imports Measure_Space Borel_Space

(* Title: HOL/Analysis/Nonnegative_Lebesgue_Integration.thy Author: Johannes Hölzl, TU München Author: Armin Heller, TU München *) section ‹Lebesgue Integration for Nonnegative Functions› theory Nonnegative_Lebesgue_Integration imports Measure_Space Borel_Space begin subsection ‹Approximating functions› lemma AE_upper_bound_inf_ennreal: fixes F G::"'a ⇒ ennreal" assumes "⋀e. (e::real) > 0 ⟹ AE x in M. F x ≤ G x + e" shows "AE x in M. F x ≤ G x" proof - have "AE x in M. ∀n::nat. F x ≤ G x + ennreal (1 / Suc n)" using assms by (auto simp: AE_all_countable) then show ?thesis proof (eventually_elim) fix x assume x: "∀n::nat. F x ≤ G x + ennreal (1 / Suc n)" show "F x ≤ G x" proof (rule ennreal_le_epsilon) fix e :: real assume "0 < e" then obtain n where n: "1 / Suc n < e" by (blast elim: nat_approx_posE) have "F x ≤ G x + 1 / Suc n" using x by simp also have "… ≤ G x + e" using n by (intro add_mono ennreal_leI) auto finally show "F x ≤ G x + ennreal e" . qed qed qed lemma AE_upper_bound_inf: fixes F G::"'a ⇒ real" assumes "⋀e. e > 0 ⟹ AE x in M. F x ≤ G x + e" shows "AE x in M. F x ≤ G x" proof - have "AE x in M. F x ≤ G x + 1/real (n+1)" for n::nat by (rule assms, auto) then have "AE x in M. ∀n::nat ∈ UNIV. F x ≤ G x + 1/real (n+1)" by (rule AE_ball_countable', auto) moreover { fix x assume i: "∀n::nat ∈ UNIV. F x ≤ G x + 1/real (n+1)" have "(λn. G x + 1/real (n+1)) ⇢ G x + 0" by (rule tendsto_add, simp, rule LIMSEQ_ignore_initial_segment[OF lim_1_over_n, of 1]) then have "F x ≤ G x" using i LIMSEQ_le_const by fastforce } ultimately show ?thesis by auto qed lemma not_AE_zero_ennreal_E: fixes f::"'a ⇒ ennreal" assumes "¬ (AE x in M. f x = 0)" and [measurable]: "f ∈ borel_measurable M" shows "∃A∈sets M. ∃e::real>0. emeasure M A > 0 ∧ (∀x ∈ A. f x ≥ e)" proof - { assume "¬ (∃e::real>0. {x ∈ space M. f x ≥ e} ∉ null_sets M)" then have "0 < e ⟹ AE x in M. f x ≤ e" for e :: real by (auto simp: not_le less_imp_le dest!: AE_not_in) then have "AE x in M. f x ≤ 0" by (intro AE_upper_bound_inf_ennreal[where G="λ_. 0"]) simp then have False using assms by auto } then obtain e::real where e: "e > 0" "{x ∈ space M. f x ≥ e} ∉ null_sets M" by auto define A where "A = {x ∈ space M. f x ≥ e}" have 1 [measurable]: "A ∈ sets M" unfolding A_def by auto have 2: "emeasure M A > 0" using e(2) A_def ‹A ∈ sets M› by auto have 3: "⋀x. x ∈ A ⟹ f x ≥ e" unfolding A_def by auto show ?thesis using e(1) 1 2 3 by blast qed lemma not_AE_zero_E: fixes f::"'a ⇒ real" assumes "AE x in M. f x ≥ 0" "¬(AE x in M. f x = 0)" and [measurable]: "f ∈ borel_measurable M" shows "∃A e. A ∈ sets M ∧ e>0 ∧ emeasure M A > 0 ∧ (∀x ∈ A. f x ≥ e)" proof - have "∃e. e > 0 ∧ {x ∈ space M. f x ≥ e} ∉ null_sets M" proof (rule ccontr) assume *: "¬(∃e. e > 0 ∧ {x ∈ space M. f x ≥ e} ∉ null_sets M)" { fix e::real assume "e > 0" then have "{x ∈ space M. f x ≥ e} ∈ null_sets M" using * by blast then have "AE x in M. x ∉ {x ∈ space M. f x ≥ e}" using AE_not_in by blast then have "AE x in M. f x ≤ e" by auto } then have "AE x in M. f x ≤ 0" by (rule AE_upper_bound_inf, auto) then have "AE x in M. f x = 0" using assms(1) by auto then show False using assms(2) by auto qed then obtain e where e: "e>0" "{x ∈ space M. f x ≥ e} ∉ null_sets M" by auto define A where "A = {x ∈ space M. f x ≥ e}" have 1 [measurable]: "A ∈ sets M" unfolding A_def by auto have 2: "emeasure M A > 0" using e(2) A_def ‹A ∈ sets M› by auto have 3: "⋀x. x ∈ A ⟹ f x ≥ e" unfolding A_def by auto show ?thesis using e(1) 1 2 3 by blast qed subsection "Simple function" text ‹ Our simple functions are not restricted to nonnegative real numbers. Instead they are just functions with a finite range and are measurable when singleton sets are measurable. › definition "simple_function M g ⟷ finite (g ` space M) ∧ (∀x ∈ g ` space M. g -` {x} ∩ space M ∈ sets M)" lemma simple_functionD: assumes "simple_function M g" shows "finite (g ` space M)" and "g -` X ∩ space M ∈ sets M" proof - show "finite (g ` space M)" using assms unfolding simple_function_def by auto have "g -` X ∩ space M = g -` (X ∩ g`space M) ∩ space M" by auto also have "… = (⋃x∈X ∩ g`space M. g-`{x} ∩ space M)" by auto finally show "g -` X ∩ space M ∈ sets M" using assms by (auto simp del: UN_simps simp: simple_function_def) qed lemma measurable_simple_function[measurable_dest]: "simple_function M f ⟹ f ∈ measurable M (count_space UNIV)" unfolding simple_function_def measurable_def proof safe fix A assume "finite (f ` space M)" "∀x∈f ` space M. f -` {x} ∩ space M ∈ sets M" then have "(⋃x∈f ` space M. if x ∈ A then f -` {x} ∩ space M else {}) ∈ sets M" by (intro sets.finite_UN) auto also have "(⋃x∈f ` space M. if x ∈ A then f -` {x} ∩ space M else {}) = f -` A ∩ space M" by (auto split: if_split_asm) finally show "f -` A ∩ space M ∈ sets M" . qed simp lemma borel_measurable_simple_function: "simple_function M f ⟹ f ∈ borel_measurable M" by (auto dest!: measurable_simple_function simp: measurable_def) lemma simple_function_measurable2[intro]: assumes "simple_function M f" "simple_function M g" shows "f -` A ∩ g -` B ∩ space M ∈ sets M" proof - have "f -` A ∩ g -` B ∩ space M = (f -` A ∩ space M) ∩ (g -` B ∩ space M)" by auto then show ?thesis using assms[THEN simple_functionD(2)] by auto qed lemma simple_function_indicator_representation: fixes f ::"'a ⇒ ennreal" assumes f: "simple_function M f" and x: "x ∈ space M" shows "f x = (∑y ∈ f ` space M. y * indicator (f -` {y} ∩ space M) x)" (is "?l = ?r") proof - have "?r = (∑y ∈ f ` space M. (if y = f x then y * indicator (f -` {y} ∩ space M) x else 0))" by (auto intro!: sum.cong) also have "... = f x * indicator (f -` {f x} ∩ space M) x" using assms by (auto dest: simple_functionD simp: sum.delta) also have "... = f x" using x by (auto simp: indicator_def) finally show ?thesis by auto qed lemma simple_function_notspace: "simple_function M (λx. h x * indicator (- space M) x::ennreal)" (is "simple_function M ?h") proof - have "?h ` space M ⊆ {0}" unfolding indicator_def by auto hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset) have "?h -` {0} ∩ space M = space M" by auto thus ?thesis unfolding simple_function_def by auto qed lemma simple_function_cong: assumes "⋀t. t ∈ space M ⟹ f t = g t" shows "simple_function M f ⟷ simple_function M g" proof - have "⋀x. f -` {x} ∩ space M = g -` {x} ∩ space M" using assms by auto with assms show ?thesis by (simp add: simple_function_def cong: image_cong) qed lemma simple_function_cong_algebra: assumes "sets N = sets M" "space N = space M" shows "simple_function M f ⟷ simple_function N f" unfolding simple_function_def assms .. lemma simple_function_borel_measurable: fixes f :: "'a ⇒ 'x::{t2_space}" assumes "f ∈ borel_measurable M" and "finite (f ` space M)" shows "simple_function M f" using assms unfolding simple_function_def by (auto intro: borel_measurable_vimage) lemma simple_function_iff_borel_measurable: fixes f :: "'a ⇒ 'x::{t2_space}" shows "simple_function M f ⟷ finite (f ` space M) ∧ f ∈ borel_measurable M" by (metis borel_measurable_simple_function simple_functionD(1) simple_function_borel_measurable) lemma simple_function_eq_measurable: "simple_function M f ⟷ finite (f`space M) ∧ f ∈ measurable M (count_space UNIV)" using measurable_simple_function[of M f] by (fastforce simp: simple_function_def) lemma simple_function_const[intro, simp]: "simple_function M (λx. c)" by (auto intro: finite_subset simp: simple_function_def) lemma simple_function_compose[intro, simp]: assumes "simple_function M f" shows "simple_function M (g ∘ f)" unfolding simple_function_def proof safe show "finite ((g ∘ f) ` space M)" using assms unfolding simple_function_def by (auto simp: image_comp [symmetric]) next fix x assume "x ∈ space M" let ?G = "g -` {g (f x)} ∩ (f`space M)" have *: "(g ∘ f) -` {(g ∘ f) x} ∩ space M = (⋃x∈?G. f -` {x} ∩ space M)" by auto show "(g ∘ f) -` {(g ∘ f) x} ∩ space M ∈ sets M" using assms unfolding simple_function_def * by (rule_tac sets.finite_UN) auto qed lemma simple_function_indicator[intro, simp]: assumes "A ∈ sets M" shows "simple_function M (indicator A)" proof - have "indicator A ` space M ⊆ {0, 1}" (is "?S ⊆ _") by (auto simp: indicator_def) hence "finite ?S" by (rule finite_subset) simp moreover have "- A ∩ space M = space M - A" by auto ultimately show ?thesis unfolding simple_function_def using assms by (auto simp: indicator_def [abs_def]) qed lemma simple_function_Pair[intro, simp]: assumes "simple_function M f" assumes "simple_function M g" shows "simple_function M (λx. (f x, g x))" (is "simple_function M ?p") unfolding simple_function_def proof safe show "finite (?p ` space M)" using assms unfolding simple_function_def by (rule_tac finite_subset[of _ "f`space M × g`space M"]) auto next fix x assume "x ∈ space M" have "(λx. (f x, g x)) -` {(f x, g x)} ∩ space M = (f -` {f x} ∩ space M) ∩ (g -` {g x} ∩ space M)" by auto with ‹x ∈ space M› show "(λx. (f x, g x)) -` {(f x, g x)} ∩ space M ∈ sets M" using assms unfolding simple_function_def by auto qed lemma simple_function_compose1: assumes "simple_function M f" shows "simple_function M (λx. g (f x))" using simple_function_compose[OF assms, of g] by (simp add: comp_def) lemma simple_function_compose2: assumes "simple_function M f" and "simple_function M g" shows "simple_function M (λx. h (f x) (g x))" proof - have "simple_function M ((λ(x, y). h x y) ∘ (λx. (f x, g x)))" using assms by auto thus ?thesis by (simp_all add: comp_def) qed lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="(+)"] and simple_function_diff[intro, simp] = simple_function_compose2[where h="(-)"] and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"] and simple_function_mult[intro, simp] = simple_function_compose2[where h="( * )"] and simple_function_div[intro, simp] = simple_function_compose2[where h="(/)"] and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"] and simple_function_max[intro, simp] = simple_function_compose2[where h=max] lemma simple_function_sum[intro, simp]: assumes "⋀i. i ∈ P ⟹ simple_function M (f i)" shows "simple_function M (λx. ∑i∈P. f i x)" proof cases assume "finite P" from this assms show ?thesis by induct auto qed auto lemma simple_function_ennreal[intro, simp]: fixes f g :: "'a ⇒ real" assumes sf: "simple_function M f" shows "simple_function M (λx. ennreal (f x))" by (rule simple_function_compose1[OF sf]) lemma simple_function_real_of_nat[intro, simp]: fixes f g :: "'a ⇒ nat" assumes sf: "simple_function M f" shows "simple_function M (λx. real (f x))" by (rule simple_function_compose1[OF sf]) lemma borel_measurable_implies_simple_function_sequence: fixes u :: "'a ⇒ ennreal" assumes u[measurable]: "u ∈ borel_measurable M" shows "∃f. incseq f ∧ (∀i. (∀x. f i x < top) ∧ simple_function M (f i)) ∧ u = (SUP i. f i)" proof - define f where [abs_def]: "f i x = real_of_int (floor (enn2real (min i (u x)) * 2^i)) / 2^i" for i x have [simp]: "0 ≤ f i x" for i x by (auto simp: f_def intro!: divide_nonneg_nonneg mult_nonneg_nonneg enn2real_nonneg) have *: "2^n * real_of_int x = real_of_int (2^n * x)" for n x by simp have "real_of_int ⌊real i * 2 ^ i⌋ = real_of_int ⌊i * 2 ^ i⌋" for i by (intro arg_cong[where f=real_of_int]) simp then have [simp]: "real_of_int ⌊real i * 2 ^ i⌋ = i * 2 ^ i" for i unfolding floor_of_nat by simp have "incseq f" proof (intro monoI le_funI) fix m n :: nat and x assume "m ≤ n" moreover { fix d :: nat have "⌊2^d::real⌋ * ⌊2^m * enn2real (min (of_nat m) (u x))⌋ ≤ ⌊2^d * (2^m * enn2real (min (of_nat m) (u x)))⌋" by (rule le_mult_floor) (auto simp: enn2real_nonneg) also have "… ≤ ⌊2^d * (2^m * enn2real (min (of_nat d + of_nat m) (u x)))⌋" by (intro floor_mono mult_mono enn2real_mono min.mono) (auto simp: enn2real_nonneg min_less_iff_disj of_nat_less_top) finally have "f m x ≤ f (m + d) x" unfolding f_def by (auto simp: field_simps power_add * simp del: of_int_mult) } ultimately show "f m x ≤ f n x" by (auto simp add: le_iff_add) qed then have inc_f: "incseq (λi. ennreal (f i x))" for x by (auto simp: incseq_def le_fun_def) then have "incseq (λi x. ennreal (f i x))" by (auto simp: incseq_def le_fun_def) moreover have "simple_function M (f i)" for i proof (rule simple_function_borel_measurable) have "⌊enn2real (min (of_nat i) (u x)) * 2 ^ i⌋ ≤ ⌊int i * 2 ^ i⌋" for x by (cases "u x" rule: ennreal_cases) (auto split: split_min intro!: floor_mono) then have "f i ` space M ⊆ (λn. real_of_int n / 2^i) ` {0 .. of_nat i * 2^i}" unfolding floor_of_int by (auto simp: f_def enn2real_nonneg intro!: imageI) then show "finite (f i ` space M)" by (rule finite_subset) auto show "f i ∈ borel_measurable M" unfolding f_def enn2real_def by measurable qed moreover { fix x have "(SUP i. ennreal (f i x)) = u x" proof (cases "u x" rule: ennreal_cases) case top then show ?thesis by (simp add: f_def inf_min[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_SUP_of_nat_eq_top) next case (real r) obtain n where "r ≤ of_nat n" using real_arch_simple by auto then have min_eq_r: "∀⇩_{F}x in sequentially. min (real x) r = r" by (auto simp: eventually_sequentially intro!: exI[of _ n] split: split_min) have "(λi. real_of_int ⌊min (real i) r * 2^i⌋ / 2^i) ⇢ r" proof (rule tendsto_sandwich) show "(λn. r - (1/2)^n) ⇢ r" by (auto intro!: tendsto_eq_intros LIMSEQ_power_zero) show "∀⇩_{F}n in sequentially. real_of_int ⌊min (real n) r * 2 ^ n⌋ / 2 ^ n ≤ r" using min_eq_r by eventually_elim (auto simp: field_simps) have *: "r * (2 ^ n * 2 ^ n) ≤ 2^n + 2^n * real_of_int ⌊r * 2 ^ n⌋" for n using real_of_int_floor_ge_diff_one[of "r * 2^n", THEN mult_left_mono, of "2^n"] by (auto simp: field_simps) show "∀⇩_{F}n in sequentially. r - (1/2)^n ≤ real_of_int ⌊min (real n) r * 2 ^ n⌋ / 2 ^ n" using min_eq_r by eventually_elim (insert *, auto simp: field_simps) qed auto then have "(λi. ennreal (f i x)) ⇢ ennreal r" by (simp add: real f_def ennreal_of_nat_eq_real_of_nat min_ennreal) from LIMSEQ_unique[OF LIMSEQ_SUP[OF inc_f] this] show ?thesis by (simp add: real) qed } ultimately show ?thesis by (intro exI[of _ "λi x. ennreal (f i x)"]) auto qed lemma borel_measurable_implies_simple_function_sequence': fixes u :: "'a ⇒ ennreal" assumes u: "u ∈ borel_measurable M" obtains f where "⋀i. simple_function M (f i)" "incseq f" "⋀i x. f i x < top" "⋀x. (SUP i. f i x) = u x" using borel_measurable_implies_simple_function_sequence[OF u] by (auto simp: fun_eq_iff) blast lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]: fixes u :: "'a ⇒ ennreal" assumes u: "simple_function M u" assumes cong: "⋀f g. simple_function M f ⟹ simple_function M g ⟹ (AE x in M. f x = g x) ⟹ P f ⟹ P g" assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)" assumes mult: "⋀u c. P u ⟹ P (λx. c * u x)" assumes add: "⋀u v. P u ⟹ P v ⟹ P (λx. v x + u x)" shows "P u" proof (rule cong) from AE_space show "AE x in M. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x" proof eventually_elim fix x assume x: "x ∈ space M" from simple_function_indicator_representation[OF u x] show "(∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x" .. qed next from u have "finite (u ` space M)" unfolding simple_function_def by auto then show "P (λx. ∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x)" proof induct case empty show ?case using set[of "{}"] by (simp add: indicator_def[abs_def]) qed (auto intro!: add mult set simple_functionD u) next show "simple_function M (λx. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x))" apply (subst simple_function_cong) apply (rule simple_function_indicator_representation[symmetric]) apply (auto intro: u) done qed fact lemma simple_function_induct_nn[consumes 1, case_names cong set mult add]: fixes u :: "'a ⇒ ennreal" assumes u: "simple_function M u" assumes cong: "⋀f g. simple_function M f ⟹ simple_function M g ⟹ (⋀x. x ∈ space M ⟹ f x = g x) ⟹ P f ⟹ P g" assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)" assumes mult: "⋀u c. simple_function M u ⟹ P u ⟹ P (λx. c * u x)" assumes add: "⋀u v. simple_function M u ⟹ P u ⟹ simple_function M v ⟹ (⋀x. x ∈ space M ⟹ u x = 0 ∨ v x = 0) ⟹ P v ⟹ P (λx. v x + u x)" shows "P u" proof - show ?thesis proof (rule cong) fix x assume x: "x ∈ space M" from simple_function_indicator_representation[OF u x] show "(∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x) = u x" .. next show "simple_function M (λx. (∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x))" apply (subst simple_function_cong) apply (rule simple_function_indicator_representation[symmetric]) apply (auto intro: u) done next from u have "finite (u ` space M)" unfolding simple_function_def by auto then show "P (λx. ∑y∈u ` space M. y * indicator (u -` {y} ∩ space M) x)" proof induct case empty show ?case using set[of "{}"] by (simp add: indicator_def[abs_def]) next case (insert x S) { fix z have "(∑y∈S. y * indicator (u -` {y} ∩ space M) z) = 0 ∨ x * indicator (u -` {x} ∩ space M) z = 0" using insert by (subst sum_eq_0_iff) (auto simp: indicator_def) } note disj = this from insert show ?case by (auto intro!: add mult set simple_functionD u simple_function_sum disj) qed qed fact qed lemma borel_measurable_induct[consumes 1, case_names cong set mult add seq, induct set: borel_measurable]: fixes u :: "'a ⇒ ennreal" assumes u: "u ∈ borel_measurable M" assumes cong: "⋀f g. f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ f x = g x) ⟹ P g ⟹ P f" assumes set: "⋀A. A ∈ sets M ⟹ P (indicator A)" assumes mult': "⋀u c. c < top ⟹ u ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ u x < top) ⟹ P u ⟹ P (λx. c * u x)" assumes add: "⋀u v. u ∈ borel_measurable M⟹ (⋀x. x ∈ space M ⟹ u x < top) ⟹ P u ⟹ v ∈ borel_measurable M ⟹ (⋀x. x ∈ space M ⟹ v x < top) ⟹ (⋀x. x ∈ space M ⟹ u x = 0 ∨ v x = 0) ⟹ P v ⟹ P (λx. v x + u x)" assumes seq: "⋀U. (⋀i. U i ∈ borel_measurable M) ⟹ (⋀i x. x ∈ space M ⟹ U i x < top) ⟹ (⋀i. P (U i)) ⟹ incseq U ⟹ u = (SUP i. U i) ⟹ P (SUP i. U i)" shows "P u" using u proof (induct rule: borel_measurable_implies_simple_function_sequence') fix U assume U: "⋀i. simple_function M (U i)" "incseq U" "⋀i x. U i x < top" and sup: "⋀x. (SUP i. U i x) = u x" have u_eq: "u = (SUP i. U i)" using u sup by auto have not_inf: "⋀x i. x ∈ space M ⟹ U i x < top" using U by (auto simp: image_iff eq_commute) from U have "⋀i. U i ∈ borel_measurable M" by (simp add: borel_measurable_simple_function) show "P u" unfolding u_eq proof (rule seq) fix i show "P (U i)" using ‹simple_function M (U i)› not_inf[of _ i] proof (induct rule: simple_function_induct_nn) case (mult u c) show ?case proof cases assume "c = 0 ∨ space M = {} ∨ (∀x∈space M. u x = 0)" with mult(1) show ?thesis by (intro cong[of "λx. c * u x" "indicator {}"] set) (auto dest!: borel_measurable_simple_function) next assume "¬ (c = 0 ∨ space M = {} ∨ (∀x∈space M. u x = 0))" then obtain x where "space M ≠ {}" and x: "x ∈ space M" "u x ≠ 0" "c ≠ 0" by auto with mult(3)[of x] have "c < top" by (auto simp: ennreal_mult_less_top) then have u_fin: "x' ∈ space M ⟹ u x' < top" for x' using mult(3)[of x'] ‹c ≠ 0› by (auto simp: ennreal_mult_less_top) then have "P u" by (rule mult) with u_fin ‹c < top› mult(1) show ?thesis by (intro mult') (auto dest!: borel_measurable_simple_function) qed qed (auto intro: cong intro!: set add dest!: borel_measurable_simple_function) qed fact+ qed lemma simple_function_If_set: assumes sf: "simple_function M f" "simple_function M g" and A: "A ∩ space M ∈ sets M" shows "simple_function M (λx. if x ∈ A then f x else g x)" (is "simple_function M ?IF") proof - define F where "F x = f -` {x} ∩ space M" for x define G where "G x = g -` {x} ∩ space M" for x show ?thesis unfolding simple_function_def proof safe have "?IF ` space M ⊆ f ` space M ∪ g ` space M" by auto from finite_subset[OF this] assms show "finite (?IF ` space M)" unfolding simple_function_def by auto next fix x assume "x ∈ space M" then have *: "?IF -` {?IF x} ∩ space M = (if x ∈ A then ((F (f x) ∩ (A ∩ space M)) ∪ (G (f x) - (G (f x) ∩ (A ∩ space M)))) else ((F (g x) ∩ (A ∩ space M)) ∪ (G (g x) - (G (g x) ∩ (A ∩ space M)))))" using sets.sets_into_space[OF A] by (auto split: if_split_asm simp: G_def F_def) have [intro]: "⋀x. F x ∈ sets M" "⋀x. G x ∈ sets M" unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto show "?IF -` {?IF x} ∩ space M ∈ sets M" unfolding * using A by auto qed qed lemma simple_function_If: assumes sf: "simple_function M f" "simple_function M g" and P: "{x∈space M. P x} ∈ sets M" shows "simple_function M (λx. if P x then f x else g x)" proof - have "{x∈space M. P x} = {x. P x} ∩ space M" by auto with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp qed lemma simple_function_subalgebra: assumes "simple_function N f" and N_subalgebra: "sets N ⊆ sets M" "space N = space M" shows "simple_function M f" using assms unfolding simple_function_def by auto lemma simple_function_comp: assumes T: "T ∈ measurable M M'" and f: "simple_function M' f" shows "simple_function M (λx. f (T x))" proof (intro simple_function_def[THEN iffD2] conjI ballI) have "(λx. f (T x)) ` space M ⊆ f ` space M'" using T unfolding measurable_def by auto then show "finite ((λx. f (T x)) ` space M)" using f unfolding simple_function_def by (auto intro: finite_subset) fix i assume i: "i ∈ (λx. f (T x)) ` space M" then have "i ∈ f ` space M'" using T unfolding measurable_def by auto then have "f -` {i} ∩ space M' ∈ sets M'" using f unfolding simple_function_def by auto then have "T -` (f -` {i} ∩ space M') ∩ space M ∈ sets M" using T unfolding measurable_def by auto also have "T -` (f -` {i} ∩ space M') ∩ space M = (λx. f (T x)) -` {i} ∩ space M" using T unfolding measurable_def by auto finally show "(λx. f (T x)) -` {i} ∩ space M ∈ sets M" . qed subsection "Simple integral" definition simple_integral :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ ennreal" ("integral⇧^{S}") where "integral⇧^{S}M f = (∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M))" syntax "_simple_integral" :: "pttrn ⇒ ennreal ⇒ 'a measure ⇒ ennreal" ("∫⇧^{S}_. _ ∂_" [60,61] 110) translations "∫⇧^{S}x. f ∂M" == "CONST simple_integral M (%x. f)" lemma simple_integral_cong: assumes "⋀t. t ∈ space M ⟹ f t = g t" shows "integral⇧^{S}M f = integral⇧^{S}M g" proof - have "f ` space M = g ` space M" "⋀x. f -` {x} ∩ space M = g -` {x} ∩ space M" using assms by (auto intro!: image_eqI) thus ?thesis unfolding simple_integral_def by simp qed lemma simple_integral_const[simp]: "(∫⇧^{S}x. c ∂M) = c * (emeasure M) (space M)" proof (cases "space M = {}") case True thus ?thesis unfolding simple_integral_def by simp next case False hence "(λx. c) ` space M = {c}" by auto thus ?thesis unfolding simple_integral_def by simp qed lemma simple_function_partition: assumes f: "simple_function M f" and g: "simple_function M g" assumes sub: "⋀x y. x ∈ space M ⟹ y ∈ space M ⟹ g x = g y ⟹ f x = f y" assumes v: "⋀x. x ∈ space M ⟹ f x = v (g x)" shows "integral⇧^{S}M f = (∑y∈g ` space M. v y * emeasure M {x∈space M. g x = y})" (is "_ = ?r") proof - from f g have [simp]: "finite (f`space M)" "finite (g`space M)" by (auto simp: simple_function_def) from f g have [measurable]: "f ∈ measurable M (count_space UNIV)" "g ∈ measurable M (count_space UNIV)" by (auto intro: measurable_simple_function) { fix y assume "y ∈ space M" then have "f ` space M ∩ {i. ∃x∈space M. i = f x ∧ g y = g x} = {v (g y)}" by (auto cong: sub simp: v[symmetric]) } note eq = this have "integral⇧^{S}M f = (∑y∈f`space M. y * (∑z∈g`space M. if ∃x∈space M. y = f x ∧ z = g x then emeasure M {x∈space M. g x = z} else 0))" unfolding simple_integral_def proof (safe intro!: sum.cong ennreal_mult_left_cong) fix y assume y: "y ∈ space M" "f y ≠ 0" have [simp]: "g ` space M ∩ {z. ∃x∈space M. f y = f x ∧ z = g x} = {z. ∃x∈space M. f y = f x ∧ z = g x}" by auto have eq:"(⋃i∈{z. ∃x∈space M. f y = f x ∧ z = g x}. {x ∈ space M. g x = i}) = f -` {f y} ∩ space M" by (auto simp: eq_commute cong: sub rev_conj_cong) have "finite (g`space M)" by simp then have "finite {z. ∃x∈space M. f y = f x ∧ z = g x}" by (rule rev_finite_subset) auto then show "emeasure M (f -` {f y} ∩ space M) = (∑z∈g ` space M. if ∃x∈space M. f y = f x ∧ z = g x then emeasure M {x ∈ space M. g x = z} else 0)" apply (simp add: sum.If_cases) apply (subst sum_emeasure) apply (auto simp: disjoint_family_on_def eq) done qed also have "… = (∑y∈f`space M. (∑z∈g`space M. if ∃x∈space M. y = f x ∧ z = g x then y * emeasure M {x∈space M. g x = z} else 0))" by (auto intro!: sum.cong simp: sum_distrib_left) also have "… = ?r" by (subst sum.swap) (auto intro!: sum.cong simp: sum.If_cases scaleR_sum_right[symmetric] eq) finally show "integral⇧^{S}M f = ?r" . qed lemma simple_integral_add[simp]: assumes f: "simple_function M f" and "⋀x. 0 ≤ f x" and g: "simple_function M g" and "⋀x. 0 ≤ g x" shows "(∫⇧^{S}x. f x + g x ∂M) = integral⇧^{S}M f + integral⇧^{S}M g" proof - have "(∫⇧^{S}x. f x + g x ∂M) = (∑y∈(λx. (f x, g x))`space M. (fst y + snd y) * emeasure M {x∈space M. (f x, g x) = y})" by (intro simple_function_partition) (auto intro: f g) also have "… = (∑y∈(λx. (f x, g x))`space M. fst y * emeasure M {x∈space M. (f x, g x) = y}) + (∑y∈(λx. (f x, g x))`space M. snd y * emeasure M {x∈space M. (f x, g x) = y})" using assms(2,4) by (auto intro!: sum.cong distrib_right simp: sum.distrib[symmetric]) also have "(∑y∈(λx. (f x, g x))`space M. fst y * emeasure M {x∈space M. (f x, g x) = y}) = (∫⇧^{S}x. f x ∂M)" by (intro simple_function_partition[symmetric]) (auto intro: f g) also have "(∑y∈(λx. (f x, g x))`space M. snd y * emeasure M {x∈space M. (f x, g x) = y}) = (∫⇧^{S}x. g x ∂M)" by (intro simple_function_partition[symmetric]) (auto intro: f g) finally show ?thesis . qed lemma simple_integral_sum[simp]: assumes "⋀i x. i ∈ P ⟹ 0 ≤ f i x" assumes "⋀i. i ∈ P ⟹ simple_function M (f i)" shows "(∫⇧^{S}x. (∑i∈P. f i x) ∂M) = (∑i∈P. integral⇧^{S}M (f i))" proof cases assume "finite P" from this assms show ?thesis by induct (auto simp: simple_function_sum simple_integral_add sum_nonneg) qed auto lemma simple_integral_mult[simp]: assumes f: "simple_function M f" shows "(∫⇧^{S}x. c * f x ∂M) = c * integral⇧^{S}M f" proof - have "(∫⇧^{S}x. c * f x ∂M) = (∑y∈f ` space M. (c * y) * emeasure M {x∈space M. f x = y})" using f by (intro simple_function_partition) auto also have "… = c * integral⇧^{S}M f" using f unfolding simple_integral_def by (subst sum_distrib_left) (auto simp: mult.assoc Int_def conj_commute) finally show ?thesis . qed lemma simple_integral_mono_AE: assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g" and mono: "AE x in M. f x ≤ g x" shows "integral⇧^{S}M f ≤ integral⇧^{S}M g" proof - let ?μ = "λP. emeasure M {x∈space M. P x}" have "integral⇧^{S}M f = (∑y∈(λx. (f x, g x))`space M. fst y * ?μ (λx. (f x, g x) = y))" using f g by (intro simple_function_partition) auto also have "… ≤ (∑y∈(λx. (f x, g x))`space M. snd y * ?μ (λx. (f x, g x) = y))" proof (clarsimp intro!: sum_mono) fix x assume "x ∈ space M" let ?M = "?μ (λy. f y = f x ∧ g y = g x)" show "f x * ?M ≤ g x * ?M" proof cases assume "?M ≠ 0" then have "0 < ?M" by (simp add: less_le) also have "… ≤ ?μ (λy. f x ≤ g x)" using mono by (intro emeasure_mono_AE) auto finally have "¬ ¬ f x ≤ g x" by (intro notI) auto then show ?thesis by (intro mult_right_mono) auto qed simp qed also have "… = integral⇧^{S}M g" using f g by (intro simple_function_partition[symmetric]) auto finally show ?thesis . qed lemma simple_integral_mono: assumes "simple_function M f" and "simple_function M g" and mono: "⋀ x. x ∈ space M ⟹ f x ≤ g x" shows "integral⇧^{S}M f ≤ integral⇧^{S}M g" using assms by (intro simple_integral_mono_AE) auto lemma simple_integral_cong_AE: assumes "simple_function M f" and "simple_function M g" and "AE x in M. f x = g x" shows "integral⇧^{S}M f = integral⇧^{S}M g" using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE) lemma simple_integral_cong': assumes sf: "simple_function M f" "simple_function M g" and mea: "(emeasure M) {x∈space M. f x ≠ g x} = 0" shows "integral⇧^{S}M f = integral⇧^{S}M g" proof (intro simple_integral_cong_AE sf AE_I) show "(emeasure M) {x∈space M. f x ≠ g x} = 0" by fact show "{x ∈ space M. f x ≠ g x} ∈ sets M" using sf[THEN borel_measurable_simple_function] by auto qed simp lemma simple_integral_indicator: assumes A: "A ∈ sets M" assumes f: "simple_function M f" shows "(∫⇧^{S}x. f x * indicator A x ∂M) = (∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M ∩ A))" proof - have eq: "(λx. (f x, indicator A x)) ` space M ∩ {x. snd x = 1} = (λx. (f x, 1::ennreal))`A" using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: if_split_asm) have eq2: "⋀x. f x ∉ f ` A ⟹ f -` {f x} ∩ space M ∩ A = {}" by (auto simp: image_iff) have "(∫⇧^{S}x. f x * indicator A x ∂M) = (∑y∈(λx. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x∈space M. (f x, indicator A x) = y})" using assms by (intro simple_function_partition) auto also have "… = (∑y∈(λx. (f x, indicator A x::ennreal))`space M. if snd y = 1 then fst y * emeasure M (f -` {fst y} ∩ space M ∩ A) else 0)" by (auto simp: indicator_def split: if_split_asm intro!: arg_cong2[where f="( * )"] arg_cong2[where f=emeasure] sum.cong) also have "… = (∑y∈(λx. (f x, 1::ennreal))`A. fst y * emeasure M (f -` {fst y} ∩ space M ∩ A))" using assms by (subst sum.If_cases) (auto intro!: simple_functionD(1) simp: eq) also have "… = (∑y∈fst`(λx. (f x, 1::ennreal))`A. y * emeasure M (f -` {y} ∩ space M ∩ A))" by (subst sum.reindex [of fst]) (auto simp: inj_on_def) also have "… = (∑x ∈ f ` space M. x * emeasure M (f -` {x} ∩ space M ∩ A))" using A[THEN sets.sets_into_space] by (intro sum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2) finally show ?thesis . qed lemma simple_integral_indicator_only[simp]: assumes "A ∈ sets M" shows "integral⇧^{S}M (indicator A) = emeasure M A" using simple_integral_indicator[OF assms, of "λx. 1"] sets.sets_into_space[OF assms] by (simp_all add: image_constant_conv Int_absorb1 split: if_split_asm) lemma simple_integral_null_set: assumes "simple_function M u" "⋀x. 0 ≤ u x" and "N ∈ null_sets M" shows "(∫⇧^{S}x. u x * indicator N x ∂M) = 0" proof - have "AE x in M. indicator N x = (0 :: ennreal)" using ‹N ∈ null_sets M› by (auto simp: indicator_def intro!: AE_I[of _ _ N]) then have "(∫⇧^{S}x. u x * indicator N x ∂M) = (∫⇧^{S}x. 0 ∂M)" using assms apply (intro simple_integral_cong_AE) by auto then show ?thesis by simp qed lemma simple_integral_cong_AE_mult_indicator: assumes sf: "simple_function M f" and eq: "AE x in M. x ∈ S" and "S ∈ sets M" shows "integral⇧^{S}M f = (∫⇧^{S}x. f x * indicator S x ∂M)" using assms by (intro simple_integral_cong_AE) auto lemma simple_integral_cmult_indicator: assumes A: "A ∈ sets M" shows "(∫⇧^{S}x. c * indicator A x ∂M) = c * emeasure M A" using simple_integral_mult[OF simple_function_indicator[OF A]] unfolding simple_integral_indicator_only[OF A] by simp lemma simple_integral_nonneg: assumes f: "simple_function M f" and ae: "AE x in M. 0 ≤ f x" shows "0 ≤ integral⇧^{S}M f" proof - have "integral⇧^{S}M (λx. 0) ≤ integral⇧^{S}M f" using simple_integral_mono_AE[OF _ f ae] by auto then show ?thesis by simp qed subsection ‹Integral on nonnegative functions› definition nn_integral :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ ennreal" ("integral⇧^{N}") where "integral⇧^{N}M f = (SUP g : {g. simple_function M g ∧ g ≤ f}. integral⇧^{S}M g)" syntax "_nn_integral" :: "pttrn ⇒ ennreal ⇒ 'a measure ⇒ ennreal" ("∫⇧^{+}((2 _./ _)/ ∂_)" [60,61] 110) translations "∫⇧^{+}x. f ∂M" == "CONST nn_integral M (λx. f)" lemma nn_integral_def_finite: "integral⇧^{N}M f = (SUP g : {g. simple_function M g ∧ g ≤ f ∧ (∀x. g x < top)}. integral⇧^{S}M g)" (is "_ = SUPREMUM ?A ?f") unfolding nn_integral_def proof (safe intro!: antisym SUP_least) fix g assume g[measurable]: "simple_function M g" "g ≤ f" show "integral⇧^{S}M g ≤ SUPREMUM ?A ?f" proof cases assume ae: "AE x in M. g x ≠ top" let ?G = "{x ∈ space M. g x ≠ top}" have "integral⇧^{S}M g = integral⇧^{S}M (λx. g x * indicator ?G x)" proof (rule simple_integral_cong_AE) show "AE x in M. g x = g x * indicator ?G x" using ae AE_space by eventually_elim auto qed (insert g, auto) also have "… ≤ SUPREMUM ?A ?f" using g by (intro SUP_upper) (auto simp: le_fun_def less_top split: split_indicator) finally show ?thesis . next assume nAE: "¬ (AE x in M. g x ≠ top)" then have "emeasure M {x∈space M. g x = top} ≠ 0" (is "emeasure M ?G ≠ 0") by (subst (asm) AE_iff_measurable[OF _ refl]) auto then have "top = (SUP n. (∫⇧^{S}x. of_nat n * indicator ?G x ∂M))" by (simp add: ennreal_SUP_of_nat_eq_top ennreal_top_eq_mult_iff SUP_mult_right_ennreal[symmetric]) also have "… ≤ SUPREMUM ?A ?f" using g by (safe intro!: SUP_least SUP_upper) (auto simp: le_fun_def of_nat_less_top top_unique[symmetric] split: split_indicator intro: order_trans[of _ "g x" "f x" for x, OF order_trans[of _ top]]) finally show ?thesis by (simp add: top_unique del: SUP_eq_top_iff Sup_eq_top_iff) qed qed (auto intro: SUP_upper) lemma nn_integral_mono_AE: assumes ae: "AE x in M. u x ≤ v x" shows "integral⇧^{N}M u ≤ integral⇧^{N}M v" unfolding nn_integral_def proof (safe intro!: SUP_mono) fix n assume n: "simple_function M n" "n ≤ u" from ae[THEN AE_E] guess N . note N = this then have ae_N: "AE x in M. x ∉ N" by (auto intro: AE_not_in) let ?n = "λx. n x * indicator (space M - N) x" have "AE x in M. n x ≤ ?n x" "simple_function M ?n" using n N ae_N by auto moreover { fix x have "?n x ≤ v x" proof cases assume x: "x ∈ space M - N" with N have "u x ≤ v x" by auto with n(2)[THEN le_funD, of x] x show ?thesis by (auto simp: max_def split: if_split_asm) qed simp } then have "?n ≤ v" by (auto simp: le_funI) moreover have "integral⇧^{S}M n ≤ integral⇧^{S}M ?n" using ae_N N n by (auto intro!: simple_integral_mono_AE) ultimately show "∃m∈{g. simple_function M g ∧ g ≤ v}. integral⇧^{S}M n ≤ integral⇧^{S}M m" by force qed lemma nn_integral_mono: "(⋀x. x ∈ space M ⟹ u x ≤ v x) ⟹ integral⇧^{N}M u ≤ integral⇧^{N}M v" by (auto intro: nn_integral_mono_AE) lemma mono_nn_integral: "mono F ⟹ mono (λx. integral⇧^{N}M (F x))" by (auto simp add: mono_def le_fun_def intro!: nn_integral_mono) lemma nn_integral_cong_AE: "AE x in M. u x = v x ⟹ integral⇧^{N}M u = integral⇧^{N}M v" by (auto simp: eq_iff intro!: nn_integral_mono_AE) lemma nn_integral_cong: "(⋀x. x ∈ space M ⟹ u x = v x) ⟹ integral⇧^{N}M u = integral⇧^{N}M v" by (auto intro: nn_integral_cong_AE) lemma nn_integral_cong_simp: "(⋀x. x ∈ space M =simp=> u x = v x) ⟹ integral⇧^{N}M u = integral⇧^{N}M v" by (auto intro: nn_integral_cong simp: simp_implies_def) lemma nn_integral_cong_strong: "M = N ⟹ (⋀x. x ∈ space M ⟹ u x = v x) ⟹ integral⇧^{N}M u = integral⇧^{N}N v" by (auto intro: nn_integral_cong) lemma incseq_nn_integral: assumes "incseq f" shows "incseq (λi. integral⇧^{N}M (f i))" proof - have "⋀i x. f i x ≤ f (Suc i) x" using assms by (auto dest!: incseq_SucD simp: le_fun_def) then show ?thesis by (auto intro!: incseq_SucI nn_integral_mono) qed lemma nn_integral_eq_simple_integral: assumes f: "simple_function M f" shows "integral⇧^{N}M f = integral⇧^{S}M f" proof - let ?f = "λx. f x * indicator (space M) x" have f': "simple_function M ?f" using f by auto have "integral⇧^{N}M ?f ≤ integral⇧^{S}M ?f" using f' by (force intro!: SUP_least simple_integral_mono simp: le_fun_def nn_integral_def) moreover have "integral⇧^{S}M ?f ≤ integral⇧^{N}M ?f" unfolding nn_integral_def using f' by (auto intro!: SUP_upper) ultimately show ?thesis by (simp cong: nn_integral_cong simple_integral_cong) qed text ‹Beppo-Levi monotone convergence theorem› lemma nn_integral_monotone_convergence_SUP: assumes f: "incseq f" and [measurable]: "⋀i. f i ∈ borel_measurable M" shows "(∫⇧^{+}x. (SUP i. f i x) ∂M) = (SUP i. integral⇧^{N}M (f i))" proof (rule antisym) show "(∫⇧^{+}x. (SUP i. f i x) ∂M) ≤ (SUP i. (∫⇧^{+}x. f i x ∂M))" unfolding nn_integral_def_finite[of _ "λx. SUP i. f i x"] proof (safe intro!: SUP_least) fix u assume sf_u[simp]: "simple_function M u" and u: "u ≤ (λx. SUP i. f i x)" and u_range: "∀x. u x < top" note sf_u[THEN borel_measurable_simple_function, measurable] show "integral⇧^{S}M u ≤ (SUP j. ∫⇧^{+}x. f j x ∂M)" proof (rule ennreal_approx_unit) fix a :: ennreal assume "a < 1" let ?au = "λx. a * u x" let ?B = "λc i. {x∈space M. ?au x = c ∧ c ≤ f i x}" have "integral⇧^{S}M ?au = (∑c∈?au`space M. c * (SUP i. emeasure M (?B c i)))" unfolding simple_integral_def proof (intro sum.cong ennreal_mult_left_cong refl) fix c assume "c ∈ ?au ` space M" "c ≠ 0" { fix x' assume x': "x' ∈ space M" "?au x' = c" with ‹c ≠ 0› u_range have "?au x' < 1 * u x'" by (intro ennreal_mult_strict_right_mono ‹a < 1›) (auto simp: less_le) also have "… ≤ (SUP i. f i x')" using u by (auto simp: le_fun_def) finally have "∃i. ?au x' ≤ f i x'" by (auto simp: less_SUP_iff intro: less_imp_le) } then have *: "?au -` {c} ∩ space M = (⋃i. ?B c i)" by auto show "emeasure M (?au -` {c} ∩ space M) = (SUP i. emeasure M (?B c i))" unfolding * using f by (intro SUP_emeasure_incseq[symmetric]) (auto simp: incseq_def le_fun_def intro: order_trans) qed also have "… = (SUP i. ∑c∈?au`space M. c * emeasure M (?B c i))" unfolding SUP_mult_left_ennreal using f by (intro ennreal_SUP_sum[symmetric]) (auto intro!: mult_mono emeasure_mono simp: incseq_def le_fun_def intro: order_trans) also have "… ≤ (SUP i. integral⇧^{N}M (f i))" proof (intro SUP_subset_mono order_refl) fix i have "(∑c∈?au`space M. c * emeasure M (?B c i)) = (∫⇧^{S}x. (a * u x) * indicator {x∈space M. a * u x ≤ f i x} x ∂M)" by (subst simple_integral_indicator) (auto intro!: sum.cong ennreal_mult_left_cong arg_cong2[where f=emeasure]) also have "… = (∫⇧^{+}x. (a * u x) * indicator {x∈space M. a * u x ≤ f i x} x ∂M)" by (rule nn_integral_eq_simple_integral[symmetric]) simp also have "… ≤ (∫⇧^{+}x. f i x ∂M)" by (intro nn_integral_mono) (auto split: split_indicator) finally show "(∑c∈?au`space M. c * emeasure M (?B c i)) ≤ (∫⇧^{+}x. f i x ∂M)" . qed finally show "a * integral⇧^{S}M u ≤ (SUP i. integral⇧^{N}M (f i))" by simp qed qed qed (auto intro!: SUP_least SUP_upper nn_integral_mono) lemma sup_continuous_nn_integral[order_continuous_intros]: assumes f: "⋀y. sup_continuous (f y)" assumes [measurable]: "⋀x. (λy. f y x) ∈ borel_measurable M" shows "sup_continuous (λx. (∫⇧^{+}y. f y x ∂M))" unfolding sup_continuous_def proof safe fix C :: "nat ⇒ 'b" assume C: "incseq C" with sup_continuous_mono[OF f] show "(∫⇧^{+}y. f y (SUPREMUM UNIV C) ∂M) = (SUP i. ∫⇧^{+}y. f y (C i) ∂M)" unfolding sup_continuousD[OF f C] by (subst nn_integral_monotone_convergence_SUP) (auto simp: mono_def le_fun_def) qed lemma nn_integral_monotone_convergence_SUP_AE: assumes f: "⋀i. AE x in M. f i x ≤ f (Suc i) x" "⋀i. f i ∈ borel_measurable M" shows "(∫⇧^{+}x. (SUP i. f i x) ∂M) = (SUP i. integral⇧^{N}M (f i))" proof - from f have "AE x in M. ∀i. f i x ≤ f (Suc i) x" by (simp add: AE_all_countable) from this[THEN AE_E] guess N . note N = this let ?f = "λi x. if x ∈ space M - N then f i x else 0" have f_eq: "AE x in M. ∀i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N]) then have "(∫⇧^{+}x. (SUP i. f i x) ∂M) = (∫⇧^{+}x. (SUP i. ?f i x) ∂M)" by (auto intro!: nn_integral_cong_AE) also have "… = (SUP i. (∫⇧^{+}x. ?f i x ∂M))" proof (rule nn_integral_monotone_convergence_SUP) show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI) { fix i show "(λx. if x ∈ space M - N then f i x else 0) ∈ borel_measurable M" using f N(3) by (intro measurable_If_set) auto } qed also have "… = (SUP i. (∫⇧^{+}x. f i x ∂M))" using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] nn_integral_cong_AE ext) finally show ?thesis . qed lemma nn_integral_monotone_convergence_simple: "incseq f ⟹ (⋀i. simple_function M (f i)) ⟹ (SUP i. ∫⇧^{S}x. f i x ∂M) = (∫⇧^{+}x. (SUP i. f i x) ∂M)" using nn_integral_monotone_convergence_SUP[of f M] by (simp add: nn_integral_eq_simple_integral[symmetric] borel_measurable_simple_function) lemma SUP_simple_integral_sequences: assumes f: "incseq f" "⋀i. simple_function M (f i)" and g: "incseq g" "⋀i. simple_function M (g i)" and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)" shows "(SUP i. integral⇧^{S}M (f i)) = (SUP i. integral⇧^{S}M (g i))" (is "SUPREMUM _ ?F = SUPREMUM _ ?G") proof - have "(SUP i. integral⇧^{S}M (f i)) = (∫⇧^{+}x. (SUP i. f i x) ∂M)" using f by (rule nn_integral_monotone_convergence_simple) also have "… = (∫⇧^{+}x. (SUP i. g i x) ∂M)" unfolding eq[THEN nn_integral_cong_AE] .. also have "… = (SUP i. ?G i)" using g by (rule nn_integral_monotone_convergence_simple[symmetric]) finally show ?thesis by simp qed lemma nn_integral_const[simp]: "(∫⇧^{+}x. c ∂M) = c * emeasure M (space M)" by (subst nn_integral_eq_simple_integral) auto lemma nn_integral_linear: assumes f: "f ∈ borel_measurable M" and g: "g ∈ borel_measurable M" shows "(∫⇧^{+}x. a * f x + g x ∂M) = a * integral⇧^{N}M f + integral⇧^{N}M g" (is "integral⇧^{N}M ?L = _") proof - from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u . note u = nn_integral_monotone_convergence_simple[OF this(2,1)] this from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v . note v = nn_integral_monotone_convergence_simple[OF this(2,1)] this let ?L' = "λi x. a * u i x + v i x" have "?L ∈ borel_measurable M" using assms by auto from borel_measurable_implies_simple_function_sequence'[OF this] guess l . note l = nn_integral_monotone_convergence_simple[OF this(2,1)] this have inc: "incseq (λi. a * integral⇧^{S}M (u i))" "incseq (λi. integral⇧^{S}M (v i))" using u v by (auto simp: incseq_Suc_iff le_fun_def intro!: add_mono mult_left_mono simple_integral_mono) have l': "(SUP i. integral⇧^{S}M (l i)) = (SUP i. integral⇧^{S}M (?L' i))" proof (rule SUP_simple_integral_sequences[OF l(3,2)]) show "incseq ?L'" "⋀i. simple_function M (?L' i)" using u v unfolding incseq_Suc_iff le_fun_def by (auto intro!: add_mono mult_left_mono) { fix x have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)" using u(3) v(3) u(4)[of _ x] v(4)[of _ x] unfolding SUP_mult_left_ennreal by (auto intro!: ennreal_SUP_add simp: incseq_Suc_iff le_fun_def add_mono mult_left_mono) } then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)" unfolding l(5) using u(5) v(5) by (intro AE_I2) auto qed also have "… = (SUP i. a * integral⇧^{S}M (u i) + integral⇧^{S}M (v i))" using u(2) v(2) by auto finally show ?thesis unfolding l(5)[symmetric] l(1)[symmetric] by (simp add: ennreal_SUP_add[OF inc] v u SUP_mult_left_ennreal[symmetric]) qed lemma nn_integral_cmult: "f ∈ borel_measurable M ⟹ (∫⇧^{+}x. c * f x ∂M) = c * integral⇧^{N}M f" using nn_integral_linear[of f M "λx. 0" c] by simp lemma nn_integral_multc: "f ∈ borel_measurable M ⟹ (∫⇧^{+}x. f x * c ∂M) = integral⇧^{N}M f * c" unfolding mult.commute[of _ c] nn_integral_cmult by simp lemma nn_integral_divide: "f ∈ borel_measurable M ⟹ (∫⇧^{+}x. f x / c ∂M) = (∫⇧^{+}x. f x ∂M) / c" unfolding divide_ennreal_def by (rule nn_integral_multc) lemma nn_integral_indicator[simp]: "A ∈ sets M ⟹ (∫⇧^{+}x. indicator A x∂M) = (emeasure M) A" by (subst nn_integral_eq_simple_integral) (auto simp: simple_integral_indicator) lemma nn_integral_cmult_indicator: "A ∈ sets M ⟹ (∫⇧^{+}x. c * indicator A x ∂M) = c * emeasure M A" by (subst nn_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator) lemma nn_integral_indicator': assumes [measurable]: "A ∩ space M ∈ sets M" shows "(∫⇧^{+}x. indicator A x ∂M) = emeasure M (A ∩ space M)" proof - have "(∫⇧^{+}x. indicator A x ∂M) = (∫⇧^{+}x. indicator (A ∩ space M) x ∂M)" by (intro nn_integral_cong) (simp split: split_indicator) also have "… = emeasure M (A ∩ space M)" by simp finally show ?thesis . qed lemma nn_integral_indicator_singleton[simp]: assumes [measurable]: "{y} ∈ sets M" shows "(∫⇧^{+}x. f x * indicator {y} x ∂M) = f y * emeasure M {y}" proof - have "(∫⇧^{+}x. f x * indicator {y} x ∂M) = (∫⇧^{+}x. f y * indicator {y} x ∂M)" by (auto intro!: nn_integral_cong split: split_indicator) then show ?thesis by (simp add: nn_integral_cmult) qed lemma nn_integral_set_ennreal: "(∫⇧^{+}x. ennreal (f x) * indicator A x ∂M) = (∫⇧^{+}x. ennreal (f x * indicator A x) ∂M)" by (rule nn_integral_cong) (simp split: split_indicator) lemma nn_integral_indicator_singleton'[simp]: assumes [measurable]: "{y} ∈ sets M" shows "(∫⇧^{+}x. ennreal (f x * indicator {y} x) ∂M) = f y * emeasure M {y}" by (subst nn_integral_set_ennreal[symmetric]) (simp add: nn_integral_indicator_singleton) lemma nn_integral_add: "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ (∫⇧^{+}x. f x + g x ∂M) = integral⇧^{N}M f + integral⇧^{N}M g" using nn_integral_linear[of f M g 1] by simp lemma nn_integral_sum: "(⋀i. i ∈ P ⟹ f i ∈ borel_measurable M) ⟹ (∫⇧^{+}x. (∑i∈P. f i x) ∂M) = (∑i∈P. integral⇧^{N}M (f i))" by (induction P rule: infinite_finite_induct) (auto simp: nn_integral_add) lemma nn_integral_suminf: assumes f: "⋀i. f i ∈ borel_measurable M" shows "(∫⇧^{+}x. (∑i. f i x) ∂M) = (∑i. integral⇧^{N}M (f i))" proof - have all_pos: "AE x in M. ∀i. 0 ≤ f i x" using assms by (auto simp: AE_all_countable) have "(∑i. integral⇧^{N}M (f i)) = (SUP n. ∑i<n. integral⇧^{N}M (f i))" by (rule suminf_eq_SUP) also have "… = (SUP n. ∫⇧^{+}x. (∑i<n. f i x) ∂M)" unfolding nn_integral_sum[OF f] .. also have "… = ∫⇧^{+}x. (SUP n. ∑i<n. f i x) ∂M" using f all_pos by (intro nn_integral_monotone_convergence_SUP_AE[symmetric]) (elim AE_mp, auto simp: sum_nonneg simp del: sum_lessThan_Suc intro!: AE_I2 sum_mono2) also have "… = ∫⇧^{+}x. (∑i. f i x) ∂M" using all_pos by (intro nn_integral_cong_AE) (auto simp: suminf_eq_SUP) finally show ?thesis by simp qed lemma nn_integral_bound_simple_function: assumes bnd: "⋀x. x ∈ space M ⟹ f x < ∞" assumes f[measurable]: "simple_function M f" assumes supp: "emeasure M {x∈space M. f x ≠ 0} < ∞" shows "nn_integral M f < ∞" proof cases assume "space M = {}" then have "nn_integral M f = (∫⇧^{+}x. 0 ∂M)" by (intro nn_integral_cong) auto then show ?thesis by simp next assume "space M ≠ {}" with simple_functionD(1)[OF f] bnd have bnd: "0 ≤ Max (f`space M) ∧ Max (f`space M) < ∞" by (subst Max_less_iff) (auto simp: Max_ge_iff) have "nn_integral M f ≤ (∫⇧^{+}x. Max (f`space M) * indicator {x∈space M. f x ≠ 0} x ∂M)" proof (rule nn_integral_mono) fix x assume "x ∈ space M" with f show "f x ≤ Max (f ` space M) * indicator {x ∈ space M. f x ≠ 0} x" by (auto split: split_indicator intro!: Max_ge simple_functionD) qed also have "… < ∞" using bnd supp by (subst nn_integral_cmult) (auto simp: ennreal_mult_less_top) finally show ?thesis . qed lemma nn_integral_Markov_inequality: assumes u: "u ∈ borel_measurable M" and "A ∈ sets M" shows "(emeasure M) ({x∈space M. 1 ≤ c * u x} ∩ A) ≤ c * (∫⇧^{+}x. u x * indicator A x ∂M)" (is "(emeasure M) ?A ≤ _ * ?PI") proof - have "?A ∈ sets M" using ‹A ∈ sets M› u by auto hence "(emeasure M) ?A = (∫⇧^{+}x. indicator ?A x ∂M)" using nn_integral_indicator by simp also have "… ≤ (∫⇧^{+}x. c * (u x * indicator A x) ∂M)" using u by (auto intro!: nn_integral_mono_AE simp: indicator_def) also have "… = c * (∫⇧^{+}x. u x * indicator A x ∂M)" using assms by (auto intro!: nn_integral_cmult) finally show ?thesis . qed lemma nn_integral_noteq_infinite: assumes g: "g ∈ borel_measurable M" and "integral⇧^{N}M g ≠ ∞" shows "AE x in M. g x ≠ ∞" proof (rule ccontr) assume c: "¬ (AE x in M. g x ≠ ∞)" have "(emeasure M) {x∈space M. g x = ∞} ≠ 0" using c g by (auto simp add: AE_iff_null) then have "0 < (emeasure M) {x∈space M. g x = ∞}" by (auto simp: zero_less_iff_neq_zero) then have "∞ = ∞ * (emeasure M) {x∈space M. g x = ∞}" by (auto simp: ennreal_top_eq_mult_iff) also have "… ≤ (∫⇧^{+}x. ∞ * indicator {x∈space M. g x = ∞} x ∂M)" using g by (subst nn_integral_cmult_indicator) auto also have "… ≤ integral⇧^{N}M g" using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def) finally show False using ‹integral⇧^{N}M g ≠ ∞› by (auto simp: top_unique) qed lemma nn_integral_PInf: assumes f: "f ∈ borel_measurable M" and not_Inf: "integral⇧^{N}M f ≠ ∞" shows "emeasure M (f -` {∞} ∩ space M) = 0" proof - have "∞ * emeasure M (f -` {∞} ∩ space M) = (∫⇧^{+}x. ∞ * indicator (f -` {∞} ∩ space M) x ∂M)" using f by (subst nn_integral_cmult_indicator) (auto simp: measurable_sets) also have "… ≤ integral⇧^{N}M f" by (auto intro!: nn_integral_mono simp: indicator_def) finally have "∞ * (emeasure M) (f -` {∞} ∩ space M) ≤ integral⇧^{N}M f" by simp then show ?thesis using assms by (auto simp: ennreal_top_mult top_unique split: if_split_asm) qed lemma simple_integral_PInf: "simple_function M f ⟹ integral⇧^{S}M f ≠ ∞ ⟹ emeasure M (f -` {∞} ∩ space M) = 0" by (rule nn_integral_PInf) (auto simp: nn_integral_eq_simple_integral borel_measurable_simple_function) lemma nn_integral_PInf_AE: assumes "f ∈ borel_measurable M" "integral⇧^{N}M f ≠ ∞" shows "AE x in M. f x ≠ ∞" proof (rule AE_I) show "(emeasure M) (f -` {∞} ∩ space M) = 0" by (rule nn_integral_PInf[OF assms]) show "f -` {∞} ∩ space M ∈ sets M" using assms by (auto intro: borel_measurable_vimage) qed auto lemma nn_integral_diff: assumes f: "f ∈ borel_measurable M" and g: "g ∈ borel_measurable M" and fin: "integral⇧^{N}M g ≠ ∞" and mono: "AE x in M. g x ≤ f x" shows "(∫⇧^{+}x. f x - g x ∂M) = integral⇧^{N}M f - integral⇧^{N}M g" proof - have diff: "(λx. f x - g x) ∈ borel_measurable M" using assms by auto have "AE x in M. f x = f x - g x + g x" using diff_add_cancel_ennreal mono nn_integral_noteq_infinite[OF g fin] assms by auto then have **: "integral⇧^{N}M f = (∫⇧^{+}x. f x - g x ∂M) + integral⇧^{N}M g" unfolding nn_integral_add[OF diff g, symmetric] by (rule nn_integral_cong_AE) show ?thesis unfolding ** using fin by (cases rule: ennreal2_cases[of "∫⇧^{+}x. f x - g x ∂M" "integral⇧^{N}M g"]) auto qed lemma nn_integral_mult_bounded_inf: assumes f: "f ∈ borel_measurable M" "(∫⇧^{+}x. f x ∂M) < ∞" and c: "c ≠ ∞" and ae: "AE x in M. g x ≤ c * f x" shows "(∫⇧^{+}x. g x ∂M) < ∞" proof - have "(∫⇧^{+}x. g x ∂M) ≤ (∫⇧^{+}x. c * f x ∂M)" by (intro nn_integral_mono_AE ae) also have "(∫⇧^{+}x. c * f x ∂M) < ∞" using c f by (subst nn_integral_cmult) (auto simp: ennreal_mult_less_top top_unique not_less) finally show ?thesis . qed text ‹Fatou's lemma: convergence theorem on limes inferior› lemma nn_integral_monotone_convergence_INF_AE': assumes f: "⋀i. AE x in M. f (Suc i) x ≤ f i x" and [measurable]: "⋀i. f i ∈ borel_measurable M" and *: "(∫⇧^{+}x. f 0 x ∂M) < ∞" shows "(∫⇧^{+}x. (INF i. f i x) ∂M) = (INF i. integral⇧^{N}M (f i))" proof (rule ennreal_minus_cancel) have "integral⇧^{N}M (f 0) - (∫⇧^{+}x. (INF i. f i x) ∂M) = (∫⇧^{+}x. f 0 x - (INF i. f i x) ∂M)" proof (rule nn_integral_diff[symmetric]) have "(∫⇧^{+}x. (INF i. f i x) ∂M) ≤ (∫⇧^{+}x. f 0 x ∂M)" by (intro nn_integral_mono INF_lower) simp with * show "(∫⇧^{+}x. (INF i. f i x) ∂M) ≠ ∞" by simp qed (auto intro: INF_lower) also have "… = (∫⇧^{+}x. (SUP i. f 0 x - f i x) ∂M)" by (simp add: ennreal_INF_const_minus) also have "… = (SUP i. (∫⇧^{+}x. f 0 x - f i x ∂M))" proof (intro nn_integral_monotone_convergence_SUP_AE) show "AE x in M. f 0 x - f i x ≤ f 0 x - f (Suc i) x" for i using f[of i] by eventually_elim (auto simp: ennreal_mono_minus) qed simp also have "… = (SUP i. nn_integral M (f 0) - (∫⇧^{+}x. f i x ∂M))" proof (subst nn_integral_diff[symmetric]) fix i have dec: "AE x in M. ∀i. f (Suc i) x ≤ f i x" unfolding AE_all_countable using f by auto then show "AE x in M. f i x ≤ f 0 x" using dec by eventually_elim (auto intro: lift_Suc_antimono_le[of "λi. f i x" 0 i for x]) then have "(∫⇧^{+}x. f i x ∂M) ≤ (∫⇧^{+}x. f 0 x ∂M)" by (rule nn_integral_mono_AE) with * show "(∫⇧^{+}x. f i x ∂M) ≠ ∞" by simp qed (insert f, auto simp: decseq_def le_fun_def) finally show "integral⇧^{N}M (f 0) - (∫⇧^{+}x. (INF i. f i x) ∂M) = integral⇧^{N}M (f 0) - (INF i. ∫⇧^{+}x. f i x ∂M)" by (simp add: ennreal_INF_const_minus) qed (insert *, auto intro!: nn_integral_mono intro: INF_lower) lemma nn_integral_monotone_convergence_INF_AE: fixes f :: "nat ⇒ 'a ⇒ ennreal" assumes f: "⋀i. AE x in M. f (Suc i) x ≤ f i x" and [measurable]: "⋀i. f i ∈ borel_measurable M" and fin: "(∫⇧^{+}x. f i x ∂M) < ∞" shows "(∫⇧^{+}x. (INF i. f i x) ∂M) = (INF i. integral⇧^{N}M (f i))" proof - { fix f :: "nat ⇒ ennreal" and j assume "decseq f" then have "(INF i. f i) = (INF i. f (i + j))" apply (intro INF_eq) apply (rule_tac x="i" in bexI) apply (auto simp: decseq_def le_fun_def) done } note INF_shift = this have mono: "AE x in M. ∀i. f (Suc i) x ≤ f i x" using f by (auto simp: AE_all_countable) then have "AE x in M. (INF i. f i x) = (INF n. f (n + i) x)" by eventually_elim (auto intro!: decseq_SucI INF_shift) then have "(∫⇧^{+}x. (INF i. f i x) ∂M) = (∫⇧^{+}x. (INF n. f (n + i) x) ∂M)" by (rule nn_integral_cong_AE) also have "… = (INF n. (∫⇧^{+}x. f (n + i) x ∂M))" by (rule nn_integral_monotone_convergence_INF_AE') (insert assms, auto) also have "… = (INF n. (∫⇧^{+}x. f n x ∂M))" by (intro INF_shift[symmetric] decseq_SucI nn_integral_mono_AE f) finally show ?thesis . qed lemma nn_integral_monotone_convergence_INF_decseq: assumes f: "decseq f" and *: "⋀i. f i ∈ borel_measurable M" "(∫⇧^{+}x. f i x ∂M) < ∞" shows "(∫⇧^{+}x. (INF i. f i x) ∂M) = (INF i. integral⇧^{N}M (f i))" using nn_integral_monotone_convergence_INF_AE[of f M i, OF _ *] f by (auto simp: decseq_Suc_iff le_fun_def) lemma nn_integral_liminf: fixes u :: "nat ⇒ 'a ⇒ ennreal" assumes u: "⋀i. u i ∈ borel_measurable M" shows "(∫⇧^{+}x. liminf (λn. u n x) ∂M) ≤ liminf (λn. integral⇧^{N}M (u n))" proof - have "(∫⇧^{+}x. liminf (λn. u n x) ∂M) = (SUP n. ∫⇧^{+}x. (INF i:{n..}. u i x) ∂M)" unfolding liminf_SUP_INF using u by (intro nn_integral_monotone_convergence_SUP_AE) (auto intro!: AE_I2 intro: INF_greatest INF_superset_mono) also have "… ≤ liminf (λn. integral⇧^{N}M (u n))" by (auto simp: liminf_SUP_INF intro!: SUP_mono INF_greatest nn_integral_mono INF_lower) finally show ?thesis . qed lemma nn_integral_limsup: fixes u :: "nat ⇒ 'a ⇒ ennreal" assumes [measurable]: "⋀i. u i ∈ borel_measurable M" "w ∈ borel_measurable M" assumes bounds: "⋀i. AE x in M. u i x ≤ w x" and w: "(∫⇧^{+}x. w x ∂M) < ∞" shows "limsup (λn. integral⇧^{N}M (u n)) ≤ (∫⇧^{+}x. limsup (λn. u n x) ∂M)" proof - have bnd: "AE x in M. ∀i. u i x ≤ w x" using bounds by (auto simp: AE_all_countable) then have "(∫⇧^{+}x. (SUP n. u n x) ∂M) ≤ (∫⇧^{+}x. w x ∂M)" by (auto intro!: nn_integral_mono_AE elim: eventually_mono intro: SUP_least) then have "(∫⇧^{+}x. limsup (λn. u n x) ∂M) = (INF n. ∫⇧^{+}x. (SUP i:{n..}. u i x) ∂M)" unfolding limsup_INF_SUP using bnd w by (intro nn_integral_monotone_convergence_INF_AE') (auto intro!: AE_I2 intro: SUP_least SUP_subset_mono) also have "… ≥ limsup (λn. integral⇧^{N}M (u n))" by (auto simp: limsup_INF_SUP intro!: INF_mono SUP_least exI nn_integral_mono SUP_upper) finally (xtrans) show ?thesis . qed lemma nn_integral_LIMSEQ: assumes f: "incseq f" "⋀i. f i ∈ borel_measurable M" and u: "⋀x. (λi. f i x) ⇢ u x" shows "(λn. integral⇧^{N}M (f n)) ⇢ integral⇧^{N}M u" proof - have "(λn. integral⇧^{N}M (f n)) ⇢ (SUP n. integral⇧^{N}M (f n))" using f by (intro LIMSEQ_SUP[of "λn. integral⇧^{N}M (f n)"] incseq_nn_integral) also have "(SUP n. integral⇧^{N}M (f n)) = integral⇧^{N}M (λx. SUP n. f n x)" using f by (intro nn_integral_monotone_convergence_SUP[symmetric]) also have "integral⇧^{N}M (λx. SUP n. f n x) = integral⇧^{N}M (λx. u x)" using f by (subst LIMSEQ_SUP[THEN LIMSEQ_unique, OF _ u]) (auto simp: incseq_def le_fun_def) finally show ?thesis . qed lemma nn_integral_dominated_convergence: assumes [measurable]: "⋀i. u i ∈ borel_measurable M" "u' ∈ borel_measurable M" "w ∈ borel_measurable M" and bound: "⋀j. AE x in M. u j x ≤ w x" and w: "(∫⇧^{+}x. w x ∂M) < ∞" and u': "AE x in M. (λi. u i x) ⇢ u' x" shows "(λi. (∫⇧^{+}x. u i x ∂M)) ⇢ (∫⇧^{+}x. u' x ∂M)" proof - have "limsup (λn. integral⇧^{N}M (u n)) ≤ (∫⇧^{+}x. limsup (λn. u n x) ∂M)" by (intro nn_integral_limsup[OF _ _ bound w]) auto moreover have "(∫⇧^{+}x. limsup (λn. u n x) ∂M) = (∫⇧^{+}x. u' x ∂M)" using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot) moreover have "(∫⇧^{+}x. liminf (λn. u n x) ∂M) = (∫⇧^{+}x. u' x ∂M)" using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot) moreover have "(∫⇧^{+}x. liminf (λn. u n x) ∂M) ≤ liminf (λn. integral⇧^{N}M (u n))" by (intro nn_integral_liminf) auto moreover have "liminf (λn. integral⇧^{N}M (u n)) ≤ limsup (λn. integral⇧^{N}M (u n))" by (intro Liminf_le_Limsup sequentially_bot) ultimately show ?thesis by (intro Liminf_eq_Limsup) auto qed lemma inf_continuous_nn_integral[order_continuous_intros]: assumes f: "⋀y. inf_continuous (f y)" assumes [measurable]: "⋀x. (λy. f y x) ∈ borel_measurable M" assumes bnd: "⋀x. (∫⇧^{+}y. f y x ∂M) ≠ ∞" shows "inf_continuous (λx. (∫⇧^{+}y. f y x ∂M))" unfolding inf_continuous_def proof safe fix C :: "nat ⇒ 'b" assume C: "decseq C" then show "(∫⇧^{+}y. f y (INFIMUM UNIV C) ∂M) = (INF i. ∫⇧^{+}y. f y (C i) ∂M)" using inf_continuous_mono[OF f] bnd by (auto simp add: inf_continuousD[OF f C] fun_eq_iff antimono_def mono_def le_fun_def less_top intro!: nn_integral_monotone_convergence_INF_decseq) qed lemma nn_integral_null_set: assumes "N ∈ null_sets M" shows "(∫⇧^{+}x. u x * indicator N x ∂M) = 0" proof - have "(∫⇧^{+}x. u x * indicator N x ∂M) = (∫⇧^{+}x. 0 ∂M)" proof (intro nn_integral_cong_AE AE_I) show "{x ∈ space M. u x * indicator N x ≠ 0} ⊆ N" by (auto simp: indicator_def) show "(emeasure M) N = 0" "N ∈ sets M" using assms by auto qed then show ?thesis by simp qed lemma nn_integral_0_iff: assumes u: "u ∈ borel_measurable M" shows "integral⇧^{N}M u = 0 ⟷ emeasure M {x∈space M. u x ≠ 0} = 0" (is "_ ⟷ (emeasure M) ?A = 0") proof - have u_eq: "(∫⇧^{+}x. u x * indicator ?A x ∂M) = integral⇧^{N}M u" by (auto intro!: nn_integral_cong simp: indicator_def) show ?thesis proof assume "(emeasure M) ?A = 0" with nn_integral_null_set[of ?A M u] u show "integral⇧^{N}M u = 0" by (simp add: u_eq null_sets_def) next assume *: "integral⇧^{N}M u = 0" let ?M = "λn. {x ∈ space M. 1 ≤ real (n::nat) * u x}" have "0 = (SUP n. (emeasure M) (?M n ∩ ?A))" proof - { fix n :: nat from nn_integral_Markov_inequality[OF u, of ?A "of_nat n"] u have "(emeasure M) (?M n ∩ ?A) ≤ 0" by (simp add: ennreal_of_nat_eq_real_of_nat u_eq *) moreover have "0 ≤ (emeasure M) (?M n ∩ ?A)" using u by auto ultimately have "(emeasure M) (?M n ∩ ?A) = 0" by auto } thus ?thesis by simp qed also have "… = (emeasure M) (⋃n. ?M n ∩ ?A)" proof (safe intro!: SUP_emeasure_incseq) fix n show "?M n ∩ ?A ∈ sets M" using u by (auto intro!: sets.Int) next show "incseq (λn. {x ∈ space M. 1 ≤ real n * u x} ∩ {x ∈ space M. u x ≠ 0})" proof (safe intro!: incseq_SucI) fix n :: nat and x assume *: "1 ≤ real n * u x" also have "real n * u x ≤ real (Suc n) * u x" by (auto intro!: mult_right_mono) finally show "1 ≤ real (Suc n) * u x" by auto qed qed also have "… = (emeasure M) {x∈space M. 0 < u x}" proof (safe intro!: arg_cong[where f="(emeasure M)"]) fix x assume "0 < u x" and [simp, intro]: "x ∈ space M" show "x ∈ (⋃n. ?M n ∩ ?A)" proof (cases "u x" rule: ennreal_cases) case (real r) with ‹0 < u x› have "0 < r" by auto obtain j :: nat where "1 / r ≤ real j" using real_arch_simple .. hence "1 / r * r ≤ real j * r" unfolding mult_le_cancel_right using ‹0 < r› by auto hence "1 ≤ real j * r" using real ‹0 < r› by auto thus ?thesis using ‹0 < r› real by (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_1[symmetric] ennreal_mult[symmetric] simp del: ennreal_1) qed (insert ‹0 < u x›, auto simp: ennreal_mult_top) qed (auto simp: zero_less_iff_neq_zero) finally show "emeasure M ?A = 0" by (simp add: zero_less_iff_neq_zero) qed qed lemma nn_integral_0_iff_AE: assumes u: "u ∈ borel_measurable M" shows "integral⇧^{N}M u = 0 ⟷ (AE x in M. u x = 0)" proof - have sets: "{x∈space M. u x ≠ 0} ∈ sets M" using u by auto show "integral⇧^{N}M u = 0 ⟷ (AE x in M. u x = 0)" using nn_integral_0_iff[of u] AE_iff_null[OF sets] u by auto qed lemma AE_iff_nn_integral: "{x∈space M. P x} ∈ sets M ⟹ (AE x in M. P x) ⟷ integral⇧^{N}M (indicator {x. ¬ P x}) = 0" by (subst nn_integral_0_iff_AE) (auto simp: indicator_def[abs_def]) lemma nn_integral_less: assumes [measurable]: "f ∈ borel_measurable M" "g ∈ borel_measurable M" assumes f: "(∫⇧^{+}x. f x ∂M) ≠ ∞" assumes ord: "AE x in M. f x ≤ g x" "¬ (AE x in M. g x ≤ f x)" shows "(∫⇧^{+}x. f x ∂M) < (∫⇧^{+}x. g x ∂M)" proof - have "0 < (∫⇧^{+}x. g x - f x ∂M)" proof (intro order_le_neq_trans notI) assume "0 = (∫⇧^{+}x. g x - f x ∂M)" then have "AE x in M. g x - f x = 0" using nn_integral_0_iff_AE[of "λx. g x - f x" M] by simp with ord(1) have "AE x in M. g x ≤ f x" by eventually_elim (auto simp: ennreal_minus_eq_0) with ord show False by simp qed simp also have "… = (∫⇧^{+}x. g x ∂M) - (∫⇧^{+}x. f x ∂M)" using f by (subst nn_integral_diff) (auto simp: ord) finally show ?thesis using f by (auto dest!: ennreal_minus_pos_iff[rotated] simp: less_top) qed lemma nn_integral_subalgebra: assumes f: "f ∈ borel_measurable N" and N: "sets N ⊆ sets M" "space N = space M" "⋀A. A ∈ sets N ⟹ emeasure N A = emeasure M A" shows "integral⇧^{N}N f = integral⇧^{N}M f" proof - have [simp]: "⋀f :: 'a ⇒ ennreal. f ∈ borel_measurable N ⟹ f ∈ borel_measurable M" using N by (auto simp: measurable_def) have [simp]: "⋀P. (AE x in N. P x) ⟹ (AE x in M. P x)" using N by (auto simp add: eventually_ae_filter null_sets_def subset_eq) have [simp]: "⋀A. A ∈ sets N ⟹ A ∈ sets M" using N by auto from f show ?thesis apply induct apply (simp_all add: nn_integral_add nn_integral_cmult nn_integral_monotone_convergence_SUP N) apply (auto intro!: nn_integral_cong cong: nn_integral_cong simp: N(2)[symmetric]) done qed lemma nn_integral_nat_function: fixes f :: "'a ⇒ nat" assumes "f ∈ measurable M (count_space UNIV)" shows "(∫⇧^{+}x. of_nat (f x) ∂M) = (∑t. emeasure M {x∈space M. t < f x})" proof - define F where "F i = {x∈space M. i < f x}" for i with assms have [measurable]: "⋀i. F i ∈ sets M" by auto { fix x assume "x ∈ space M" have "(λi. if i < f x then 1 else 0) sums (of_nat (f x)::real)" using sums_If_finite[of "λi. i < f x" "λ_. 1::real"] by simp then have "(λi. ennreal (if i < f x then 1 else 0)) sums of_nat(f x)" unfolding ennreal_of_nat_eq_real_of_nat by (subst sums_ennreal) auto moreover have "⋀i. ennreal (if i < f x then 1 else 0) = indicator (F i) x" using ‹x ∈ space M› by (simp add: one_ennreal_def F_def) ultimately have "of_nat (f x) = (∑i. indicator (F i) x :: ennreal)" by (simp add: sums_iff) } then have "(∫⇧^{+}x. of_nat (f x) ∂M) = (∫⇧^{+}x. (∑i. indicator (F i) x) ∂M)" by (simp cong: nn_integral_cong) also have "… = (∑i. emeasure M (F i))" by (simp add: nn_integral_suminf) finally show ?thesis by (simp add: F_def) qed lemma nn_integral_lfp: assumes sets[simp]: "⋀s. sets (M s) = sets N" assumes f: "sup_continuous f" assumes g: "sup_continuous g" assumes meas: "⋀F. F ∈ borel_measurable N ⟹ f F ∈ borel_measurable N" assumes step: "⋀F s. F ∈ borel_measurable N ⟹ integral⇧^{N}(M s) (f F) = g (λs. integral⇧^{N}(M s) F) s" shows "(∫⇧^{+}ω. lfp f ω ∂M s) = lfp g s" proof (subst lfp_transfer_bounded[where α="λF s. ∫⇧^{+}x. F x ∂M s" and g=g and f=f and P="λf. f ∈ borel_measurable N", symmetric]) fix C :: "nat ⇒ 'b ⇒ ennreal" assume "incseq C" "⋀i. C i ∈ borel_measurable N" then show "(λs. ∫⇧^{+}x. (SUP i. C i) x ∂M s) = (SUP i. (λs. ∫⇧^{+}x. C i x ∂M s))" unfolding SUP_apply[abs_def] by (subst nn_integral_monotone_convergence_SUP) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets) qed (auto simp add: step le_fun_def SUP_apply[abs_def] bot_fun_def bot_ennreal intro!: meas f g) lemma nn_integral_gfp: assumes sets[simp]: "⋀s. sets (M s) = sets N" assumes f: "inf_continuous f" and g: "inf_continuous g" assumes meas: "⋀F. F ∈ borel_measurable N ⟹ f F ∈ borel_measurable N" assumes bound: "⋀F s. F ∈ borel_measurable N ⟹ (∫⇧^{+}x. f F x ∂M s) < ∞" assumes non_zero: "⋀s. emeasure (M s) (space (M s)) ≠ 0" assumes step: "⋀F s. F ∈ borel_measurable N ⟹ integral⇧^{N}(M s) (f F) = g (λs. integral⇧^{N}(M s) F) s" shows "(∫⇧^{+}ω. gfp f ω ∂M s) = gfp g s" proof (subst gfp_transfer_bounded[where α="λF s. ∫⇧^{+}x. F x ∂M s" and g=g and f=f and P="λF. F ∈ borel_measurable N ∧ (∀s. (∫⇧^{+}x. F x ∂M s) < ∞)", symmetric]) fix C :: "nat ⇒ 'b ⇒ ennreal" assume "decseq C" "⋀i. C i ∈ borel_measurable N ∧ (∀s. integral⇧^{N}(M s) (C i) < ∞)" then show "(λs. ∫⇧^{+}x. (INF i. C i) x ∂M s) = (INF i. (λs. ∫⇧^{+}x. C i x ∂M s))" unfolding INF_apply[abs_def] by (subst nn_integral_monotone_convergence_INF_decseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets) next show "⋀x. g x ≤ (λs. integral⇧^{N}(M s) (f top))" by (subst step) (auto simp add: top_fun_def less_le non_zero le_fun_def ennreal_top_mult cong del: if_weak_cong intro!: monoD[OF inf_continuous_mono[OF g], THEN le_funD]) next fix C assume "⋀i::nat. C i ∈ borel_measurable N ∧ (∀s. integral⇧^{N}(M s) (C i) < ∞)" "decseq C" with bound show "INFIMUM UNIV C ∈ borel_measurable N ∧ (∀s. integral⇧^{N}(M s) (INFIMUM UNIV C) < ∞)" unfolding INF_apply[abs_def] by (subst nn_integral_monotone_convergence_INF_decseq) (auto simp: INF_less_iff cong: measurable_cong_sets intro!: borel_measurable_INF) next show "⋀x. x ∈ borel_measurable N ∧ (∀s. integral⇧^{N}(M s) x < ∞) ⟹ (λs. integral⇧^{N}(M s) (f x)) = g (λs. integral⇧^{N}(M s) x)" by (subst step) auto qed (insert bound, auto simp add: le_fun_def INF_apply[abs_def] top_fun_def intro!: meas f g) subsection ‹Integral under concrete measures› lemma nn_integral_mono_measure: assumes "sets M = sets N" "M ≤ N" shows "nn_integral M f ≤ nn_integral N f" unfolding nn_integral_def proof (intro SUP_subset_mono) note ‹sets M = sets N›[simp] ‹sets M = sets N›[THEN sets_eq_imp_space_eq, simp] show "{g. simple_function M g ∧ g ≤ f} ⊆ {g. simple_function N g ∧ g ≤ f}" by (simp add: simple_function_def) show "integral⇧^{S}M x ≤ integral⇧^{S}N x" for x using le_measureD3[OF ‹M ≤ N›] by (auto simp add: simple_integral_def intro!: sum_mono mult_mono) qed lemma nn_integral_empty: assumes "space M = {}" shows "nn_integral M f = 0" proof - have "(∫⇧^{+}x. f x ∂M) = (∫⇧^{+}x. 0 ∂M)" by(rule nn_integral_cong)(simp add: assms) thus ?thesis by simp qed lemma nn_integral_bot[simp]: "nn_integral bot f = 0" by (simp add: nn_integral_empty) subsubsection ‹Distributions› lemma nn_integral_distr: assumes T: "T ∈ measurable M M'" and f: "f ∈ borel_measurable (distr M M' T)" shows "integral⇧^{N}(distr M M' T) f = (∫⇧^{+}x. f (T x) ∂M)" using f proof induct case (cong f g) with T show ?case apply (subst nn_integral_cong[of _ f g]) apply simp apply (subst nn_integral_cong[of _ "λx. f (T x)" "λx. g (T x)"]) apply (simp add: measurable_def Pi_iff) apply simp done next case (set A) then have eq: "⋀x. x ∈ space M ⟹ indicator A (T x) = indicator (T -` A ∩ space M) x" by (auto simp: indicator_def) from set T show ?case by (subst nn_integral_cong[OF eq]) (auto simp add: emeasure_distr intro!: nn_integral_indicator[symmetric] measurable_sets) qed (simp_all add: measurable_compose[OF T] T nn_integral_cmult nn_integral_add nn_integral_monotone_convergence_SUP le_fun_def incseq_def) subsubsection ‹Counting space› lemma simple_function_count_space[simp]: "simple_function (count_space A) f ⟷ finite (f ` A)" unfolding simple_function_def by simp lemma nn_integral_count_space: assumes A: "finite {a∈A. 0 < f a}" shows "integral⇧^{N}(count_space A) f = (∑a|a∈A ∧ 0 < f a. f a)" proof - have *: "(∫⇧^{+}x. max 0 (f x) ∂count_space A) = (∫⇧^{+}x. (∑a|a∈A ∧ 0 < f a. f a * indicator {a} x) ∂count_space A)" by (auto intro!: nn_integral_cong simp add: indicator_def if_distrib sum.If_cases[OF A] max_def le_less) also have "… = (∑a|a∈A ∧ 0 < f a. ∫⇧^{+}x. f a * indicator {a} x ∂count_space A)" by (subst nn_integral_sum) (simp_all add: AE_count_space less_imp_le) also have "… = (∑a|a∈A ∧ 0 < f a. f a)" by (auto intro!: sum.cong simp: one_ennreal_def[symmetric] max_def) finally show ?thesis by (simp add: max.absorb2) qed lemma nn_integral_count_space_finite: "finite A ⟹ (∫⇧^{+}x. f x ∂count_space A) = (∑a∈A. f a)" by (auto intro!: sum.mono_neutral_left simp: nn_integral_count_space less_le) lemma nn_integral_count_space': assumes "finite A" "⋀x. x ∈ B ⟹ x ∉ A ⟹ f x = 0" "A ⊆ B" shows "(∫⇧^{+}x. f x ∂count_space B) = (∑x∈A. f x)" proof - have "(∫⇧^{+}x. f x ∂count_space B) = (∑a | a ∈ B ∧ 0 < f a. f a)" using assms(2,3) by (intro nn_integral_count_space finite_subset[OF _ ‹finite A›]) (auto simp: less_le) also have "… = (∑a∈A. f a)" using assms by (intro sum.mono_neutral_cong_left) (auto simp: less_le) finally show ?thesis . qed lemma nn_integral_bij_count_space: assumes g: "bij_betw g A B" shows "(∫⇧^{+}x. f (g x) ∂count_space A) = (∫⇧^{+}x. f x ∂count_space B)" using g[THEN bij_betw_imp_funcset] by (subst distr_bij_count_space[OF g, symmetric]) (auto intro!: nn_integral_distr[symmetric]) lemma nn_integral_indicator_finite: fixes f :: "'a ⇒ ennreal" assumes f: "finite A" and [measurable]: "⋀a. a ∈ A ⟹ {a} ∈ sets M" shows "(∫⇧^{+}x. f x * indicator A x ∂M) = (∑x∈A. f x * emeasure M {x})" proof - from f have "(∫⇧^{+}x. f x * indicator A x ∂M) = (∫⇧^{+}x. (∑a∈A. f a * indicator {a} x) ∂M)" by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="λa. x * a" for x] sum.If_cases) also have "… = (∑a∈A. f a * emeasure M {a})" by (subst nn_integral_sum) auto finally show ?thesis . qed lemma nn_integral_count_space_nat: fixes f :: "nat ⇒ ennreal" shows "(∫⇧^{+}i. f i ∂count_space UNIV) = (∑i. f i)" proof - have "(∫⇧^{+}i. f i ∂count_space UNIV) = (∫⇧^{+}i. (∑j. f j * indicator {j} i) ∂count_space UNIV)" proof (intro nn_integral_cong) fix i have "f i = (∑j∈{i}. f j * indicator {j} i)" by simp also have "… = (∑j. f j * indicator {j} i)" by (rule suminf_finite[symmetric]) auto finally show "f i = (∑j. f j * indicator {j} i)" . qed also have "… = (∑j. (∫⇧^{+}i. f j * indicator {j} i ∂count_space UNIV))" by (rule nn_integral_suminf) auto finally show ?thesis by simp qed lemma nn_integral_enat_function: assumes f: "f ∈ measurable M (count_space UNIV)" shows "(∫⇧^{+}x. ennreal_of_enat (f x) ∂M) = (∑t. emeasure M {x ∈ space M. t < f x})" proof - define F where "F i = {x∈space M. i < f x}" for i :: nat with assms have [measurable]: "⋀i. F i ∈ sets M" by auto { fix x assume "x ∈ space M" have "(λi::nat. if i < f x then 1 else 0) sums ennreal_of_enat (f x)" using sums_If_finite[of "λr. r < f x" "λ_. 1 :: ennreal"] by (cases "f x") (simp_all add: sums_def of_nat_tendsto_top_ennreal) also have "(λi. (if i < f x then 1 else 0)) = (λi. indicator (F i) x)" using ‹x ∈ space M› by (simp add: one_ennreal_def F_def fun_eq_iff) finally have "ennreal_of_enat (f x) = (∑i. indicator (F i) x)" by (simp add: sums_iff) } then have "(∫⇧^{+}x. ennreal_of_enat (f x) ∂M) = (∫⇧^{+}x. (∑i. indicator (F i) x) ∂M)" by (simp cong: nn_integral_cong) also have "… = (∑i. emeasure M (F i))" by (simp add: nn_integral_suminf) finally show ?thesis by (simp add: F_def) qed lemma nn_integral_count_space_nn_integral: fixes f :: "'i ⇒ 'a ⇒ ennreal" assumes "countable I" and [measurable]: "⋀i. i ∈ I ⟹ f i ∈ borel_measurable M" shows "(∫⇧^{+}x. ∫⇧^{+}i. f i x ∂count_space I ∂M) = (∫⇧^{+}i. ∫⇧^{+}x. f i x ∂M ∂count_space I)" proof cases assume "finite I" then show ?thesis by (simp add: nn_integral_count_space_finite nn_integral_sum) next assume "infinite I" then have [simp]: "I ≠ {}" by auto note * = bij_betw_from_nat_into[OF ‹countable I› ‹infinite I›] have **: "⋀f. (⋀i. 0 ≤ f i) ⟹ (∫⇧^{+}i. f i ∂count_space I) = (∑n. f (from_nat_into I n))" by (simp add: nn_integral_bij_count_space[symmetric, OF *] nn_integral_count_space_nat) show ?thesis by (simp add: ** nn_integral_suminf from_nat_into) qed lemma of_bool_Bex_eq_nn_integral: assumes unique: "⋀x y. x ∈ X ⟹ y ∈ X ⟹ P x ⟹ P y ⟹ x = y" shows "of_bool (∃y∈X. P y) = (∫⇧^{+}y. of_bool (P y) ∂count_space X)" proof cases assume "∃y∈X. P y" then obtain y where "P y" "y ∈ X" by auto then show ?thesis by (subst nn_integral_count_space'[where A="{y}"]) (auto dest: unique) qed (auto cong: nn_integral_cong_simp) lemma emeasure_UN_countable: assumes sets[measurable]: "⋀i. i ∈ I ⟹ X i ∈ sets M" and I[simp]: "countable I" assumes disj: "disjoint_family_on X I" shows "emeasure M (UNION I X) = (∫⇧^{+}i. emeasure M (X i) ∂count_space I)" proof - have eq: "⋀x. indicator (UNION I X) x = ∫⇧^{+}i. indicator (X i) x ∂count_space I" proof cases fix x assume x: "x ∈ UNION I X" then obtain j where j: "x ∈ X j" "j ∈ I" by auto with disj have "⋀i. i ∈ I ⟹ indicator (X i) x = (indicator {j} i::ennreal)" by (auto simp: disjoint_family_on_def split: split_indicator) with x j show "?thesis x" by (simp cong: nn_integral_cong_simp) qed (auto simp: nn_integral_0_iff_AE) note sets.countable_UN'[unfolded subset_eq, measurable] have "emeasure M (UNION I X) = (∫⇧^{+}x. indicator (UNION I X) x ∂M)" by simp also have "… = (∫⇧^{+}i. ∫⇧^{+}x. indicator (X i) x ∂M ∂count_space I)" by (simp add: eq nn_integral_count_space_nn_integral) finally show ?thesis by (simp cong: nn_integral_cong_simp) qed lemma emeasure_countable_singleton: assumes sets: "⋀x. x ∈ X ⟹ {x} ∈ sets M" and X: "countable X" shows "emeasure M X = (∫⇧^{+}x. emeasure M {x} ∂count_space X)" proof - have "emeasure M (⋃i∈X. {i}) = (∫⇧^{+}x. emeasure M {x} ∂count_space X)" using assms by (intro emeasure_UN_countable) (auto simp: disjoint_family_on_def) also have "(⋃i∈X. {i}) = X" by auto finally show ?thesis . qed lemma measure_eqI_countable: assumes [simp]: "sets M = Pow A" "sets N = Pow A" and A: "countable A" assumes eq: "⋀a. a ∈ A ⟹ emeasure M {a} = emeasure N {a}" shows "M = N" proof (rule measure_eqI) fix X assume "X ∈ sets M" then have X: "X ⊆ A" by auto moreover from A X have "countable X" by (auto dest: countable_subset) ultimately have "emeasure M X = (∫⇧^{+}a. emeasure M {a} ∂count_space X)" "emeasure N X = (∫⇧^{+}a. emeasure N {a} ∂count_space X)" by (auto intro!: emeasure_countable_singleton) moreover have "(∫⇧^{+}a. emeasure M {a} ∂count_space X) = (∫⇧^{+}a. emeasure N {a} ∂count_space X)" using X by (intro nn_integral_cong eq) auto ultimately show "emeasure M X = emeasure N X" by simp qed simp lemma measure_eqI_countable_AE: assumes [simp]: "sets M = UNIV" "sets N = UNIV" assumes ae: "AE x in M. x ∈ Ω" "AE x in N. x ∈ Ω" and [simp]: "countable Ω" assumes eq: "⋀x. x ∈ Ω ⟹ emeasure M {x} = emeasure N {x}" shows "M = N" proof (rule measure_eqI) fix A have "emeasure N A = emeasure N {x∈Ω. x ∈ A}" using ae by (intro emeasure_eq_AE) auto also have "… = (∫⇧^{+}x. emeasure N {x} ∂count_space {x∈Ω. x ∈ A})" by (intro emeasure_countable_singleton) auto also have "… = (∫⇧^{+}x. emeasure M {x} ∂count_space {x∈Ω. x ∈ A})" by (intro nn_integral_cong eq[symmetric]) auto also have "… = emeasure M {x∈Ω. x ∈ A}" by (intro emeasure_countable_singleton[symmetric]) auto also have "… = emeasure M A" using ae by (intro emeasure_eq_AE) auto finally show "emeasure M A = emeasure N A" .. qed simp lemma nn_integral_monotone_convergence_SUP_nat: fixes f :: "'a ⇒ nat ⇒ ennreal" assumes chain: "Complete_Partial_Order.chain (≤) (f ` Y)" and nonempty: "Y ≠ {}" shows "(∫⇧^{+}x. (SUP i:Y. f i x) ∂count_space UNIV) = (SUP i:Y. (∫⇧^{+}x. f i x ∂count_space UNIV))" (is "?lhs = ?rhs" is "integral⇧^{N}?M _ = _") proof (rule order_class.order.antisym) show "?rhs ≤ ?lhs" by (auto intro!: SUP_least SUP_upper nn_integral_mono) next have "∃g. incseq g ∧ range g ⊆ (λi. f i x) ` Y ∧ (SUP i:Y. f i x) = (SUP i. g i)" for x by (rule ennreal_Sup_countable_SUP) (simp add: nonempty) then obtain g where incseq: "⋀x. incseq (g x)" and range: "⋀x. range (g x) ⊆ (λi. f i x) ` Y" and sup: "⋀x. (SUP i:Y. f i x) = (SUP i. g x i)" by moura from incseq have incseq': "incseq (λi x. g x i)" by(blast intro: incseq_SucI le_funI dest: incseq_SucD) have "?lhs = ∫⇧^{+}x. (SUP i. g x i) ∂?M" by(simp add: sup) also have "… = (SUP i. ∫⇧^{+}x. g x i ∂?M)" using incseq' by(rule nn_integral_monotone_convergence_SUP) simp also have "… ≤ (SUP i:Y. ∫⇧^{+}x. f i x ∂?M)" proof(rule SUP_least) fix n have "⋀x. ∃i. g x n = f i x ∧ i ∈ Y" using range by blast then obtain I where I: "⋀x. g x n = f (I x) x" "⋀x. I x ∈ Y" by moura have "(∫⇧^{+}x. g x n ∂count_space UNIV) = (∑x. g x n)" by(rule nn_integral_count_space_nat) also have "… = (SUP m. ∑x<m. g x n)" by(rule suminf_eq_SUP) also have "… ≤ (SUP i:Y. ∫⇧^{+}x. f i x ∂?M)" proof(rule SUP_mono) fix m show "∃m'∈Y. (∑x<m. g x n) ≤ (∫⇧^{+}x. f m' x ∂?M)" proof(cases "m > 0") case False thus ?thesis using nonempty by auto next case True let ?Y = "I ` {..<m}" have "f ` ?Y ⊆ f ` Y" using I by auto with chain have chain': "Complete_Partial_Order.chain (≤) (f ` ?Y)" by(rule chain_subset) hence "Sup (f ` ?Y) ∈ f ` ?Y" by(rule ccpo_class.in_chain_finite)(auto simp add: True lessThan_empty_iff) then obtain m' where "m' < m" and m': "(SUP i:?Y. f i) = f (I m')" by auto have "I m' ∈ Y" using I by blast have "(∑x<m. g x n) ≤ (∑x<m. f (I m') x)" proof(rule sum_mono) fix x assume "x ∈ {..<m}" hence "x < m" by simp have "g x n = f (I x) x" by(simp add: I) also have "… ≤ (SUP i:?Y. f i) x" unfolding Sup_fun_def image_image using ‹x ∈ {..<m}› by (rule Sup_upper [OF imageI]) also have "… = f (I m') x" unfolding m' by simp finally show "g x n ≤ f (I m') x" . qed also have "… ≤ (SUP m. (∑x<m. f (I m') x))" by(rule SUP_upper) simp also have "… = (∑x. f (I m') x)" by(rule suminf_eq_SUP[symmetric]) also have "… = (∫⇧^{+}x. f (I m') x ∂?M)" by(rule nn_integral_count_space_nat[symmetric]) finally show ?thesis using ‹I m' ∈ Y› by blast qed qed finally show "(∫⇧^{+}x. g x n ∂count_space UNIV) ≤ …" . qed finally show "?lhs ≤ ?rhs" . qed lemma power_series_tendsto_at_left: assumes nonneg: "⋀i. 0 ≤ f i" and summable: "⋀z. 0 ≤ z ⟹ z < 1 ⟹ summable (λn. f n * z^n)" shows "((λz. ennreal (∑n. f n * z^n)) ⤏ (∑n. ennreal (f n))) (at_left (1::real))" proof (intro tendsto_at_left_sequentially) show "0 < (1::real)" by simp fix S :: "nat ⇒ real" assume S: "⋀n. S n < 1" "⋀n. 0 < S n" "S ⇢ 1" "incseq S" then have S_nonneg: "⋀i. 0 ≤ S i" by (auto intro: less_imp_le) have "(λi. (∫⇧^{+}n. f n * S i^n ∂count_space UNIV)) ⇢ (∫⇧^{+}n. ennreal (f n) ∂count_space UNIV)" proof (rule nn_integral_LIMSEQ) show "incseq (λi n. ennreal (f n * S i^n))" using S by (auto intro!: mult_mono power_mono nonneg ennreal_leI simp: incseq_def le_fun_def less_imp_le) fix n have "(λi. ennreal (f n * S i^n)) ⇢ ennreal (f n * 1^n)" by (intro tendsto_intros tendsto_ennrealI S) then show "(λi. ennreal (f n * S i^n)) ⇢ ennreal (f n)" by simp qed (auto simp: S_nonneg intro!: mult_nonneg_nonneg nonneg) also have "(λi. (∫⇧^{+}n. f n * S i^n ∂count_space UNIV)) = (λi. ∑n. f n * S i^n)" by (subst nn_integral_count_space_nat) (intro ext suminf_ennreal2 mult_nonneg_nonneg nonneg S_nonneg zero_le_power summable S)+ also have "(∫⇧^{+}n. ennreal (f n) ∂count_space UNIV) = (∑n. ennreal (f n))" by (simp add: nn_integral_count_space_nat nonneg) finally show "(λn. ennreal (∑na. f na * S n ^ na)) ⇢ (∑n. ennreal (f n))" . qed subsubsection ‹Measures with Restricted Space› lemma simple_function_restrict_space_ennreal: fixes f :: "'a ⇒ ennreal" assumes "Ω ∩ space M ∈ sets M" shows "simple_function (restrict_space M Ω) f ⟷ simple_function M (λx. f x * indicator Ω x)" proof - { assume "finite (f ` space (restrict_space M Ω))" then have "finite (f ` space (restrict_space M Ω) ∪ {0})" by simp then have "finite ((λx. f x * indicator Ω x) ` space M)" by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } moreover { assume "finite ((λx. f x * indicator Ω x) ` space M)" then have "finite (f ` space (restrict_space M Ω))" by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } ultimately show ?thesis unfolding simple_function_iff_borel_measurable borel_measurable_restrict_space_iff_ennreal[OF assms] by auto qed lemma simple_function_restrict_space: fixes f :: "'a ⇒ 'b::real_normed_vector" assumes "Ω ∩ space M ∈ sets M" shows "simple_function (restrict_space M Ω) f ⟷ simple_function M (λx. indicator Ω x *⇩_{R}f x)" proof - { assume "finite (f ` space (restrict_space M Ω))" then have "finite (f ` space (restrict_space M Ω) ∪ {0})" by simp then have "finite ((λx. indicator Ω x *⇩_{R}f x) ` space M)" by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } moreover { assume "finite ((λx. indicator Ω x *⇩_{R}f x) ` space M)" then have "finite (f ` space (restrict_space M Ω))" by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } ultimately show ?thesis unfolding simple_function_iff_borel_measurable borel_measurable_restrict_space_iff[OF assms] by auto qed lemma simple_integral_restrict_space: assumes Ω: "Ω ∩ space M ∈ sets M" "simple_function (restrict_space M Ω) f" shows "simple_integral (restrict_space M Ω) f = simple_integral M (λx. f x * indicator Ω x)" using simple_function_restrict_space_ennreal[THEN iffD1, OF Ω, THEN simple_functionD(1)] by (auto simp add: space_restrict_space emeasure_restrict_space[OF Ω(1)] le_infI2 simple_integral_def split: split_indicator split_indicator_asm intro!: sum.mono_neutral_cong_left ennreal_mult_left_cong arg_cong2[where f=emeasure]) lemma nn_integral_restrict_space: assumes Ω[simp]: "Ω ∩ space M ∈ sets M" shows "nn_integral (restrict_space M Ω) f = nn_integral M (λx. f x * indicator Ω x)" proof - let ?R = "restrict_space M Ω" and ?X = "λM f. {s. simple_function M s ∧ s ≤ f ∧ (∀x. s x < top)}" have "integral⇧^{S}?R ` ?X ?R f = integral⇧^{S}M ` ?X M (λx. f x * indicator Ω x)" proof (safe intro!: image_eqI) fix s assume s: "simple_function ?R s" "s ≤ f" "∀x. s x < top" from s show "integral⇧^{S}(restrict_space M Ω) s = integral⇧^{S}M (λx. s x * indicator Ω x)" by (intro simple_integral_restrict_space) auto from s show "simple_function M (λx. s x * indicator Ω x)" by (simp add: simple_function_restrict_space_ennreal) from s show "(λx. s x * indicator Ω x) ≤ (λx. f x * indicator Ω x)" "⋀x. s x * indicator Ω x < top" by (auto split: split_indicator simp: le_fun_def image_subset_iff) next fix s assume s: "simple_function M s" "s ≤ (λx. f x * indicator Ω x)" "∀x. s x < top" then have "simple_function M (λx. s x * indicator (Ω ∩ space M) x)" (is ?s') by (intro simple_function_mult simple_function_indicator) auto also have "?s' ⟷ simple_function M (λx. s x * indicator Ω x)" by (rule simple_function_cong) (auto split: split_indicator) finally show sf: "simple_function (restrict_space M Ω) s" by (simp add: simple_function_restrict_space_ennreal) from s have s_eq: "s = (λx. s x * indicator Ω x)" by (auto simp add: fun_eq_iff le_fun_def image_subset_iff split: split_indicator split_indicator_asm intro: antisym) show "integral⇧^{S}M s = integral⇧^{S}(restrict_space M Ω) s" by (subst s_eq) (rule simple_integral_restrict_space[symmetric, OF Ω sf]) show "⋀x. s x < top" using s by (auto simp: image_subset_iff) from s show "s ≤ f" by (subst s_eq) (auto simp: image_subset_iff le_fun_def split: split_indicator split_indicator_asm) qed then show ?thesis unfolding nn_integral_def_finite by (simp cong del: strong_SUP_cong) qed lemma nn_integral_count_space_indicator: assumes "NO_MATCH (UNIV::'a set) (X::'a set)" shows "(∫⇧^{+}x. f x ∂count_space X) = (∫⇧^{+}x. f x * indicator X x ∂count_space UNIV)" by (simp add: nn_integral_restrict_space[symmetric] restrict_count_space) lemma nn_integral_count_space_eq: "(⋀x. x ∈ A - B ⟹ f x = 0) ⟹ (⋀x. x ∈ B - A ⟹ f x = 0) ⟹ (∫⇧^{+}x. f x ∂count_space A) = (∫⇧^{+}x. f x ∂count_space B)" by (auto simp: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator) lemma nn_integral_ge_point: assumes "x ∈ A" shows "p x ≤ ∫⇧^{+}x. p x ∂count_space A" proof - from assms have "p x ≤ ∫⇧^{+}x. p x ∂count_space {x}" by(auto simp add: nn_integral_count_space_finite max_def) also have "… = ∫⇧^{+}x'. p x' * indicator {x} x' ∂count_space A" using assms by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong) also have "… ≤ ∫⇧^{+}x. p x ∂count_space A" by(rule nn_integral_mono)(simp add: indicator_def) finally show ?thesis . qed subsubsection ‹Measure spaces with an associated density› definition density :: "'a measure ⇒ ('a ⇒ ennreal) ⇒ 'a measure" where "density M f = measure_of (space M) (sets M) (λA. ∫⇧^{+}x. f x * indicator A x ∂M)" lemma shows sets_density[simp, measurable_cong]: "sets (density M f) = sets M" and space_density[simp]: "space (density M f) = space M" by (auto simp: density_def) (* FIXME: add conversion to simplify space, sets and measurable *) lemma space_density_imp[measurable_dest]: "⋀x M f. x ∈ space (density M f) ⟹ x ∈ space M" by auto lemma shows measurable_density_eq1[simp]: "g ∈ measurable (density Mg f) Mg' ⟷ g ∈ measurable Mg Mg'" and measurable_density_eq2[simp]: "h ∈ measurable Mh (density Mh' f) ⟷ h ∈ measurable Mh Mh'" and simple_function_density_eq[simp]: "simple_function (density Mu f) u ⟷ simple_function Mu u" unfolding measurable_def simple_function_def by simp_all lemma density_cong: "f ∈ borel_measurable M ⟹ f' ∈ borel_measurable M ⟹ (AE x in M. f x = f' x) ⟹ density M f = density M f'" unfolding density_def by (auto intro!: measure_of_eq nn_integral_cong_AE sets.space_closed) lemma emeasure_density: assumes f[measurable]: "f ∈ borel_measurable M" and A[measurable]: "A ∈ sets M" shows "emeasure (density M f) A = (∫⇧^{+}x. f x * indicator A x ∂M)" (is "_ = ?μ A") unfolding density_def proof (rule emeasure_measure_of_sigma) show "sigma_algebra (space M) (sets M)" .. show "positive (sets M) ?μ" using f by (auto simp: positive_def) show "countably_additive (sets M) ?μ" proof (intro countably_additiveI) fix A :: "nat ⇒ 'a set" assume "range A ⊆ sets M" then have "⋀i. A i ∈ sets M" by auto then have *: "⋀i. (λx. f x * indicator (A i) x) ∈ borel_measurable M" by auto assume disj: "disjoint_family A" then have "(∑n. ?μ (A n)) = (∫⇧^{+}x. (∑n. f x * indicator (A n) x) ∂M)" using f * by (subst nn_integral_suminf) auto also have "(∫⇧^{+}x. (∑n. f x * indicator (A n) x) ∂M) = (∫⇧^{+}x. f x * (∑n. indicator (A n) x) ∂M)" using f by (auto intro!: ennreal_suminf_cmult nn_integral_cong_AE) also have "… = (∫⇧^{+}x. f x * indicator (⋃n. A n) x ∂M)" unfolding suminf_indicator[OF disj] .. finally show "(∑i. ∫⇧^{+}x. f x * indicator (A i) x ∂M) = ∫⇧^{+}x. f x * indicator (⋃i. A i) x ∂M" . qed qed fact lemma null_sets_density_iff: assumes f: "f ∈ borel_measurable M" shows "A ∈ null_sets (density M f) ⟷ A ∈ sets M ∧ (AE x in M. x ∈ A ⟶ f x = 0)" proof - { assume "A ∈ sets M" have "(∫⇧^{+}x. f x * indicator A x ∂M) = 0 ⟷ emeasure M {x ∈ space M. f x * indicator A x ≠ 0} = 0" using f ‹A ∈ sets M› by (intro nn_integral_0_iff) auto also have "… ⟷ (AE x in M. f x * indicator A x = 0)" using f ‹A ∈ sets M› by (intro AE_iff_measurable[OF _ refl, symmetric]) auto also have "(AE x in M. f x * indicator A x = 0) ⟷ (AE x in M. x ∈ A ⟶ f x ≤ 0)" by (auto simp add: indicator_def max_def split: if_split_asm) finally have "(∫⇧^{+}x. f x * indicator A x ∂M) = 0 ⟷ (AE x in M. x ∈ A ⟶ f x ≤ 0)" . } with f show ?thesis by (simp add: null_sets_def emeasure_density cong: conj_cong) qed lemma AE_density: assumes f: "f ∈ borel_measurable M" shows "(AE x in density M f. P x) ⟷ (AE x in M. 0 < f x ⟶ P x)" proof assume "AE x in density M f. P x" with f obtain N where "{x ∈ space M. ¬ P x} ⊆ N" "N ∈ sets M" and ae: "AE x in M. x ∈ N ⟶ f x = 0" by (auto simp: eventually_ae_filter null_sets_density_iff) then have "AE x in M. x ∉ N ⟶ P x" by auto with ae show "AE x in M. 0 < f x ⟶ P x" by (rule eventually_elim2) auto next fix N assume ae: "AE x in M. 0 < f x ⟶ P x" then obtain N where "{x ∈ space M. ¬ (0 < f x ⟶ P x)} ⊆ N" "N ∈ null_sets M" by (auto simp: eventually_ae_filter) then have *: "{x ∈ space (density M f). ¬ P x} ⊆ N ∪ {x∈space M. f x = 0}" "N ∪ {x∈space M. f x = 0} ∈ sets M" and ae2: "AE x in M. x ∉ N" using f by (auto simp: subset_eq zero_less_iff_neq_zero intro!: AE_not_in) show "AE x in density M f. P x" using ae2 unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f] by (intro exI[of _ "N ∪ {x∈space M. f x = 0}"] conjI *) (auto elim: eventually_elim2) qed lemma nn_integral_density: assumes f: "f ∈ borel_measurable M" assumes g: "g ∈ borel_measurable M" shows "integral⇧^{N}(density M f) g = (∫⇧^{+}x. f x * g x ∂M)" using g proof induct case (cong u v) then show ?case apply (subst nn_integral_cong[OF cong(3)]) apply (simp_all cong: nn_integral_cong) done next case (set A) then show ?case by (simp add: emeasure_density f) next case (mult u c) moreover have "⋀x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps) ultimately show ?case using f by (simp add: nn_integral_cmult) next case (add u v) then have "⋀x. f x * (v x + u x) = f x * v x + f x * u x" by (simp add: distrib_left) with add f show ?case by (auto simp add: nn_integral_add intro!: nn_integral_add[symmetric]) next case (seq U) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)" by eventually_elim (simp add: SUP_mult_left_ennreal seq) from seq f show ?case apply (simp add: nn_integral_monotone_convergence_SUP) apply (subst nn_integral_cong_AE[OF eq]) apply (subst nn_integral_monotone_convergence_SUP_AE) apply (auto simp: incseq_def le_fun_def intro!: mult_left_mono) done qed lemma density_distr: assumes [measurable]: "f ∈ borel_measurable N" "X ∈ measurable M N" shows "density (distr M N X) f = distr (density M (λx. f (X x))) N X" by (intro measure_eqI) (auto simp add: emeasure_density nn_integral_distr emeasure_distr split: split_indicator intro!: nn_integral_cong) lemma emeasure_restricted: assumes S: "S ∈ sets M" and X: "X ∈ sets M" shows "emeasure (density M (indicator S)) X = emeasure M (S ∩ X)" proof - have "emeasure (density M (indicator S)) X = (∫⇧^{+}x. indicator S x * indicator X x ∂M)" using S X by (simp add: emeasure_density) also have "… = (∫⇧^{+}x. indicator (S ∩ X) x ∂M)" by (auto intro!: nn_integral_cong simp: indicator_def) also have "… = emeasure M (S ∩ X)" using S X by (simp add: sets.Int) finally show ?thesis . qed lemma measure_restricted: "S ∈ sets M ⟹ X ∈ sets M ⟹ measure (density M (indicator S)) X = measure M (S ∩ X)" by (simp add: emeasure_restricted measure_def) lemma (in finite_measure) finite_measure_restricted: "S ∈ sets M ⟹ finite_measure (density M (indicator S))" by standard (simp add: emeasure_restricted) lemma emeasure_density_const: "A ∈ sets M ⟹ emeasure (density M (λ_. c)) A = c * emeasure M A" by (auto simp: nn_integral_cmult_indicator emeasure_density) lemma measure_density_const: "A ∈ sets M ⟹ c ≠ ∞ ⟹ measure (density M (λ_. c)) A = enn2real c * measure M A" by (auto simp: emeasure_density_const measure_def enn2real_mult) lemma density_density_eq: "f ∈ borel_measurable M ⟹ g ∈ borel_measurable M ⟹ density (density M f) g = density M (λx. f x * g x)" by (auto intro!: measure_eqI simp: emeasure_density nn_integral_density ac_simps) lemma distr_density_distr: assumes T: "T ∈ measurable M M'" and T': "T' ∈ measurable M' M" and inv: "∀x∈space M. T' (T x) = x" assumes f: "f ∈ borel_measurable M'" shows "distr (density (distr M M' T) f) M T' = density M (f ∘ T)" (is "?R = ?L") proof (rule measure_eqI) fix A assume A: "A ∈ sets ?R" { fix x assume "x ∈ space M" with sets.sets_into_space[OF A] have "indicator (T' -` A ∩ space M') (T x) = (indicator A x :: ennreal)" using T inv by (auto simp: indicator_def measurable_space) } with A T T' f show "emeasure ?R A = emeasure ?L A" by (simp add: measurable_comp emeasure_density emeasure_distr nn_integral_distr measurable_sets cong: nn_integral_cong) qed simp lemma density_density_divide: fixes f g :: "'a ⇒ real" assumes f: "f ∈ borel_measurable M" "AE x in M. 0 ≤ f x" assumes g: "g ∈ borel_measurable M" "AE x in M. 0 ≤ g x" assumes ac: "AE x in M. f x = 0 ⟶ g x = 0" shows "density (density M f) (λx. g x / f x) = density M g" proof - have "density M g = density M (λx. ennreal (f x) * ennreal (g x / f x))" using f g ac by (auto intro!: density_cong measurable_If simp: ennreal_mult[symmetric]) then show ?thesis using f g by (subst density_density_eq) auto qed lemma density_1: "density M (λ_. 1) = M" by (intro measure_eqI) (auto simp: emeasure_density) lemma emeasure_density_add: assumes X: "X ∈ sets M" assumes Mf[measurable]: "f ∈ borel_measurable M" assumes Mg[measurable]: "g ∈ borel_measurable M" shows "emeasure (density M f) X + emeasure (density M g) X = emeasure (density M (λx. f x + g x)) X" using assms apply (subst (1 2 3) emeasure_density, simp_all) [] apply (subst nn_integral_add[symmetric], simp_all) [] apply (intro nn_integral_cong, simp split: split_indicator) done subsubsection ‹Point measure› definition point_measure :: "'a set ⇒ ('a ⇒ ennreal) ⇒ 'a measure" where "point_measure A f = density (count_space A) f" lemma shows space_point_measure: "space (point_measure A f) = A" and sets_point_measure: "sets (point_measure A f) = Pow A" by (auto simp: point_measure_def) lemma sets_point_measure_count_space[measurable_cong]: "sets (point_measure A f) = sets (count_space A)" by (simp add: sets_point_measure) lemma measurable_point_measure_eq1[simp]: "g ∈ measurable (point_measure A f) M ⟷ g ∈ A → space M" unfolding point_measure_def by simp lemma measurable_point_measure_eq2_finite[simp]: "finite A ⟹ g ∈ measurable M (point_measure A f) ⟷ (g ∈ space M → A ∧ (∀a∈A. g -` {a} ∩ space M ∈ sets M))" unfolding point_measure_def by (simp add: measurable_count_space_eq2) lemma simple_function_point_measure[simp]: "simple_function (point_measure A f) g ⟷ finite (g ` A)" by (simp add: point_measure_def) lemma emeasure_point_measure: assumes A: "finite {a∈X. 0 < f a}" "X ⊆ A" shows "emeasure (point_measure A f) X = (∑a|a∈X ∧ 0 < f a. f a)" proof - have "{a. (a ∈ X ⟶ a ∈ A ∧ 0 < f a) ∧ a ∈ X} = {a∈X. 0 < f a}" using ‹X ⊆ A› by auto with A show ?thesis by (simp add: emeasure_density nn_integral_count_space point_measure_def indicator_def) qed lemma emeasure_point_measure_finite: "finite A ⟹ X ⊆ A ⟹ emeasure (point_measure A f) X = (∑a∈X. f a)" by (subst emeasure_point_measure) (auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le) lemma emeasure_point_measure_finite2: "X ⊆ A ⟹ finite X ⟹ emeasure (point_measure A f) X = (∑a∈X. f a)" by (subst emeasure_point_measure) (auto dest: finite_subset intro!: sum.mono_neutral_left simp: less_le) lemma null_sets_point_measure_iff: "X ∈ null_sets (point_measure A f) ⟷ X ⊆ A ∧ (∀x∈X. f x = 0)" by (auto simp: AE_count_space null_sets_density_iff point_measure_def) lemma AE_point_measure: "(AE x in point_measure A f. P x) ⟷ (∀x∈A. 0 < f x ⟶ P x)" unfolding point_measure_def by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def) lemma nn_integral_point_measure: "finite {a∈A. 0 < f a ∧ 0 < g a} ⟹ integral⇧^{N}(point_measure A f) g = (∑a|a∈A ∧ 0 < f a ∧ 0 < g a. f a * g a)" unfolding point_measure_def by (subst nn_integral_density) (simp_all add: nn_integral_density nn_integral_count_space ennreal_zero_less_mult_iff) lemma nn_integral_point_measure_finite: "finite A ⟹ integral⇧^{N}(point_measure A f) g = (∑a∈A. f a * g a)" by (subst nn_integral_point_measure) (auto intro!: sum.mono_neutral_left simp: less_le) subsubsection ‹Uniform measure› definition "uniform_measure M A = density M (λx. indicator A x / emeasure M A)" lemma shows sets_uniform_measure[simp, measurable_cong]: "sets (uniform_measure M A) = sets M" and space_uniform_measure[simp]: "space (uniform_measure M A) = space M" by (auto simp: uniform_measure_def) lemma emeasure_uniform_measure[simp]: assumes A: "A ∈ sets M" and B: "B ∈ sets M" shows "emeasure (uniform_measure M A) B = emeasure M (A ∩ B) / emeasure M A" proof - from A B have "emeasure (uniform_measure M A) B = (∫⇧^{+}x. (1 / emeasure M A) * indicator (A ∩ B) x ∂M)" by (auto simp add: uniform_measure_def emeasure_density divide_ennreal_def split: split_indicator intro!: nn_integral_cong) also have "… = emeasure M (A ∩ B) / emeasure M A" using A B by (subst nn_integral_cmult_indicator) (simp_all add: sets.Int divide_ennreal_def mult.commute) finally show ?thesis . qed lemma measure_uniform_measure[simp]: assumes A: "emeasure M A ≠ 0" "emeasure M A ≠ ∞" and B: "B ∈ sets M" shows "measure (uniform_measure M A) B = measure M (A ∩ B) / measure M A" using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A by (cases "emeasure M A" "emeasure M (A ∩ B)" rule: ennreal2_cases) (simp_all add: measure_def divide_ennreal top_ennreal.rep_eq top_ereal_def ennreal_top_divide) lemma AE_uniform_measureI: "A ∈ sets M ⟹ (AE x in M. x ∈ A ⟶ P x) ⟹ (AE x in uniform_measure M A. P x)" unfolding uniform_measure_def by (auto simp: AE_density divide_ennreal_def) lemma emeasure_uniform_measure_1: "emeasure M S ≠ 0 ⟹ emeasure M S ≠ ∞ ⟹ emeasure (uniform_measure M S) S = 1" by (subst emeasure_uniform_measure) (simp_all add: emeasure_neq_0_sets emeasure_eq_ennreal_measure divide_ennreal zero_less_iff_neq_zero[symmetric]) lemma nn_integral_uniform_measure: assumes f[measurable]: "f ∈ borel_measurable M" and S[measurable]: "S ∈ sets M" shows "(∫⇧^{+}x. f x ∂uniform_measure M S) = (∫⇧^{+}x. f x * indicator S x ∂M) / emeasure M S" proof - { assume "emeasure M S = ∞" then have ?thesis by (simp add: uniform_measure_def nn_integral_density f) } moreover { assume [simp]: "emeasure M S = 0" then have ae: "AE x in M. x ∉ S" using sets.sets_into_space[OF S] by (subst AE_iff_measurable[OF _ refl]) (simp_all add: subset_eq cong: rev_conj_cong) from ae have "(∫⇧^{+}x. indicator S x / 0 * f x ∂M) = 0" by (subst nn_integral_0_iff_AE) auto moreover from ae have "(∫⇧^{+}x. f x * indicator S x ∂M) = 0" by (subst nn_integral_0_iff_AE) auto ultimately have ?thesis by (simp add: uniform_measure_def nn_integral_density f) } moreover have "emeasure M S ≠ 0 ⟹ emeasure M S ≠ ∞ ⟹ ?thesis" unfolding uniform_measure_def by (subst nn_integral_density) (auto simp: ennreal_times_divide f nn_integral_divide[symmetric] mult.commute) ultimately show ?thesis by blast qed lemma AE_uniform_measure: assumes "emeasure M A ≠ 0" "emeasure M A < ∞" shows "(AE x in uniform_measure M A. P x) ⟷ (AE x in M. x ∈ A ⟶ P x)" proof - have "A ∈ sets M" using ‹emeasure M A ≠ 0› by (metis emeasure_notin_sets) moreover have "⋀x. 0 < indicator A x / emeasure M A ⟷ x ∈ A" using assms by (cases "emeasure M A") (auto split: split_indicator simp: ennreal_zero_less_divide) ultimately show ?thesis unfolding uniform_measure_def by (simp add: AE_density) qed subsubsection ‹Null measure› lemma null_measure_eq_density: "null_measure M = density M (λ_. 0)" by (intro measure_eqI) (simp_all add: emeasure_density) lemma nn_integral_null_measure[simp]: "(∫⇧^{+}x. f x ∂null_measure M) = 0" by (auto simp add: nn_integral_def simple_integral_def SUP_constant bot_ennreal_def le_fun_def intro!: exI[of _ "λx. 0"]) lemma density_null_measure[simp]: "density (null_measure M) f = null_measure M" proof (intro measure_eqI) fix A show "emeasure (density (null_measure M) f) A = emeasure (null_measure M) A" by (simp add: density_def) (simp only: null_measure_def[symmetric] emeasure_null_measure) qed simp subsubsection ‹Uniform count measure› definition "uniform_count_measure A = point_measure A (λx. 1 / card A)" lemma shows space_uniform_count_measure: "space (uniform_count_measure A) = A" and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A" unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure) lemma sets_uniform_count_measure_count_space[measurable_cong]: "sets (uniform_count_measure A) = sets (count_space A)" by (simp add: sets_uniform_count_measure) lemma emeasure_uniform_count_measure: "finite A ⟹ X ⊆ A ⟹ emeasure (uniform_count_measure A) X = card X / card A" by (simp add: emeasure_point_measure_finite uniform_count_measure_def divide_inverse ennreal_mult ennreal_of_nat_eq_real_of_nat) lemma measure_uniform_count_measure: "finite A ⟹ X ⊆ A ⟹ measure (uniform_count_measure A) X = card X / card A" by (simp add: emeasure_point_measure_finite uniform_count_measure_def measure_def enn2real_mult) lemma space_uniform_count_measure_empty_iff [simp]: "space (uniform_count_measure X) = {} ⟷ X = {}" by(simp add: space_uniform_count_measure) lemma sets_uniform_count_measure_eq_UNIV [simp]: "sets (uniform_count_measure UNIV) = UNIV ⟷ True" "UNIV = sets (uniform_count_measure UNIV) ⟷ True" by(simp_all add: sets_uniform_count_measure) subsubsection ‹Scaled measure› lemma nn_integral_scale_measure: assumes f: "f ∈ borel_measurable M" shows "nn_integral (scale_measure r M) f = r * nn_integral M f" using f proof induction case (cong f g) thus ?case by(simp add: cong.hyps space_scale_measure cong: nn_integral_cong_simp) next case (mult f c) thus ?case by(simp add: nn_integral_cmult max_def mult.assoc mult.left_commute) next case (add f g) thus ?case by(simp add: nn_integral_add distrib_left) next case (seq U) thus ?case by(simp add: nn_integral_monotone_convergence_SUP SUP_mult_left_ennreal) qed simp end