auto intro!: positive_integral_subalgebra[symmetric]) then show ?thesis unfolding positive_integral_max_0 . qed lemma lebesgue_integral_eq_borel: assumes "f \ borel_measurable borel" shows "integrable lebesgue f \ integrable lborel f" (is ?P) and "integral\<^sup>L lebesgue f = integral\<^sup>L lborel f" (is ?I) proof - have "sets lborel \ sets lebesgue" by auto from integral_subalgebra[of f lborel, OF _ this _ _] assms show ?P ?I by auto qed lemma borel_integral_has_integral: fixes f::"'a::ordered_euclidean_space => real" assumes f:"integrable lborel f" shows "(f has_integral (integral\<^sup>L lborel f)) UNIV" proof - have borel: "f \ borel_measurable borel" using f unfolding integrable_def by auto from f show ?thesis using lebesgue_integral_has_integral[of f] unfolding lebesgue_integral_eq_borel[OF borel] by simp qed lemma positive_integral_lebesgue_has_integral: fixes f :: "'a::ordered_euclidean_space \ real" assumes f_borel: "f \ borel_measurable lebesgue" and nonneg: "\x. 0 \ f x" assumes I: "(f has_integral I) UNIV" shows "(\\<^sup>+x. f x \lebesgue) = I" proof - from f_borel have "(\x. ereal (f x)) \ borel_measurable lebesgue" by auto from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this have "(\\<^sup>+ x. ereal (f x) \lebesgue) = (SUP i. integral\<^sup>S lebesgue (F i))" using F by (subst positive_integral_monotone_convergence_simple) (simp_all add: positive_integral_max_0 simple_function_def) also have "\ \ ereal I" proof (rule SUP_least) fix i :: nat { fix z from F(4)[of z] have "F i z \ ereal (f z)" by (metis SUP_upper UNIV_I ereal_max_0 max.absorb2 nonneg) with F(5)[of i z] have "real (F i z) \ f z" by (cases "F i z") simp_all } note F_bound = this { fix x :: ereal assume x: "x \ 0" "x \ range (F i)" with F(3,5)[of i] have [simp]: "real x \ 0" by (metis image_iff order_eq_iff real_of_ereal_le_0) let ?s = "(\n z. real x * indicator (F i -` {x} \ cube n) z) :: nat \ 'a \ real" have "(\z::'a. real x * indicator (F i -` {x}) z) integrable_on UNIV" proof (rule dominated_convergence(1)) fix n :: nat have "(\z. indicator (F i -` {x} \ cube n) z :: real) integrable_on cube n" using x F(1)[of i] by (intro lebesgueD) (auto simp: simple_function_def) then have cube: "?s n integrable_on cube n" by (simp add: integrable_on_cmult_iff) show "?s n integrable_on UNIV" by (rule integrable_on_superset[OF _ _ cube]) auto next show "f integrable_on UNIV" unfolding integrable_on_def using I by auto next fix n from F_bound show "\x\UNIV. norm (?s n x) \ f x" using nonneg F(5) by (auto split: split_indicator) next show "\z\UNIV. (\n. ?s n z) ----> real x * indicator (F i -` {x}) z" proof fix z :: 'a from mem_big_cube[of z] guess j . then have x: "eventually (\n. ?s n z = real x * indicator (F i -` {x}) z) sequentially" by (auto intro!: eventually_sequentiallyI[where c=j] dest!: cube_subset split: split_indicator) then show "(\n. ?s n z) ----> real x * indicator (F i -` {x}) z" by (rule Lim_eventually) qed qed then have "(indicator (F i -` {x}) :: 'a \ real) integrable_on UNIV" by (simp add: integrable_on_cmult_iff) } note F_finite = lmeasure_finite[OF this] have "((\x. real (F i x)) has_integral real (integral\<^sup>S lebesgue (F i))) UNIV" proof (rule simple_function_has_integral[of "F i"]) show "simple_function lebesgue (F i)" using F(1) by (simp add: simple_function_def) show "range (F i) \ {0..<\}" using F(3,5)[of i] by (auto simp: image_iff) metis next fix x assume "x \ range (F i)" "emeasure lebesgue (F i -` {x} \ UNIV) = \" with F_finite[of x] show "x = 0" by auto qed from this I have "real (integral\<^sup>S lebesgue (F i)) \ I" by (rule has_integral_le) (intro ballI F_bound) moreover { fix x assume x: "x \ range (F i)" with F(3,5)[of i] have "x = 0 \ (0 < x \ x \ \)" by (auto simp: image_iff le_less) metis with F_finite[OF _ x] x have "x * emeasure lebesgue (F i -` {x} \ UNIV) \ \" by auto } then have "integral\<^sup>S lebesgue (F i) \ \" unfolding simple_integral_def setsum_Pinfty space_lebesgue by blast moreover have "0 \ integral\<^sup>S lebesgue (F i)" using F(1,5) by (intro simple_integral_positive) (auto simp: simple_function_def) ultimately show "integral\<^sup>S lebesgue (F i) \ ereal I" by (cases "integral\<^sup>S lebesgue (F i)") auto qed also have "\ < \" by simp finally have finite: "(\\<^sup>+ x. ereal (f x) \lebesgue) \ \" by simp have borel: "(\x. ereal (f x)) \ borel_measurable lebesgue" using f_borel by (auto intro: borel_measurable_lebesgueI) from positive_integral_has_integral[OF borel _ finite] have "(f has_integral real (\\<^sup>+ x. ereal (f x) \lebesgue)) UNIV" using nonneg by (simp add: subset_eq) with I have "I = real (\\<^sup>+ x. ereal (f x) \lebesgue)" by (rule has_integral_unique) with finite positive_integral_positive[of _ "\x. ereal (f x)"] show ?thesis by (cases "\\<^sup>+ x. ereal (f x) \lebesgue") auto qed lemma has_integral_iff_positive_integral_lebesgue: fixes f :: "'a::ordered_euclidean_space \ real" assumes f: "f \ borel_measurable lebesgue" "\x. 0 \ f x" shows "(f has_integral I) UNIV \ integral\<^sup>P lebesgue f = I" using f positive_integral_lebesgue_has_integral[of f I] positive_integral_has_integral[of f] by (auto simp: subset_eq) lemma has_integral_iff_positive_integral_lborel: fixes f :: "'a::ordered_euclidean_space \ real" assumes f: "f \ borel_measurable borel" "\x. 0 \ f x" shows "(f has_integral I) UNIV \ integral\<^sup>P lborel f = I" using assms by (subst has_integral_iff_positive_integral_lebesgue) (auto simp: borel_measurable_lebesgueI lebesgue_positive_integral_eq_borel) subsection {* Equivalence between product spaces and euclidean spaces *} definition e2p :: "'a::ordered_euclidean_space \ ('a \ real)" where "e2p x = (\i\Basis. x \ i)" definition p2e :: "('a \ real) \ 'a::ordered_euclidean_space" where "p2e x = (\i\Basis. x i *\<^sub>R i)" lemma e2p_p2e[simp]: "x \ extensional Basis \ e2p (p2e x::'a::ordered_euclidean_space) = x" by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) lemma p2e_e2p[simp]: "p2e (e2p x) = (x::'a::ordered_euclidean_space)" by (auto simp: euclidean_eq_iff[where 'a='a] p2e_def e2p_def) interpretation lborel_product: product_sigma_finite "\x. lborel::real measure" by default interpretation lborel_space: finite_product_sigma_finite "\x. lborel::real measure" "Basis" by default auto lemma sets_product_borel: assumes I: "finite I" shows "sets (\\<^sub>M i\I. lborel) = sigma_sets (\\<^sub>E i\I. UNIV) { \\<^sub>E i\I. {..< x i :: real} | x. True}" (is "_ = ?G") proof (subst sigma_prod_algebra_sigma_eq[where S="\_ i::nat. {..M I (\i. lborel))) {Pi\<^sub>E I F |F. \i\I. F i \ range lessThan} = ?G" by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff) qed (auto simp: borel_eq_lessThan eucl_lessThan reals_Archimedean2) lemma measurable_e2p[measurable]: "e2p \ measurable (borel::'a::ordered_euclidean_space measure) (\\<^sub>M (i::'a)\Basis. (lborel :: real measure))" proof (rule measurable_sigma_sets[OF sets_product_borel]) fix A :: "('a \ real) set" assume "A \ {\\<^sub>E (i::'a)\Basis. {..\<^sub>E (i::'a)\Basis. {..i\Basis. x i *\<^sub>R i)}" using DIM_positive by (auto simp add: set_eq_iff e2p_def eucl_less_def) then show "e2p -` A \ space (borel::'a measure) \ sets borel" by simp qed (auto simp: e2p_def) (* FIXME: conversion in measurable prover *) lemma lborelD_Collect[measurable (raw)]: "{x\space borel. P x} \ sets borel \ {x\space lborel. P x} \ sets lborel" by simp lemma lborelD[measurable (raw)]: "A \ sets borel \ A \ sets lborel" by simp lemma measurable_p2e[measurable]: "p2e \ measurable (\\<^sub>M (i::'a)\Basis. (lborel :: real measure)) (borel :: 'a::ordered_euclidean_space measure)" (is "p2e \ measurable ?P _") proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2]) fix x and i :: 'a let ?A = "{w \ space ?P. (p2e w :: 'a) \ i \ x}" assume "i \ Basis" then have "?A = (\\<^sub>E j\Basis. if i = j then {.. x} else UNIV)" using DIM_positive by (auto simp: space_PiM p2e_def PiE_def split: split_if_asm) then show "?A \ sets ?P" by auto qed lemma lborel_eq_lborel_space: "(lborel :: 'a measure) = distr (\\<^sub>M (i::'a::ordered_euclidean_space)\Basis. lborel) borel p2e" (is "?B = ?D") proof (rule lborel_eqI) show "sets ?D = sets borel" by simp let ?P = "(\\<^sub>M (i::'a)\Basis. lborel)" fix a b :: 'a have *: "p2e -` {a .. b} \ space ?P = (\\<^sub>E i\Basis. {a \ i .. b \ i})" by (auto simp: eucl_le[where 'a='a] p2e_def space_PiM PiE_def Pi_iff) have "emeasure ?P (p2e -` {a..b} \ space ?P) = content {a..b}" proof cases assume "{a..b} \ {}" then have "a \ b" by (simp add: interval_ne_empty eucl_le[where 'a='a]) then have "emeasure lborel {a..b} = (\x\Basis. emeasure lborel {a \ x .. b \ x})" by (auto simp: content_closed_interval eucl_le[where 'a='a] intro!: setprod_ereal[symmetric]) also have "\ = emeasure ?P (p2e -` {a..b} \ space ?P)" unfolding * by (subst lborel_space.measure_times) auto finally show ?thesis by simp qed simp then show "emeasure ?D {a .. b} = content {a .. b}" by (simp add: emeasure_distr measurable_p2e) qed lemma borel_fubini_positiv_integral: fixes f :: "'a::ordered_euclidean_space \ ereal" assumes f: "f \ borel_measurable borel" shows "integral\<^sup>P lborel f = \\<^sup>+x. f (p2e x) \(\\<^sub>M (i::'a)\Basis. lborel)" by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f) lemma borel_fubini_integrable: fixes f :: "'a::ordered_euclidean_space \ real" shows "integrable lborel f \ integrable (\\<^sub>M (i::'a)\Basis. lborel) (\x. f (p2e x))" (is "_ \ integrable ?B ?f") proof assume *: "integrable lborel f" then have f: "f \ borel_measurable borel" by auto with measurable_p2e have "f \ p2e \ borel_measurable ?B" by (rule measurable_comp) with * f show "integrable ?B ?f" by (simp add: comp_def borel_fubini_positiv_integral integrable_def) next assume *: "integrable ?B ?f" then have "?f \ e2p \ borel_measurable (borel::'a measure)" by (auto intro!: measurable_e2p) then have "f \ borel_measurable borel" by (simp cong: measurable_cong) with * show "integrable lborel f" by (simp add: borel_fubini_positiv_integral integrable_def) qed lemma borel_fubini: fixes f :: "'a::ordered_euclidean_space \ real" assumes f: "f \ borel_measurable borel" shows "integral\<^sup>L lborel f = \x. f (p2e x) \((\\<^sub>M (i::'a)\Basis. lborel))" using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def) lemma integrable_on_borel_integrable: fixes f :: "'a::ordered_euclidean_space \ real" assumes f_borel: "f \ borel_measurable borel" and nonneg: "\x. 0 \ f x" assumes f: "f integrable_on UNIV" shows "integrable lborel f" proof - have "(\\<^sup>+ x. ereal (f x) \lborel) \ \" using has_integral_iff_positive_integral_lborel[OF f_borel nonneg] f by (auto simp: integrable_on_def) moreover have "(\\<^sup>+ x. ereal (- f x) \lborel) = 0" using f_borel nonneg by (subst positive_integral_0_iff_AE) auto ultimately show ?thesis using f_borel by (auto simp: integrable_def) qed subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *} lemma borel_integrable_atLeastAtMost: fixes a b :: real assumes f: "\x. a \ x \ x \ b \ isCont f x" shows "integrable lborel (\x. f x * indicator {a .. b} x)" (is "integrable _ ?f") proof cases assume "a \ b" from isCont_Lb_Ub[OF `a \ b`, of f] f obtain M L where bounds: "\x. a \ x \ x \ b \ f x \ M" "\x. a \ x \ x \ b \ L \ f x" by metis show ?thesis proof (rule integrable_bound) show "integrable lborel (\x. max \M\ \L\ * indicator {a..b} x)" by (rule integral_cmul_indicator) simp_all show "AE x in lborel. \?f x\ \ max \M\ \L\ * indicator {a..b} x" proof (rule AE_I2) fix x show "\?f x\ \ max \M\ \L\ * indicator {a..b} x" using bounds[of x] by (auto split: split_indicator) qed let ?g = "\x. if x = a then f a else if x = b then f b else if x \ {a <..< b} then f x else 0" from f have "continuous_on {a <..< b} f" by (subst continuous_on_eq_continuous_at) auto then have "?g \ borel_measurable borel" using borel_measurable_continuous_on_open[of "{a <..< b }" f "\x. x" borel 0] by (auto intro!: measurable_If[where P="\x. x = a"] measurable_If[where P="\x. x = b"]) also have "?g = ?f" using `a \ b` by (intro ext) (auto split: split_indicator) finally show "?f \ borel_measurable lborel" by simp qed qed simp lemma has_field_derivative_subset: "(f has_field_derivative y) (at x within s) \ t \ s \ (f has_field_derivative y) (at x within t)" unfolding has_field_derivative_def by (rule has_derivative_subset) lemma integral_FTC_atLeastAtMost: fixes a b :: real assumes "a \ b" and F: "\x. a \ x \ x \ b \ DERIV F x :> f x" and f: "\x. a \ x \ x \ b \ isCont f x" shows "integral\<^sup>L lborel (\x. f x * indicator {a .. b} x) = F b - F a" proof - let ?f = "\x. f x * indicator {a .. b} x" have "(?f has_integral (\x. ?f x \lborel)) UNIV" using borel_integrable_atLeastAtMost[OF f] by (rule borel_integral_has_integral) moreover have "(f has_integral F b - F a) {a .. b}" by (intro fundamental_theorem_of_calculus) (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] intro: has_field_derivative_subset[OF F] assms(1)) then have "(?f has_integral F b - F a) {a .. b}" by (subst has_integral_eq_eq[where g=f]) auto then have "(?f has_integral F b - F a) UNIV" by (intro has_integral_on_superset[where t=UNIV and s="{a..b}"]) auto ultimately show "integral\<^sup>L lborel ?f = F b - F a" by (rule has_integral_unique) qed text {* For the positive integral we replace continuity with Borel-measurability. *} lemma positive_integral_FTC_atLeastAtMost: assumes f_borel: "f \ borel_measurable borel" assumes f: "\x. x \ {a..b} \ DERIV F x :> f x" "\x. x \ {a..b} \ 0 \ f x" and "a \ b" shows "(\\<^sup>+x. f x * indicator {a .. b} x \lborel) = F b - F a" proof - have i: "(f has_integral F b - F a) {a..b}" by (intro fundamental_theorem_of_calculus) (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] intro: has_field_derivative_subset[OF f(1)] `a \ b`) have i: "((\x. f x * indicator {a..b} x) has_integral F b - F a) {a..b}" by (rule has_integral_eq[OF _ i]) auto have i: "((\x. f x * indicator {a..b} x) has_integral F b - F a) UNIV" by (rule has_integral_on_superset[OF _ _ i]) auto then have "(\\<^sup>+x. ereal (f x * indicator {a .. b} x) \lborel) = F b - F a" using f f_borel by (subst has_integral_iff_positive_integral_lborel[symmetric]) (auto split: split_indicator) also have "(\\<^sup>+x. ereal (f x * indicator {a .. b} x) \lborel) = (\\<^sup>+x. ereal (f x) * indicator {a .. b} x \lborel)" by (auto intro!: positive_integral_cong simp: indicator_def) finally show ?thesis by simp qed lemma positive_integral_FTC_atLeast: fixes f :: "real \ real" assumes f_borel: "f \ borel_measurable borel" assumes f: "\x. a \ x \ DERIV F x :> f x" assumes nonneg: "\x. a \ x \ 0 \ f x" assumes lim: "(F ---> T) at_top" shows "(\\<^sup>+x. ereal (f x) * indicator {a ..} x \lborel) = T - F a" proof - let ?f = "\(i::nat) (x::real). ereal (f x) * indicator {a..a + real i} x" let ?fR = "\x. ereal (f x) * indicator {a ..} x" have "\x. (SUP i::nat. ?f i x) = ?fR x" proof (rule SUP_Lim_ereal) show "\x. incseq (\i. ?f i x)" using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator) fix x from reals_Archimedean2[of "x - a"] guess n .. then have "eventually (\n. ?f n x = ?fR x) sequentially" by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator) then show "(\n. ?f n x) ----> ?fR x" by (rule Lim_eventually) qed then have "integral\<^sup>P lborel ?fR = (\\<^sup>+ x. (SUP i::nat. ?f i x) \lborel)" by simp also have "\ = (SUP i::nat. (\\<^sup>+ x. ?f i x \lborel))" proof (rule positive_integral_monotone_convergence_SUP) show "incseq ?f" using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator) show "\i. (?f i) \ borel_measurable lborel" using f_borel by auto show "\i x. 0 \ ?f i x" using nonneg by (auto split: split_indicator) qed also have "\ = (SUP i::nat. ereal (F (a + real i) - F a))" by (subst positive_integral_FTC_atLeastAtMost[OF f_borel f nonneg]) auto also have "\ = T - F a" proof (rule SUP_Lim_ereal) show "incseq (\n. ereal (F (a + real n) - F a))" proof (simp add: incseq_def, safe) fix m n :: nat assume "m \ n" with f nonneg show "F (a + real m) \ F (a + real n)" by (intro DERIV_nonneg_imp_nondecreasing[where f=F]) (simp, metis add_increasing2 order_refl order_trans real_of_nat_ge_zero) qed have "(\x. F (a + real x)) ----> T" apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top]) apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl]) apply (rule filterlim_real_sequentially) done then show "(\n. ereal (F (a + real n) - F a)) ----> ereal (T - F a)" unfolding lim_ereal by (intro tendsto_diff) auto qed finally show ?thesis . qed end