src/HOL/Probability/Lebesgue.thy
author hoelzl
Mon May 03 14:35:10 2010 +0200 (2010-05-03)
changeset 36624 25153c08655e
parent 35977 30d42bfd0174
child 36725 34c36a5cb808
permissions -rw-r--r--
Cleanup information theory
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header {*Lebesgue Integration*}
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theory Lebesgue
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imports Measure Borel
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begin
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text{*From the HOL4 Hurd/Coble Lebesgue integration, translated by Armin Heller and Johannes Hoelzl.*}
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definition
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  "pos_part f = (\<lambda>x. max 0 (f x))"
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definition
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  "neg_part f = (\<lambda>x. - min 0 (f x))"
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definition
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  "nonneg f = (\<forall>x. 0 \<le> f x)"
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lemma nonneg_pos_part[intro!]:
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  fixes f :: "'c \<Rightarrow> 'd\<Colon>{linorder,zero}"
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  shows "nonneg (pos_part f)"
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  unfolding nonneg_def pos_part_def by auto
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lemma nonneg_neg_part[intro!]:
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  fixes f :: "'c \<Rightarrow> 'd\<Colon>{linorder,ordered_ab_group_add}"
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  shows "nonneg (neg_part f)"
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  unfolding nonneg_def neg_part_def min_def by auto
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lemma pos_neg_part_abs:
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  fixes f :: "'a \<Rightarrow> real"
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  shows "pos_part f x + neg_part f x = \<bar>f x\<bar>"
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unfolding real_abs_def pos_part_def neg_part_def by auto
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lemma pos_part_abs:
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  fixes f :: "'a \<Rightarrow> real"
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  shows "pos_part (\<lambda> x. \<bar>f x\<bar>) y = \<bar>f y\<bar>"
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unfolding pos_part_def real_abs_def by auto
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lemma neg_part_abs:
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  fixes f :: "'a \<Rightarrow> real"
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  shows "neg_part (\<lambda> x. \<bar>f x\<bar>) y = 0"
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unfolding neg_part_def real_abs_def by auto
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lemma (in measure_space)
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  assumes "f \<in> borel_measurable M"
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  shows pos_part_borel_measurable: "pos_part f \<in> borel_measurable M"
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  and neg_part_borel_measurable: "neg_part f \<in> borel_measurable M"
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using assms
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proof -
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  { fix a :: real
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    { assume asm: "0 \<le> a"
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      from asm have pp: "\<And> w. (pos_part f w \<le> a) = (f w \<le> a)" unfolding pos_part_def by auto
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      have "{w | w. w \<in> space M \<and> f w \<le> a} \<in> sets M"
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        unfolding pos_part_def using assms borel_measurable_le_iff by auto
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      hence "{w . w \<in> space M \<and> pos_part f w \<le> a} \<in> sets M" using pp by auto }
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    moreover have "a < 0 \<Longrightarrow> {w \<in> space M. pos_part f w \<le> a} \<in> sets M"
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      unfolding pos_part_def using empty_sets by auto
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    ultimately have "{w . w \<in> space M \<and> pos_part f w \<le> a} \<in> sets M"
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      using le_less_linear by auto
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  } hence pos: "pos_part f \<in> borel_measurable M" using borel_measurable_le_iff by auto
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  { fix a :: real
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    { assume asm: "0 \<le> a"
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      from asm have pp: "\<And> w. (neg_part f w \<le> a) = (f w \<ge> - a)" unfolding neg_part_def by auto
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      have "{w | w. w \<in> space M \<and> f w \<ge> - a} \<in> sets M"
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        unfolding neg_part_def using assms borel_measurable_ge_iff by auto
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      hence "{w . w \<in> space M \<and> neg_part f w \<le> a} \<in> sets M" using pp by auto }
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    moreover have "a < 0 \<Longrightarrow> {w \<in> space M. neg_part f w \<le> a} = {}" unfolding neg_part_def by auto
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    moreover hence "a < 0 \<Longrightarrow> {w \<in> space M. neg_part f w \<le> a} \<in> sets M" by (simp only: empty_sets)
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    ultimately have "{w . w \<in> space M \<and> neg_part f w \<le> a} \<in> sets M"
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      using le_less_linear by auto
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  } hence neg: "neg_part f \<in> borel_measurable M" using borel_measurable_le_iff by auto
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  from pos neg show "pos_part f \<in> borel_measurable M" and "neg_part f \<in> borel_measurable M" by auto
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qed
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lemma (in measure_space) pos_part_neg_part_borel_measurable_iff:
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  "f \<in> borel_measurable M \<longleftrightarrow>
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  pos_part f \<in> borel_measurable M \<and> neg_part f \<in> borel_measurable M"
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proof -
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  { fix x
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    have "f x = pos_part f x - neg_part f x"
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      unfolding pos_part_def neg_part_def unfolding max_def min_def
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      by auto }
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  hence "(\<lambda> x. f x) = (\<lambda> x. pos_part f x - neg_part f x)" by auto
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  hence "f = (\<lambda> x. pos_part f x - neg_part f x)" by blast
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  from pos_part_borel_measurable[of f] neg_part_borel_measurable[of f]
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    borel_measurable_diff_borel_measurable[of "pos_part f" "neg_part f"]
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    this
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  show ?thesis by auto
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qed
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context measure_space
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begin
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section "Simple discrete step function"
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definition
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 "pos_simple f =
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  { (s :: nat set, a, x).
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    finite s \<and> nonneg f \<and> nonneg x \<and> a ` s \<subseteq> sets M \<and>
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    (\<forall>t \<in> space M. (\<exists>!i\<in>s. t\<in>a i) \<and> (\<forall>i\<in>s. t \<in> a i \<longrightarrow> f t = x i)) }"
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definition
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  "pos_simple_integral \<equiv> \<lambda>(s, a, x). \<Sum> i \<in> s. x i * measure M (a i)"
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definition
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  "psfis f = pos_simple_integral ` (pos_simple f)"
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lemma pos_simpleE[consumes 1]:
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  assumes ps: "(s, a, x) \<in> pos_simple f"
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  obtains "finite s" and "nonneg f" and "nonneg x"
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    and "a ` s \<subseteq> sets M" and "(\<Union>i\<in>s. a i) = space M"
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    and "disjoint_family_on a s"
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    and "\<And>t. t \<in> space M \<Longrightarrow> (\<exists>!i. i \<in> s \<and> t \<in> a i)"
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    and "\<And>t i. \<lbrakk> t \<in> space M ; i \<in> s ; t \<in> a i \<rbrakk> \<Longrightarrow> f t = x i"
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proof
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  show "finite s" and "nonneg f" and "nonneg x"
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    and as_in_M: "a ` s \<subseteq> sets M"
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    and *: "\<And>t. t \<in> space M \<Longrightarrow> (\<exists>!i. i \<in> s \<and> t \<in> a i)"
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    and **: "\<And>t i. \<lbrakk> t \<in> space M ; i \<in> s ; t \<in> a i \<rbrakk> \<Longrightarrow> f t = x i"
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    using ps unfolding pos_simple_def by auto
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  thus t: "(\<Union>i\<in>s. a i) = space M"
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  proof safe
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    fix x assume "x \<in> space M"
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    from *[OF this] show "x \<in> (\<Union>i\<in>s. a i)" by auto
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  next
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    fix t i assume "i \<in> s"
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    hence "a i \<in> sets M" using as_in_M by auto
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    moreover assume "t \<in> a i"
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    ultimately show "t \<in> space M" using sets_into_space by blast
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  qed
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  show "disjoint_family_on a s" unfolding disjoint_family_on_def
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  proof safe
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    fix i j and t assume "i \<in> s" "t \<in> a i" and "j \<in> s" "t \<in> a j" and "i \<noteq> j"
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    with t * show "t \<in> {}" by auto
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  qed
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qed
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lemma pos_simple_cong:
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  assumes "nonneg f" and "nonneg g" and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
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  shows "pos_simple f = pos_simple g"
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  unfolding pos_simple_def using assms by auto
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lemma psfis_cong:
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  assumes "nonneg f" and "nonneg g" and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
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  shows "psfis f = psfis g"
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  unfolding psfis_def using pos_simple_cong[OF assms] by simp
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lemma psfis_0: "0 \<in> psfis (\<lambda>x. 0)"
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  unfolding psfis_def pos_simple_def pos_simple_integral_def
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  by (auto simp: nonneg_def
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      intro: image_eqI[where x="({0}, (\<lambda>i. space M), (\<lambda>i. 0))"])
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lemma pos_simple_setsum_indicator_fn:
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  assumes ps: "(s, a, x) \<in> pos_simple f"
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  and "t \<in> space M"
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  shows "(\<Sum>i\<in>s. x i * indicator_fn (a i) t) = f t"
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proof -
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  from assms obtain i where *: "i \<in> s" "t \<in> a i"
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    and "finite s" and xi: "x i = f t" by (auto elim!: pos_simpleE)
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  have **: "(\<Sum>i\<in>s. x i * indicator_fn (a i) t) =
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    (\<Sum>j\<in>s. if j \<in> {i} then x i else 0)"
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    unfolding indicator_fn_def using assms *
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    by (auto intro!: setsum_cong elim!: pos_simpleE)
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  show ?thesis unfolding ** setsum_cases[OF `finite s`] xi
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    using `i \<in> s` by simp
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qed
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lemma pos_simple_common_partition:
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  assumes psf: "(s, a, x) \<in> pos_simple f"
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  assumes psg: "(s', b, y) \<in> pos_simple g"
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  obtains z z' c k where "(k, c, z) \<in> pos_simple f" "(k, c, z') \<in> pos_simple g"
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proof -
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  (* definitions *)
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  def k == "{0 ..< card (s \<times> s')}"
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  have fs: "finite s" "finite s'" "finite k" unfolding k_def
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    using psf psg unfolding pos_simple_def by auto
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  hence "finite (s \<times> s')" by simp
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  then obtain p where p: "p ` k = s \<times> s'" "inj_on p k"
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    using ex_bij_betw_nat_finite[of "s \<times> s'"] unfolding bij_betw_def k_def by blast
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  def c == "\<lambda> i. a (fst (p i)) \<inter> b (snd (p i))"
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  def z == "\<lambda> i. x (fst (p i))"
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  def z' == "\<lambda> i. y (snd (p i))"
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  have "finite k" unfolding k_def by simp
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  have "nonneg z" and "nonneg z'"
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    using psf psg unfolding z_def z'_def pos_simple_def nonneg_def by auto
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  have ck_subset_M: "c ` k \<subseteq> sets M"
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  proof
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    fix x assume "x \<in> c ` k"
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    then obtain j where "j \<in> k" and "x = c j" by auto
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    hence "p j \<in> s \<times> s'" using p(1) by auto
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    hence "a (fst (p j)) \<in> sets M" (is "?a \<in> _")
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      and "b (snd (p j)) \<in> sets M" (is "?b \<in> _")
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      using psf psg unfolding pos_simple_def by auto
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    thus "x \<in> sets M" unfolding `x = c j` c_def using Int by simp
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  qed
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  { fix t assume "t \<in> space M"
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    hence ex1s: "\<exists>!i\<in>s. t \<in> a i" and ex1s': "\<exists>!i\<in>s'. t \<in> b i"
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      using psf psg unfolding pos_simple_def by auto
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    then obtain j j' where j: "j \<in> s" "t \<in> a j" and j': "j' \<in> s'" "t \<in> b j'"
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      by auto
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    then obtain i :: nat where i: "(j,j') = p i" and "i \<in> k" using p by auto
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    have "\<exists>!i\<in>k. t \<in> c i"
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    proof (rule ex1I[of _ i])
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      show "\<And>x. x \<in> k \<Longrightarrow> t \<in> c x \<Longrightarrow> x = i"
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      proof -
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        fix l assume "l \<in> k" "t \<in> c l"
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        hence "p l \<in> s \<times> s'" and t_in: "t \<in> a (fst (p l))" "t \<in> b (snd (p l))"
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          using p unfolding c_def by auto
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        hence "fst (p l) \<in> s" and "snd (p l) \<in> s'" by auto
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        with t_in j j' ex1s ex1s' have "p l = (j, j')" by (cases "p l", auto)
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        thus "l = i"
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          using `(j, j') = p i` p(2)[THEN inj_onD] `l \<in> k` `i \<in> k` by auto
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      qed
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      show "i \<in> k \<and> t \<in> c i"
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        using `i \<in> k` `t \<in> a j` `t \<in> b j'` c_def i[symmetric] by auto
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    qed auto
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  } note ex1 = this
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  show thesis
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  proof (rule that)
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    { fix t i assume "t \<in> space M" and "i \<in> k"
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      hence "p i \<in> s \<times> s'" using p(1) by auto
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      hence "fst (p i) \<in> s" by auto
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      moreover
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      assume "t \<in> c i"
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      hence "t \<in> a (fst (p i))" unfolding c_def by auto
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      ultimately have "f t = z i" using psf `t \<in> space M`
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        unfolding z_def pos_simple_def by auto
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    }
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    thus "(k, c, z) \<in> pos_simple f"
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      using psf `finite k` `nonneg z` ck_subset_M ex1
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      unfolding pos_simple_def by auto
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    { fix t i assume "t \<in> space M" and "i \<in> k"
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      hence "p i \<in> s \<times> s'" using p(1) by auto
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      hence "snd (p i) \<in> s'" by auto
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      moreover
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      assume "t \<in> c i"
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      hence "t \<in> b (snd (p i))" unfolding c_def by auto
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      ultimately have "g t = z' i" using psg `t \<in> space M`
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        unfolding z'_def pos_simple_def by auto
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    }
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    thus "(k, c, z') \<in> pos_simple g"
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      using psg `finite k` `nonneg z'` ck_subset_M ex1
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      unfolding pos_simple_def by auto
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  qed
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qed
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lemma pos_simple_integral_equal:
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  assumes psx: "(s, a, x) \<in> pos_simple f"
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  assumes psy: "(s', b, y) \<in> pos_simple f"
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  shows "pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
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  unfolding pos_simple_integral_def
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proof simp
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  have "(\<Sum>i\<in>s. x i * measure M (a i)) =
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    (\<Sum>i\<in>s. (\<Sum>j \<in> s'. x i * measure M (a i \<inter> b j)))" (is "?left = _")
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    using psy psx unfolding setsum_right_distrib[symmetric]
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    by (auto intro!: setsum_cong measure_setsum_split elim!: pos_simpleE)
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   267
  also have "... = (\<Sum>i\<in>s. (\<Sum>j \<in> s'. y j * measure M (a i \<inter> b j)))"
hoelzl@35582
   268
  proof (rule setsum_cong, simp, rule setsum_cong, simp_all)
hoelzl@35582
   269
    fix i j assume i: "i \<in> s" and j: "j \<in> s'"
hoelzl@35582
   270
    hence "a i \<in> sets M" using psx by (auto elim!: pos_simpleE)
hoelzl@35582
   271
hoelzl@35582
   272
    show "measure M (a i \<inter> b j) = 0 \<or> x i = y j"
hoelzl@35582
   273
    proof (cases "a i \<inter> b j = {}")
hoelzl@35582
   274
      case True thus ?thesis using empty_measure by simp
hoelzl@35582
   275
    next
hoelzl@35582
   276
      case False then obtain t where t: "t \<in> a i" "t \<in> b j" by auto
hoelzl@35582
   277
      hence "t \<in> space M" using `a i \<in> sets M` sets_into_space by auto
hoelzl@35582
   278
      with psx psy t i j have "x i = f t" and "y j = f t"
hoelzl@35582
   279
        unfolding pos_simple_def by auto
hoelzl@35582
   280
      thus ?thesis by simp
hoelzl@35582
   281
    qed
hoelzl@35582
   282
  qed
hoelzl@35582
   283
  also have "... = (\<Sum>j\<in>s'. (\<Sum>i\<in>s. y j * measure M (a i \<inter> b j)))"
hoelzl@35582
   284
    by (subst setsum_commute) simp
hoelzl@35582
   285
  also have "... = (\<Sum>i\<in>s'. y i * measure M (b i))" (is "?sum_sum = ?right")
hoelzl@35582
   286
  proof (rule setsum_cong)
hoelzl@35582
   287
    fix j assume "j \<in> s'"
hoelzl@35582
   288
    have "y j * measure M (b j) = (\<Sum>i\<in>s. y j * measure M (b j \<inter> a i))"
hoelzl@35582
   289
      using psx psy `j \<in> s'` unfolding setsum_right_distrib[symmetric]
hoelzl@35582
   290
      by (auto intro!: measure_setsum_split elim!: pos_simpleE)
hoelzl@35582
   291
    thus "(\<Sum>i\<in>s. y j * measure M (a i \<inter> b j)) = y j * measure M (b j)"
hoelzl@35582
   292
      by (auto intro!: setsum_cong arg_cong[where f="measure M"])
hoelzl@35582
   293
  qed simp
hoelzl@35582
   294
  finally show "?left = ?right" .
hoelzl@35582
   295
qed
hoelzl@35582
   296
hoelzl@35692
   297
lemma psfis_present:
hoelzl@35582
   298
  assumes "A \<in> psfis f"
hoelzl@35582
   299
  assumes "B \<in> psfis g"
hoelzl@35582
   300
  obtains z z' c k where
hoelzl@35582
   301
  "A = pos_simple_integral (k, c, z)"
hoelzl@35582
   302
  "B = pos_simple_integral (k, c, z')"
hoelzl@35582
   303
  "(k, c, z) \<in> pos_simple f"
hoelzl@35582
   304
  "(k, c, z') \<in> pos_simple g"
hoelzl@35582
   305
using assms
hoelzl@35582
   306
proof -
hoelzl@35582
   307
  from assms obtain s a x s' b y where
hoelzl@35582
   308
    ps: "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple g" and
hoelzl@35582
   309
    A: "A = pos_simple_integral (s, a, x)" and
hoelzl@35582
   310
    B: "B = pos_simple_integral (s', b, y)"
hoelzl@35582
   311
    unfolding psfis_def pos_simple_integral_def by auto
hoelzl@35582
   312
hoelzl@35582
   313
  guess z z' c k using pos_simple_common_partition[OF ps] . note part = this
hoelzl@35582
   314
  show thesis
hoelzl@35582
   315
  proof (rule that[of k c z z', OF _ _ part])
hoelzl@35582
   316
    show "A = pos_simple_integral (k, c, z)"
hoelzl@35582
   317
      unfolding A by (rule pos_simple_integral_equal[OF ps(1) part(1)])
hoelzl@35582
   318
    show "B = pos_simple_integral (k, c, z')"
hoelzl@35582
   319
      unfolding B by (rule pos_simple_integral_equal[OF ps(2) part(2)])
hoelzl@35582
   320
  qed
hoelzl@35582
   321
qed
hoelzl@35582
   322
hoelzl@35692
   323
lemma pos_simple_integral_add:
hoelzl@35582
   324
  assumes "(s, a, x) \<in> pos_simple f"
hoelzl@35582
   325
  assumes "(s', b, y) \<in> pos_simple g"
hoelzl@35582
   326
  obtains s'' c z where
hoelzl@35582
   327
    "(s'', c, z) \<in> pos_simple (\<lambda>x. f x + g x)"
hoelzl@35582
   328
    "(pos_simple_integral (s, a, x) +
hoelzl@35582
   329
      pos_simple_integral (s', b, y) =
hoelzl@35582
   330
      pos_simple_integral (s'', c, z))"
hoelzl@35582
   331
using assms
hoelzl@35582
   332
proof -
hoelzl@35582
   333
  guess z z' c k
hoelzl@35582
   334
    by (rule pos_simple_common_partition[OF `(s, a, x) \<in> pos_simple f ` `(s', b, y) \<in> pos_simple g`])
hoelzl@35582
   335
  note kczz' = this
hoelzl@35582
   336
  have x: "pos_simple_integral (s, a, x) = pos_simple_integral (k, c, z)"
hoelzl@35582
   337
    by (rule pos_simple_integral_equal) (fact, fact)
hoelzl@35582
   338
  have y: "pos_simple_integral (s', b, y) = pos_simple_integral (k, c, z')"
hoelzl@35582
   339
    by (rule pos_simple_integral_equal) (fact, fact)
hoelzl@35582
   340
hoelzl@35582
   341
  have "pos_simple_integral (k, c, (\<lambda> x. z x + z' x))
hoelzl@35582
   342
    = (\<Sum> x \<in> k. (z x + z' x) * measure M (c x))"
hoelzl@35582
   343
    unfolding pos_simple_integral_def by auto
hoelzl@35582
   344
  also have "\<dots> = (\<Sum> x \<in> k. z x * measure M (c x) + z' x * measure M (c x))" using real_add_mult_distrib by auto
hoelzl@35582
   345
  also have "\<dots> = (\<Sum> x \<in> k. z x * measure M (c x)) + (\<Sum> x \<in> k. z' x * measure M (c x))" using setsum_addf by auto
hoelzl@35582
   346
  also have "\<dots> = pos_simple_integral (k, c, z) + pos_simple_integral (k, c, z')" unfolding pos_simple_integral_def by auto
hoelzl@35582
   347
  finally have ths: "pos_simple_integral (s, a, x) + pos_simple_integral (s', b, y) =
hoelzl@35582
   348
    pos_simple_integral (k, c, (\<lambda> x. z x + z' x))" using x y by auto
hoelzl@35582
   349
  show ?thesis
hoelzl@35582
   350
    apply (rule that[of k c "(\<lambda> x. z x + z' x)", OF _ ths])
hoelzl@35582
   351
    using kczz' unfolding pos_simple_def nonneg_def by (auto intro!: add_nonneg_nonneg)
hoelzl@35582
   352
qed
hoelzl@35582
   353
hoelzl@35582
   354
lemma psfis_add:
hoelzl@35582
   355
  assumes "a \<in> psfis f" "b \<in> psfis g"
hoelzl@35582
   356
  shows "a + b \<in> psfis (\<lambda>x. f x + g x)"
hoelzl@35582
   357
using assms pos_simple_integral_add
hoelzl@35582
   358
unfolding psfis_def by auto blast
hoelzl@35582
   359
hoelzl@35582
   360
lemma pos_simple_integral_mono_on_mspace:
hoelzl@35582
   361
  assumes f: "(s, a, x) \<in> pos_simple f"
hoelzl@35582
   362
  assumes g: "(s', b, y) \<in> pos_simple g"
hoelzl@35582
   363
  assumes mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
hoelzl@35582
   364
  shows "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
hoelzl@35582
   365
using assms
hoelzl@35582
   366
proof -
hoelzl@35582
   367
  guess z z' c k by (rule pos_simple_common_partition[OF f g])
hoelzl@35582
   368
  note kczz' = this
hoelzl@35582
   369
  (* w = z and w' = z'  except where c _ = {} or undef *)
hoelzl@35582
   370
  def w == "\<lambda> i. if i \<notin> k \<or> c i = {} then 0 else z i"
hoelzl@35582
   371
  def w' == "\<lambda> i. if i \<notin> k \<or> c i = {} then 0 else z' i"
hoelzl@35582
   372
  { fix i
hoelzl@35582
   373
    have "w i \<le> w' i"
hoelzl@35582
   374
    proof (cases "i \<notin> k \<or> c i = {}")
hoelzl@35582
   375
      case False hence "i \<in> k" "c i \<noteq> {}" by auto
hoelzl@35582
   376
      then obtain v where v: "v \<in> c i" and "c i \<in> sets M"
hoelzl@35582
   377
        using kczz'(1) unfolding pos_simple_def by blast
hoelzl@35582
   378
      hence "v \<in> space M" using sets_into_space by blast
hoelzl@35582
   379
      with mono[OF `v \<in> space M`] kczz' `i \<in> k` `v \<in> c i`
hoelzl@35582
   380
      have "z i \<le> z' i" unfolding pos_simple_def by simp
hoelzl@35582
   381
      thus ?thesis unfolding w_def w'_def using False by auto
hoelzl@35582
   382
    next
hoelzl@35582
   383
      case True thus ?thesis unfolding w_def w'_def by auto
hoelzl@35582
   384
   qed
hoelzl@35582
   385
  } note w_mn = this
hoelzl@35582
   386
hoelzl@35582
   387
  (* some technical stuff for the calculation*)
hoelzl@35582
   388
  have "\<And> i. i \<in> k \<Longrightarrow> c i \<in> sets M" using kczz' unfolding pos_simple_def by auto
hoelzl@35582
   389
  hence "\<And> i. i \<in> k \<Longrightarrow> measure M (c i) \<ge> 0" using positive by auto
hoelzl@35582
   390
  hence w_mono: "\<And> i. i \<in> k \<Longrightarrow> w i * measure M (c i) \<le> w' i * measure M (c i)"
hoelzl@35582
   391
    using mult_right_mono w_mn by blast
hoelzl@35582
   392
hoelzl@35582
   393
  { fix i have "\<lbrakk>i \<in> k ; z i \<noteq> w i\<rbrakk> \<Longrightarrow> measure M (c i) = 0"
hoelzl@35582
   394
      unfolding w_def by (cases "c i = {}") auto }
hoelzl@35582
   395
  hence zw: "\<And> i. i \<in> k \<Longrightarrow> z i * measure M (c i) = w i * measure M (c i)" by auto
hoelzl@35582
   396
hoelzl@35582
   397
  { fix i have "i \<in> k \<Longrightarrow> z i * measure M (c i) = w i * measure M (c i)"
hoelzl@35582
   398
      unfolding w_def by (cases "c i = {}") simp_all }
hoelzl@35582
   399
  note zw = this
hoelzl@35582
   400
hoelzl@35582
   401
  { fix i have "i \<in> k \<Longrightarrow> z' i * measure M (c i) = w' i * measure M (c i)"
hoelzl@35582
   402
      unfolding w'_def by (cases "c i = {}") simp_all }
hoelzl@35582
   403
  note z'w' = this
hoelzl@35582
   404
hoelzl@35582
   405
  (* the calculation *)
hoelzl@35582
   406
  have "pos_simple_integral (s, a, x) = pos_simple_integral (k, c, z)"
hoelzl@35582
   407
    using f kczz'(1) by (rule pos_simple_integral_equal)
hoelzl@35582
   408
  also have "\<dots> = (\<Sum> i \<in> k. z i * measure M (c i))"
hoelzl@35582
   409
    unfolding pos_simple_integral_def by auto
hoelzl@35582
   410
  also have "\<dots> = (\<Sum> i \<in> k. w i * measure M (c i))"
hoelzl@35582
   411
    using setsum_cong2[of k, OF zw] by auto
hoelzl@35582
   412
  also have "\<dots> \<le> (\<Sum> i \<in> k. w' i * measure M (c i))"
hoelzl@35582
   413
    using setsum_mono[OF w_mono, of k] by auto
hoelzl@35582
   414
  also have "\<dots> = (\<Sum> i \<in> k. z' i * measure M (c i))"
hoelzl@35582
   415
    using setsum_cong2[of k, OF z'w'] by auto
hoelzl@35582
   416
  also have "\<dots> = pos_simple_integral (k, c, z')"
hoelzl@35582
   417
    unfolding pos_simple_integral_def by auto
hoelzl@35582
   418
  also have "\<dots> = pos_simple_integral (s', b, y)"
hoelzl@35582
   419
    using kczz'(2) g by (rule pos_simple_integral_equal)
hoelzl@35582
   420
  finally show "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
hoelzl@35582
   421
    by simp
hoelzl@35582
   422
qed
hoelzl@35582
   423
hoelzl@35582
   424
lemma pos_simple_integral_mono:
hoelzl@35582
   425
  assumes a: "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple g"
hoelzl@35582
   426
  assumes "\<And> z. f z \<le> g z"
hoelzl@35582
   427
  shows "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
hoelzl@35582
   428
using assms pos_simple_integral_mono_on_mspace by auto
hoelzl@35582
   429
hoelzl@35582
   430
lemma psfis_mono:
hoelzl@35582
   431
  assumes "a \<in> psfis f" "b \<in> psfis g"
hoelzl@35582
   432
  assumes "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
hoelzl@35582
   433
  shows "a \<le> b"
hoelzl@35582
   434
using assms pos_simple_integral_mono_on_mspace unfolding psfis_def by auto
hoelzl@35582
   435
hoelzl@35582
   436
lemma pos_simple_fn_integral_unique:
hoelzl@35582
   437
  assumes "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple f"
hoelzl@35582
   438
  shows "pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
hoelzl@35582
   439
using assms real_le_antisym real_le_refl pos_simple_integral_mono by metis
hoelzl@35582
   440
hoelzl@35582
   441
lemma psfis_unique:
hoelzl@35582
   442
  assumes "a \<in> psfis f" "b \<in> psfis f"
hoelzl@35582
   443
  shows "a = b"
hoelzl@35582
   444
using assms real_le_antisym real_le_refl psfis_mono by metis
hoelzl@35582
   445
hoelzl@35582
   446
lemma pos_simple_integral_indicator:
hoelzl@35582
   447
  assumes "A \<in> sets M"
hoelzl@35582
   448
  obtains s a x where
hoelzl@35582
   449
  "(s, a, x) \<in> pos_simple (indicator_fn A)"
hoelzl@35582
   450
  "measure M A = pos_simple_integral (s, a, x)"
hoelzl@35582
   451
using assms
hoelzl@35582
   452
proof -
hoelzl@35582
   453
  def s == "{0, 1} :: nat set"
hoelzl@35582
   454
  def a == "\<lambda> i :: nat. if i = 0 then A else space M - A"
hoelzl@35582
   455
  def x == "\<lambda> i :: nat. if i = 0 then 1 else (0 :: real)"
hoelzl@35582
   456
  have eq: "pos_simple_integral (s, a, x) = measure M A"
hoelzl@35582
   457
    unfolding s_def a_def x_def pos_simple_integral_def by auto
hoelzl@35582
   458
  have "(s, a, x) \<in> pos_simple (indicator_fn A)"
hoelzl@35582
   459
    unfolding pos_simple_def indicator_fn_def s_def a_def x_def nonneg_def
hoelzl@35582
   460
    using assms sets_into_space by auto
hoelzl@35582
   461
   from that[OF this] eq show thesis by auto
hoelzl@35582
   462
qed
hoelzl@35582
   463
hoelzl@35582
   464
lemma psfis_indicator:
hoelzl@35582
   465
  assumes "A \<in> sets M"
hoelzl@35582
   466
  shows "measure M A \<in> psfis (indicator_fn A)"
hoelzl@35582
   467
using pos_simple_integral_indicator[OF assms]
hoelzl@35582
   468
  unfolding psfis_def image_def by auto
hoelzl@35582
   469
hoelzl@35582
   470
lemma pos_simple_integral_mult:
hoelzl@35582
   471
  assumes f: "(s, a, x) \<in> pos_simple f"
hoelzl@35582
   472
  assumes "0 \<le> z"
hoelzl@35582
   473
  obtains s' b y where
hoelzl@35582
   474
  "(s', b, y) \<in> pos_simple (\<lambda>x. z * f x)"
hoelzl@35582
   475
  "pos_simple_integral (s', b, y) = z * pos_simple_integral (s, a, x)"
hoelzl@35582
   476
  using assms that[of s a "\<lambda>i. z * x i"]
hoelzl@35582
   477
  by (simp add: pos_simple_def pos_simple_integral_def setsum_right_distrib algebra_simps nonneg_def mult_nonneg_nonneg)
hoelzl@35582
   478
hoelzl@35582
   479
lemma psfis_mult:
hoelzl@35582
   480
  assumes "r \<in> psfis f"
hoelzl@35582
   481
  assumes "0 \<le> z"
hoelzl@35582
   482
  shows "z * r \<in> psfis (\<lambda>x. z * f x)"
hoelzl@35582
   483
using assms
hoelzl@35582
   484
proof -
hoelzl@35582
   485
  from assms obtain s a x
hoelzl@35582
   486
    where sax: "(s, a, x) \<in> pos_simple f"
hoelzl@35582
   487
    and r: "r = pos_simple_integral (s, a, x)"
hoelzl@35582
   488
    unfolding psfis_def image_def by auto
hoelzl@35582
   489
  obtain s' b y where
hoelzl@35582
   490
    "(s', b, y) \<in> pos_simple (\<lambda>x. z * f x)"
hoelzl@35582
   491
    "z * pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
hoelzl@35582
   492
    using pos_simple_integral_mult[OF sax `0 \<le> z`, of thesis] by auto
hoelzl@35582
   493
  thus ?thesis using r unfolding psfis_def image_def by auto
hoelzl@35582
   494
qed
hoelzl@35582
   495
hoelzl@35582
   496
lemma psfis_setsum_image:
hoelzl@35582
   497
  assumes "finite P"
hoelzl@35582
   498
  assumes "\<And>i. i \<in> P \<Longrightarrow> a i \<in> psfis (f i)"
hoelzl@35582
   499
  shows "(setsum a P) \<in> psfis (\<lambda>t. \<Sum>i \<in> P. f i t)"
hoelzl@35582
   500
using assms
hoelzl@35582
   501
proof (induct P)
hoelzl@35582
   502
  case empty
hoelzl@35582
   503
  let ?s = "{0 :: nat}"
hoelzl@35582
   504
  let ?a = "\<lambda> i. if i = (0 :: nat) then space M else {}"
hoelzl@35582
   505
  let ?x = "\<lambda> (i :: nat). (0 :: real)"
hoelzl@35582
   506
  have "(?s, ?a, ?x) \<in> pos_simple (\<lambda> t. (0 :: real))"
hoelzl@35582
   507
    unfolding pos_simple_def image_def nonneg_def by auto
hoelzl@35582
   508
  moreover have "(\<Sum> i \<in> ?s. ?x i * measure M (?a i)) = 0" by auto
hoelzl@35582
   509
  ultimately have "0 \<in> psfis (\<lambda> t. 0)"
hoelzl@35582
   510
    unfolding psfis_def image_def pos_simple_integral_def nonneg_def
hoelzl@35582
   511
    by (auto intro!:bexI[of _ "(?s, ?a, ?x)"])
hoelzl@35582
   512
  thus ?case by auto
hoelzl@35582
   513
next
hoelzl@35582
   514
  case (insert x P) note asms = this
hoelzl@35582
   515
  have "finite P" by fact
hoelzl@35582
   516
  have "x \<notin> P" by fact
hoelzl@35582
   517
  have "(\<And>i. i \<in> P \<Longrightarrow> a i \<in> psfis (f i)) \<Longrightarrow>
hoelzl@35582
   518
    setsum a P \<in> psfis (\<lambda>t. \<Sum>i\<in>P. f i t)" by fact
hoelzl@35582
   519
  have "setsum a (insert x P) = a x + setsum a P" using `finite P` `x \<notin> P` by auto
hoelzl@35582
   520
  also have "\<dots> \<in> psfis (\<lambda> t. f x t + (\<Sum> i \<in> P. f i t))"
hoelzl@35582
   521
    using asms psfis_add by auto
hoelzl@35582
   522
  also have "\<dots> = psfis (\<lambda> t. \<Sum> i \<in> insert x P. f i t)"
hoelzl@35582
   523
    using `x \<notin> P` `finite P` by auto
hoelzl@35582
   524
  finally show ?case by simp
hoelzl@35582
   525
qed
hoelzl@35582
   526
hoelzl@35582
   527
lemma psfis_intro:
hoelzl@35582
   528
  assumes "a ` P \<subseteq> sets M" and "nonneg x" and "finite P"
hoelzl@35582
   529
  shows "(\<Sum>i\<in>P. x i * measure M (a i)) \<in> psfis (\<lambda>t. \<Sum>i\<in>P. x i * indicator_fn (a i) t)"
hoelzl@35582
   530
using assms psfis_mult psfis_indicator
hoelzl@35582
   531
unfolding image_def nonneg_def
hoelzl@35582
   532
by (auto intro!:psfis_setsum_image)
hoelzl@35582
   533
hoelzl@35582
   534
lemma psfis_nonneg: "a \<in> psfis f \<Longrightarrow> nonneg f"
hoelzl@35582
   535
unfolding psfis_def pos_simple_def by auto
hoelzl@35582
   536
hoelzl@35582
   537
lemma pos_psfis: "r \<in> psfis f \<Longrightarrow> 0 \<le> r"
hoelzl@35582
   538
unfolding psfis_def pos_simple_integral_def image_def pos_simple_def nonneg_def
hoelzl@35582
   539
using positive[unfolded positive_def] by (auto intro!:setsum_nonneg mult_nonneg_nonneg)
hoelzl@35582
   540
hoelzl@35582
   541
lemma psfis_borel_measurable:
hoelzl@35582
   542
  assumes "a \<in> psfis f"
hoelzl@35582
   543
  shows "f \<in> borel_measurable M"
hoelzl@35582
   544
using assms
hoelzl@35582
   545
proof -
hoelzl@35582
   546
  from assms obtain s a' x where sa'x: "(s, a', x) \<in> pos_simple f" and sa'xa: "pos_simple_integral (s, a', x) = a"
hoelzl@35582
   547
    and fs: "finite s"
hoelzl@35582
   548
    unfolding psfis_def pos_simple_integral_def image_def
hoelzl@35582
   549
    by (auto elim:pos_simpleE)
hoelzl@35582
   550
  { fix w assume "w \<in> space M"
hoelzl@35582
   551
    hence "(f w \<le> a) = ((\<Sum> i \<in> s. x i * indicator_fn (a' i) w) \<le> a)"
hoelzl@35582
   552
      using pos_simple_setsum_indicator_fn[OF sa'x, of w] by simp
hoelzl@35582
   553
  } hence "\<And> w. (w \<in> space M \<and> f w \<le> a) = (w \<in> space M \<and> (\<Sum> i \<in> s. x i * indicator_fn (a' i) w) \<le> a)"
hoelzl@35582
   554
    by auto
hoelzl@35582
   555
  { fix i assume "i \<in> s"
hoelzl@35582
   556
    hence "indicator_fn (a' i) \<in> borel_measurable M"
hoelzl@35582
   557
      using borel_measurable_indicator using sa'x[unfolded pos_simple_def] by auto
hoelzl@35582
   558
    hence "(\<lambda> w. x i * indicator_fn (a' i) w) \<in> borel_measurable M"
hoelzl@35582
   559
      using affine_borel_measurable[of "\<lambda> w. indicator_fn (a' i) w" 0 "x i"]
hoelzl@35582
   560
        real_mult_commute by auto }
hoelzl@35582
   561
  from borel_measurable_setsum_borel_measurable[OF fs this] affine_borel_measurable
hoelzl@35582
   562
  have "(\<lambda> w. (\<Sum> i \<in> s. x i * indicator_fn (a' i) w)) \<in> borel_measurable M" by auto
hoelzl@35582
   563
  from borel_measurable_cong[OF pos_simple_setsum_indicator_fn[OF sa'x]] this
hoelzl@35582
   564
  show ?thesis by simp
hoelzl@35582
   565
qed
hoelzl@35582
   566
hoelzl@35582
   567
lemma psfis_mono_conv_mono:
hoelzl@35582
   568
  assumes mono: "mono_convergent u f (space M)"
hoelzl@35582
   569
  and ps_u: "\<And>n. x n \<in> psfis (u n)"
hoelzl@35582
   570
  and "x ----> y"
hoelzl@35582
   571
  and "r \<in> psfis s"
hoelzl@35582
   572
  and f_upper_bound: "\<And>t. t \<in> space M \<Longrightarrow> s t \<le> f t"
hoelzl@35582
   573
  shows "r <= y"
hoelzl@35582
   574
proof (rule field_le_mult_one_interval)
hoelzl@35582
   575
  fix z :: real assume "0 < z" and "z < 1"
hoelzl@35582
   576
  hence "0 \<le> z" by auto
hoelzl@35582
   577
  let "?B' n" = "{w \<in> space M. z * s w <= u n w}"
hoelzl@35582
   578
hoelzl@35582
   579
  have "incseq x" unfolding incseq_def
hoelzl@35582
   580
  proof safe
hoelzl@35582
   581
    fix m n :: nat assume "m \<le> n"
hoelzl@35582
   582
    show "x m \<le> x n"
hoelzl@35582
   583
    proof (rule psfis_mono[OF `x m \<in> psfis (u m)` `x n \<in> psfis (u n)`])
hoelzl@35582
   584
      fix t assume "t \<in> space M"
hoelzl@35582
   585
      with mono_convergentD[OF mono this] `m \<le> n` show "u m t \<le> u n t"
hoelzl@35582
   586
        unfolding incseq_def by auto
hoelzl@35582
   587
    qed
hoelzl@35582
   588
  qed
hoelzl@35582
   589
hoelzl@35582
   590
  from `r \<in> psfis s`
hoelzl@35582
   591
  obtain s' a x' where r: "r = pos_simple_integral (s', a, x')"
hoelzl@35582
   592
    and ps_s: "(s', a, x') \<in> pos_simple s"
hoelzl@35582
   593
    unfolding psfis_def by auto
hoelzl@35582
   594
hoelzl@35582
   595
  { fix t i assume "i \<in> s'" "t \<in> a i"
hoelzl@35582
   596
    hence "t \<in> space M" using ps_s by (auto elim!: pos_simpleE) }
hoelzl@35582
   597
  note t_in_space = this
hoelzl@35582
   598
hoelzl@35582
   599
  { fix n
hoelzl@35582
   600
    from psfis_borel_measurable[OF `r \<in> psfis s`]
hoelzl@35582
   601
       psfis_borel_measurable[OF ps_u]
hoelzl@35582
   602
    have "?B' n \<in> sets M"
hoelzl@35582
   603
      by (auto intro!:
hoelzl@35582
   604
        borel_measurable_leq_borel_measurable
hoelzl@35582
   605
        borel_measurable_times_borel_measurable
hoelzl@35582
   606
        borel_measurable_const) }
hoelzl@35582
   607
  note B'_in_M = this
hoelzl@35582
   608
hoelzl@35582
   609
  { fix n have "(\<lambda>i. a i \<inter> ?B' n) ` s' \<subseteq> sets M" using B'_in_M ps_s
hoelzl@35582
   610
      by (auto intro!: Int elim!: pos_simpleE) }
hoelzl@35582
   611
  note B'_inter_a_in_M = this
hoelzl@35582
   612
hoelzl@35582
   613
  let "?sum n" = "(\<Sum>i\<in>s'. x' i * measure M (a i \<inter> ?B' n))"
hoelzl@35582
   614
  { fix n
hoelzl@35582
   615
    have "z * ?sum n \<le> x n"
hoelzl@35582
   616
    proof (rule psfis_mono[OF _ ps_u])
hoelzl@35582
   617
      have *: "\<And>i t. indicator_fn (?B' n) t * (x' i * indicator_fn (a i) t) =
hoelzl@35582
   618
        x' i * indicator_fn (a i \<inter> ?B' n) t" unfolding indicator_fn_def by auto
hoelzl@35582
   619
      have ps': "?sum n \<in> psfis (\<lambda>t. indicator_fn (?B' n) t * (\<Sum>i\<in>s'. x' i * indicator_fn (a i) t))"
hoelzl@35582
   620
        unfolding setsum_right_distrib * using B'_in_M ps_s
hoelzl@35582
   621
        by (auto intro!: psfis_intro Int elim!: pos_simpleE)
hoelzl@35582
   622
      also have "... = psfis (\<lambda>t. indicator_fn (?B' n) t * s t)" (is "psfis ?l = psfis ?r")
hoelzl@35582
   623
      proof (rule psfis_cong)
hoelzl@35582
   624
        show "nonneg ?l" using psfis_nonneg[OF ps'] .
hoelzl@35582
   625
        show "nonneg ?r" using psfis_nonneg[OF `r \<in> psfis s`] unfolding nonneg_def indicator_fn_def by auto
hoelzl@35582
   626
        fix t assume "t \<in> space M"
hoelzl@35582
   627
        show "?l t = ?r t" unfolding pos_simple_setsum_indicator_fn[OF ps_s `t \<in> space M`] ..
hoelzl@35582
   628
      qed
hoelzl@35582
   629
      finally show "z * ?sum n \<in> psfis (\<lambda>t. z * ?r t)" using psfis_mult[OF _ `0 \<le> z`] by simp
hoelzl@35582
   630
    next
hoelzl@35582
   631
      fix t assume "t \<in> space M"
hoelzl@35582
   632
      show "z * (indicator_fn (?B' n) t * s t) \<le> u n t"
hoelzl@35582
   633
         using psfis_nonneg[OF ps_u] unfolding nonneg_def indicator_fn_def by auto
hoelzl@35582
   634
    qed }
hoelzl@35582
   635
  hence *: "\<exists>N. \<forall>n\<ge>N. z * ?sum n \<le> x n" by (auto intro!: exI[of _ 0])
hoelzl@35582
   636
hoelzl@35582
   637
  show "z * r \<le> y" unfolding r pos_simple_integral_def
hoelzl@35582
   638
  proof (rule LIMSEQ_le[OF mult_right.LIMSEQ `x ----> y` *],
hoelzl@35582
   639
         simp, rule LIMSEQ_setsum, rule mult_right.LIMSEQ)
hoelzl@35582
   640
    fix i assume "i \<in> s'"
hoelzl@35582
   641
    from psfis_nonneg[OF `r \<in> psfis s`, unfolded nonneg_def]
hoelzl@35582
   642
    have "\<And>t. 0 \<le> s t" by simp
hoelzl@35582
   643
hoelzl@35582
   644
    have *: "(\<Union>j. a i \<inter> ?B' j) = a i"
hoelzl@35582
   645
    proof (safe, simp, safe)
hoelzl@35582
   646
      fix t assume "t \<in> a i"
hoelzl@35582
   647
      thus "t \<in> space M" using t_in_space[OF `i \<in> s'`] by auto
hoelzl@35582
   648
      show "\<exists>j. z * s t \<le> u j t"
hoelzl@35582
   649
      proof (cases "s t = 0")
hoelzl@35582
   650
        case True thus ?thesis
hoelzl@35582
   651
          using psfis_nonneg[OF ps_u] unfolding nonneg_def by auto
hoelzl@35582
   652
      next
hoelzl@35582
   653
        case False with `0 \<le> s t`
hoelzl@35582
   654
        have "0 < s t" by auto
hoelzl@35582
   655
        hence "z * s t < 1 * s t" using `0 < z` `z < 1`
hoelzl@35582
   656
          by (auto intro!: mult_strict_right_mono simp del: mult_1_left)
hoelzl@35582
   657
        also have "... \<le> f t" using f_upper_bound `t \<in> space M` by auto
hoelzl@35582
   658
        finally obtain b where "\<And>j. b \<le> j \<Longrightarrow> z * s t < u j t" using `t \<in> space M`
hoelzl@35582
   659
          using mono_conv_outgrow[of "\<lambda>n. u n t" "f t" "z * s t"]
hoelzl@35582
   660
          using mono_convergentD[OF mono] by auto
hoelzl@35582
   661
        from this[of b] show ?thesis by (auto intro!: exI[of _ b])
hoelzl@35582
   662
      qed
hoelzl@35582
   663
    qed
hoelzl@35582
   664
hoelzl@35582
   665
    show "(\<lambda>n. measure M (a i \<inter> ?B' n)) ----> measure M (a i)"
hoelzl@35582
   666
    proof (safe intro!:
hoelzl@35582
   667
        monotone_convergence[of "\<lambda>n. a i \<inter> ?B' n", unfolded comp_def *])
hoelzl@35582
   668
      fix n show "a i \<inter> ?B' n \<in> sets M"
hoelzl@35582
   669
        using B'_inter_a_in_M[of n] `i \<in> s'` by auto
hoelzl@35582
   670
    next
hoelzl@35582
   671
      fix j t assume "t \<in> space M" and "z * s t \<le> u j t"
hoelzl@35582
   672
      thus "z * s t \<le> u (Suc j) t"
hoelzl@35582
   673
        using mono_convergentD(1)[OF mono, unfolded incseq_def,
hoelzl@35582
   674
          rule_format, of t j "Suc j"]
hoelzl@35582
   675
        by auto
hoelzl@35582
   676
    qed
hoelzl@35582
   677
  qed
hoelzl@35582
   678
qed
hoelzl@35582
   679
hoelzl@35692
   680
section "Continuous posititve integration"
hoelzl@35692
   681
hoelzl@35692
   682
definition
hoelzl@35692
   683
  "nnfis f = { y. \<exists>u x. mono_convergent u f (space M) \<and>
hoelzl@35692
   684
                        (\<forall>n. x n \<in> psfis (u n)) \<and> x ----> y }"
hoelzl@35692
   685
hoelzl@35582
   686
lemma psfis_nnfis:
hoelzl@35582
   687
  "a \<in> psfis f \<Longrightarrow> a \<in> nnfis f"
hoelzl@35582
   688
  unfolding nnfis_def mono_convergent_def incseq_def
hoelzl@35582
   689
  by (auto intro!: exI[of _ "\<lambda>n. f"] exI[of _ "\<lambda>n. a"] LIMSEQ_const)
hoelzl@35582
   690
hoelzl@35748
   691
lemma nnfis_0: "0 \<in> nnfis (\<lambda> x. 0)"
hoelzl@35748
   692
  by (rule psfis_nnfis[OF psfis_0])
hoelzl@35748
   693
hoelzl@35582
   694
lemma nnfis_times:
hoelzl@35582
   695
  assumes "a \<in> nnfis f" and "0 \<le> z"
hoelzl@35582
   696
  shows "z * a \<in> nnfis (\<lambda>t. z * f t)"
hoelzl@35582
   697
proof -
hoelzl@35582
   698
  obtain u x where "mono_convergent u f (space M)" and
hoelzl@35582
   699
    "\<And>n. x n \<in> psfis (u n)" "x ----> a"
hoelzl@35582
   700
    using `a \<in> nnfis f` unfolding nnfis_def by auto
hoelzl@35582
   701
  with `0 \<le> z`show ?thesis unfolding nnfis_def mono_convergent_def incseq_def
hoelzl@35582
   702
    by (auto intro!: exI[of _ "\<lambda>n w. z * u n w"] exI[of _ "\<lambda>n. z * x n"]
hoelzl@35582
   703
      LIMSEQ_mult LIMSEQ_const psfis_mult mult_mono1)
hoelzl@35582
   704
qed
hoelzl@35582
   705
hoelzl@35582
   706
lemma nnfis_add:
hoelzl@35582
   707
  assumes "a \<in> nnfis f" and "b \<in> nnfis g"
hoelzl@35582
   708
  shows "a + b \<in> nnfis (\<lambda>t. f t + g t)"
hoelzl@35582
   709
proof -
hoelzl@35582
   710
  obtain u x w y
hoelzl@35582
   711
    where "mono_convergent u f (space M)" and
hoelzl@35582
   712
    "\<And>n. x n \<in> psfis (u n)" "x ----> a" and
hoelzl@35582
   713
    "mono_convergent w g (space M)" and
hoelzl@35582
   714
    "\<And>n. y n \<in> psfis (w n)" "y ----> b"
hoelzl@35582
   715
    using `a \<in> nnfis f` `b \<in> nnfis g` unfolding nnfis_def by auto
hoelzl@35582
   716
  thus ?thesis unfolding nnfis_def mono_convergent_def incseq_def
hoelzl@35582
   717
    by (auto intro!: exI[of _ "\<lambda>n a. u n a + w n a"] exI[of _ "\<lambda>n. x n + y n"]
hoelzl@35582
   718
      LIMSEQ_add LIMSEQ_const psfis_add add_mono)
hoelzl@35582
   719
qed
hoelzl@35582
   720
hoelzl@35582
   721
lemma nnfis_mono:
hoelzl@35582
   722
  assumes nnfis: "a \<in> nnfis f" "b \<in> nnfis g"
hoelzl@35582
   723
  and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
hoelzl@35582
   724
  shows "a \<le> b"
hoelzl@35582
   725
proof -
hoelzl@35582
   726
  obtain u x w y where
hoelzl@35582
   727
    mc: "mono_convergent u f (space M)" "mono_convergent w g (space M)" and
hoelzl@35582
   728
    psfis: "\<And>n. x n \<in> psfis (u n)" "\<And>n. y n \<in> psfis (w n)" and
hoelzl@35582
   729
    limseq: "x ----> a" "y ----> b" using nnfis unfolding nnfis_def by auto
hoelzl@35582
   730
  show ?thesis
hoelzl@35582
   731
  proof (rule LIMSEQ_le_const2[OF limseq(1)], rule exI[of _ 0], safe)
hoelzl@35582
   732
    fix n
hoelzl@35582
   733
    show "x n \<le> b"
hoelzl@35582
   734
    proof (rule psfis_mono_conv_mono[OF mc(2) psfis(2) limseq(2) psfis(1)])
hoelzl@35582
   735
      fix t assume "t \<in> space M"
hoelzl@35582
   736
      from mono_convergent_le[OF mc(1) this] mono[OF this]
hoelzl@35582
   737
      show "u n t \<le> g t" by (rule order_trans)
hoelzl@35582
   738
    qed
hoelzl@35582
   739
  qed
hoelzl@35582
   740
qed
hoelzl@35582
   741
hoelzl@35582
   742
lemma nnfis_unique:
hoelzl@35582
   743
  assumes a: "a \<in> nnfis f" and b: "b \<in> nnfis f"
hoelzl@35582
   744
  shows "a = b"
hoelzl@35582
   745
  using nnfis_mono[OF a b] nnfis_mono[OF b a]
hoelzl@35582
   746
  by (auto intro!: real_le_antisym[of a b])
hoelzl@35582
   747
hoelzl@35582
   748
lemma psfis_equiv:
hoelzl@35582
   749
  assumes "a \<in> psfis f" and "nonneg g"
hoelzl@35582
   750
  and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
hoelzl@35582
   751
  shows "a \<in> psfis g"
hoelzl@35582
   752
  using assms unfolding psfis_def pos_simple_def by auto
hoelzl@35582
   753
hoelzl@35582
   754
lemma psfis_mon_upclose:
hoelzl@35582
   755
  assumes "\<And>m. a m \<in> psfis (u m)"
hoelzl@35582
   756
  shows "\<exists>c. c \<in> psfis (mon_upclose n u)"
hoelzl@35582
   757
proof (induct n)
hoelzl@35582
   758
  case 0 thus ?case unfolding mon_upclose.simps using assms ..
hoelzl@35582
   759
next
hoelzl@35582
   760
  case (Suc n)
hoelzl@35582
   761
  then obtain sn an xn where ps: "(sn, an, xn) \<in> pos_simple (mon_upclose n u)"
hoelzl@35582
   762
    unfolding psfis_def by auto
hoelzl@35582
   763
  obtain ss as xs where ps': "(ss, as, xs) \<in> pos_simple (u (Suc n))"
hoelzl@35582
   764
    using assms[of "Suc n"] unfolding psfis_def by auto
hoelzl@35582
   765
  from pos_simple_common_partition[OF ps ps'] guess x x' a s .
hoelzl@35582
   766
  hence "(s, a, upclose x x') \<in> pos_simple (mon_upclose (Suc n) u)"
hoelzl@35582
   767
    by (simp add: upclose_def pos_simple_def nonneg_def max_def)
hoelzl@35582
   768
  thus ?case unfolding psfis_def by auto
hoelzl@35582
   769
qed
hoelzl@35582
   770
hoelzl@35582
   771
text {* Beppo-Levi monotone convergence theorem *}
hoelzl@35582
   772
lemma nnfis_mon_conv:
hoelzl@35582
   773
  assumes mc: "mono_convergent f h (space M)"
hoelzl@35582
   774
  and nnfis: "\<And>n. x n \<in> nnfis (f n)"
hoelzl@35582
   775
  and "x ----> z"
hoelzl@35582
   776
  shows "z \<in> nnfis h"
hoelzl@35582
   777
proof -
hoelzl@35582
   778
  have "\<forall>n. \<exists>u y. mono_convergent u (f n) (space M) \<and> (\<forall>m. y m \<in> psfis (u m)) \<and>
hoelzl@35582
   779
    y ----> x n"
hoelzl@35582
   780
    using nnfis unfolding nnfis_def by auto
hoelzl@35582
   781
  from choice[OF this] guess u ..
hoelzl@35582
   782
  from choice[OF this] guess y ..
hoelzl@35582
   783
  hence mc_u: "\<And>n. mono_convergent (u n) (f n) (space M)"
hoelzl@35582
   784
    and psfis: "\<And>n m. y n m \<in> psfis (u n m)" and "\<And>n. y n ----> x n"
hoelzl@35582
   785
    by auto
hoelzl@35582
   786
hoelzl@35582
   787
  let "?upclose n" = "mon_upclose n (\<lambda>m. u m n)"
hoelzl@35582
   788
hoelzl@35582
   789
  have "\<exists>c. \<forall>n. c n \<in> psfis (?upclose n)"
hoelzl@35582
   790
    by (safe intro!: choice psfis_mon_upclose) (rule psfis)
hoelzl@35582
   791
  then guess c .. note c = this[rule_format]
hoelzl@35582
   792
hoelzl@35582
   793
  show ?thesis unfolding nnfis_def
hoelzl@35582
   794
  proof (safe intro!: exI)
hoelzl@35582
   795
    show mc_upclose: "mono_convergent ?upclose h (space M)"
hoelzl@35582
   796
      by (rule mon_upclose_mono_convergent[OF mc_u mc])
hoelzl@35582
   797
    show psfis_upclose: "\<And>n. c n \<in> psfis (?upclose n)"
hoelzl@35582
   798
      using c .
hoelzl@35582
   799
hoelzl@35582
   800
    { fix n m :: nat assume "n \<le> m"
hoelzl@35582
   801
      hence "c n \<le> c m"
hoelzl@35582
   802
        using psfis_mono[OF c c]
hoelzl@35582
   803
        using mono_convergentD(1)[OF mc_upclose, unfolded incseq_def]
hoelzl@35582
   804
        by auto }
hoelzl@35582
   805
    hence "incseq c" unfolding incseq_def by auto
hoelzl@35582
   806
hoelzl@35582
   807
    { fix n
hoelzl@35582
   808
      have c_nnfis: "c n \<in> nnfis (?upclose n)" using c by (rule psfis_nnfis)
hoelzl@35582
   809
      from nnfis_mono[OF c_nnfis nnfis]
hoelzl@35582
   810
        mon_upclose_le_mono_convergent[OF mc_u]
hoelzl@35582
   811
        mono_convergentD(1)[OF mc]
hoelzl@35582
   812
      have "c n \<le> x n" by fast }
hoelzl@35582
   813
    note c_less_x = this
hoelzl@35582
   814
hoelzl@35582
   815
    { fix n
hoelzl@35582
   816
      note c_less_x[of n]
hoelzl@35582
   817
      also have "x n \<le> z"
hoelzl@35582
   818
      proof (rule incseq_le)
hoelzl@35582
   819
        show "x ----> z" by fact
hoelzl@35582
   820
        from mono_convergentD(1)[OF mc]
hoelzl@35582
   821
        show "incseq x" unfolding incseq_def
hoelzl@35582
   822
          by (auto intro!: nnfis_mono[OF nnfis nnfis])
hoelzl@35582
   823
      qed
hoelzl@35582
   824
      finally have "c n \<le> z" . }
hoelzl@35582
   825
    note c_less_z = this
hoelzl@35582
   826
hoelzl@35582
   827
    have "convergent c"
hoelzl@35582
   828
    proof (rule Bseq_mono_convergent[unfolded incseq_def[symmetric]])
hoelzl@35582
   829
      show "Bseq c"
hoelzl@35582
   830
        using pos_psfis[OF c] c_less_z
hoelzl@35582
   831
        by (auto intro!: BseqI'[of _ z])
hoelzl@35582
   832
      show "incseq c" by fact
hoelzl@35582
   833
    qed
hoelzl@35582
   834
    then obtain l where l: "c ----> l" unfolding convergent_def by auto
hoelzl@35582
   835
hoelzl@35582
   836
    have "l \<le> z" using c_less_x l
hoelzl@35582
   837
      by (auto intro!: LIMSEQ_le[OF _ `x ----> z`])
hoelzl@35582
   838
    moreover have "z \<le> l"
hoelzl@35582
   839
    proof (rule LIMSEQ_le_const2[OF `x ----> z`], safe intro!: exI[of _ 0])
hoelzl@35582
   840
      fix n
hoelzl@35582
   841
      have "l \<in> nnfis h"
hoelzl@35582
   842
        unfolding nnfis_def using l mc_upclose psfis_upclose by auto
hoelzl@35582
   843
      from nnfis this mono_convergent_le[OF mc]
hoelzl@35582
   844
      show "x n \<le> l" by (rule nnfis_mono)
hoelzl@35582
   845
    qed
hoelzl@35582
   846
    ultimately have "l = z" by (rule real_le_antisym)
hoelzl@35582
   847
    thus "c ----> z" using `c ----> l` by simp
hoelzl@35582
   848
  qed
hoelzl@35582
   849
qed
hoelzl@35582
   850
hoelzl@35582
   851
lemma nnfis_pos_on_mspace:
hoelzl@35582
   852
  assumes "a \<in> nnfis f" and "x \<in>space M"
hoelzl@35582
   853
  shows "0 \<le> f x"
hoelzl@35582
   854
using assms
hoelzl@35582
   855
proof -
hoelzl@35582
   856
  from assms[unfolded nnfis_def] guess u y by auto note uy = this
hoelzl@35748
   857
  hence "\<And> n. 0 \<le> u n x"
hoelzl@35582
   858
    unfolding nnfis_def psfis_def pos_simple_def nonneg_def mono_convergent_def
hoelzl@35582
   859
    by auto
hoelzl@35582
   860
  thus "0 \<le> f x" using uy[rule_format]
hoelzl@35582
   861
    unfolding nnfis_def psfis_def pos_simple_def nonneg_def mono_convergent_def
hoelzl@35582
   862
    using incseq_le[of "\<lambda> n. u n x" "f x"] real_le_trans
hoelzl@35582
   863
    by fast
hoelzl@35582
   864
qed
hoelzl@35582
   865
hoelzl@35582
   866
lemma nnfis_borel_measurable:
hoelzl@35582
   867
  assumes "a \<in> nnfis f"
hoelzl@35582
   868
  shows "f \<in> borel_measurable M"
hoelzl@35582
   869
using assms unfolding nnfis_def
hoelzl@35582
   870
apply auto
hoelzl@35582
   871
apply (rule mono_convergent_borel_measurable)
hoelzl@35582
   872
using psfis_borel_measurable
hoelzl@35582
   873
by auto
hoelzl@35582
   874
hoelzl@35582
   875
lemma borel_measurable_mon_conv_psfis:
hoelzl@35582
   876
  assumes f_borel: "f \<in> borel_measurable M"
hoelzl@35582
   877
  and nonneg: "\<And>t. t \<in> space M \<Longrightarrow> 0 \<le> f t"
hoelzl@35582
   878
  shows"\<exists>u x. mono_convergent u f (space M) \<and> (\<forall>n. x n \<in> psfis (u n))"
hoelzl@35582
   879
proof (safe intro!: exI)
hoelzl@35582
   880
  let "?I n" = "{0<..<n * 2^n}"
hoelzl@35582
   881
  let "?A n i" = "{w \<in> space M. real (i :: nat) / 2^(n::nat) \<le> f w \<and> f w < real (i + 1) / 2^n}"
hoelzl@35582
   882
  let "?u n t" = "\<Sum>i\<in>?I n. real i / 2^n * indicator_fn (?A n i) t"
hoelzl@35582
   883
  let "?x n" = "\<Sum>i\<in>?I n. real i / 2^n * measure M (?A n i)"
hoelzl@35582
   884
hoelzl@35582
   885
  let "?w n t" = "if f t < real n then real (natfloor (f t * 2^n)) / 2^n else 0"
hoelzl@35582
   886
hoelzl@35582
   887
  { fix t n assume t: "t \<in> space M"
hoelzl@35582
   888
    have "?u n t = ?w n t" (is "_ = (if _ then real ?i / _ else _)")
hoelzl@35582
   889
    proof (cases "f t < real n")
hoelzl@35582
   890
      case True
hoelzl@35582
   891
      with nonneg t
hoelzl@35582
   892
      have i: "?i < n * 2^n"
hoelzl@35582
   893
        by (auto simp: real_of_nat_power[symmetric]
hoelzl@35582
   894
                 intro!: less_natfloor mult_nonneg_nonneg)
hoelzl@35582
   895
hoelzl@35582
   896
      hence t_in_A: "t \<in> ?A n ?i"
hoelzl@35582
   897
        unfolding divide_le_eq less_divide_eq
hoelzl@35582
   898
        using nonneg t True
hoelzl@35582
   899
        by (auto intro!: real_natfloor_le
hoelzl@35582
   900
          real_natfloor_gt_diff_one[unfolded diff_less_eq]
hoelzl@35582
   901
          simp: real_of_nat_Suc zero_le_mult_iff)
hoelzl@35582
   902
hoelzl@35582
   903
      hence *: "real ?i / 2^n \<le> f t"
hoelzl@35582
   904
        "f t < real (?i + 1) / 2^n" by auto
hoelzl@35582
   905
      { fix j assume "t \<in> ?A n j"
hoelzl@35582
   906
        hence "real j / 2^n \<le> f t"
hoelzl@35582
   907
          and "f t < real (j + 1) / 2^n" by auto
hoelzl@35582
   908
        with * have "j \<in> {?i}" unfolding divide_le_eq less_divide_eq
hoelzl@35582
   909
          by auto }
hoelzl@35582
   910
      hence *: "\<And>j. t \<in> ?A n j \<longleftrightarrow> j \<in> {?i}" using t_in_A by auto
hoelzl@35582
   911
hoelzl@35582
   912
      have "?u n t = real ?i / 2^n"
hoelzl@35582
   913
        unfolding indicator_fn_def if_distrib *
hoelzl@35582
   914
          setsum_cases[OF finite_greaterThanLessThan]
hoelzl@35582
   915
        using i by (cases "?i = 0") simp_all
hoelzl@35582
   916
      thus ?thesis using True by auto
hoelzl@35582
   917
    next
hoelzl@35582
   918
      case False
hoelzl@35582
   919
      have "?u n t = (\<Sum>i \<in> {0 <..< n*2^n}. 0)"
hoelzl@35582
   920
      proof (rule setsum_cong)
hoelzl@35582
   921
        fix i assume "i \<in> {0 <..< n*2^n}"
hoelzl@35582
   922
        hence "i + 1 \<le> n * 2^n" by simp
hoelzl@35582
   923
        hence "real (i + 1) \<le> real n * 2^n"
hoelzl@35582
   924
          unfolding real_of_nat_le_iff[symmetric]
hoelzl@35582
   925
          by (auto simp: real_of_nat_power[symmetric])
hoelzl@35582
   926
        also have "... \<le> f t * 2^n"
hoelzl@35582
   927
          using False by (auto intro!: mult_nonneg_nonneg)
hoelzl@35582
   928
        finally have "t \<notin> ?A n i"
hoelzl@35582
   929
          by (auto simp: divide_le_eq less_divide_eq)
hoelzl@35582
   930
        thus "real i / 2^n * indicator_fn (?A n i) t = 0"
hoelzl@35582
   931
          unfolding indicator_fn_def by auto
hoelzl@35582
   932
      qed simp
hoelzl@35582
   933
      thus ?thesis using False by auto
hoelzl@35582
   934
    qed }
hoelzl@35582
   935
  note u_at_t = this
hoelzl@35582
   936
hoelzl@35582
   937
  show "mono_convergent ?u f (space M)" unfolding mono_convergent_def
hoelzl@35582
   938
  proof safe
hoelzl@35582
   939
    fix t assume t: "t \<in> space M"
hoelzl@35582
   940
    { fix m n :: nat assume "m \<le> n"
hoelzl@35582
   941
      hence *: "(2::real)^n = 2^m * 2^(n - m)" unfolding class_semiring.mul_pwr by auto
hoelzl@35582
   942
      have "real (natfloor (f t * 2^m) * natfloor (2^(n-m))) \<le> real (natfloor (f t * 2 ^ n))"
hoelzl@35582
   943
        apply (subst *)
hoelzl@35582
   944
        apply (subst class_semiring.mul_a)
hoelzl@35582
   945
        apply (subst real_of_nat_le_iff)
hoelzl@35582
   946
        apply (rule le_mult_natfloor)
hoelzl@35582
   947
        using nonneg[OF t] by (auto intro!: mult_nonneg_nonneg)
hoelzl@35582
   948
      hence "real (natfloor (f t * 2^m)) * 2^n \<le> real (natfloor (f t * 2^n)) * 2^m"
hoelzl@35582
   949
        apply (subst *)
hoelzl@35582
   950
        apply (subst (3) class_semiring.mul_c)
hoelzl@35582
   951
        apply (subst class_semiring.mul_a)
hoelzl@35582
   952
        by (auto intro: mult_right_mono simp: natfloor_power real_of_nat_power[symmetric]) }
hoelzl@35582
   953
    thus "incseq (\<lambda>n. ?u n t)" unfolding u_at_t[OF t] unfolding incseq_def
hoelzl@35582
   954
      by (auto simp add: le_divide_eq divide_le_eq less_divide_eq)
hoelzl@35582
   955
hoelzl@35582
   956
    show "(\<lambda>i. ?u i t) ----> f t" unfolding u_at_t[OF t]
hoelzl@35582
   957
    proof (rule LIMSEQ_I, safe intro!: exI)
hoelzl@35582
   958
      fix r :: real and n :: nat
hoelzl@35582
   959
      let ?N = "natfloor (1/r) + 1"
hoelzl@35582
   960
      assume "0 < r" and N: "max ?N (natfloor (f t) + 1) \<le> n"
hoelzl@35582
   961
      hence "?N \<le> n" by auto
hoelzl@35582
   962
hoelzl@35582
   963
      have "1 / r < real (natfloor (1/r) + 1)" using real_natfloor_add_one_gt
hoelzl@35582
   964
        by (simp add: real_of_nat_Suc)
hoelzl@35582
   965
      also have "... < 2^?N" by (rule two_realpow_gt)
hoelzl@35582
   966
      finally have less_r: "1 / 2^?N < r" using `0 < r`
hoelzl@35582
   967
        by (auto simp: less_divide_eq divide_less_eq algebra_simps)
hoelzl@35582
   968
hoelzl@35582
   969
      have "f t < real (natfloor (f t) + 1)" using real_natfloor_add_one_gt[of "f t"] by auto
hoelzl@35582
   970
      also have "... \<le> real n" unfolding real_of_nat_le_iff using N by auto
hoelzl@35582
   971
      finally have "f t < real n" .
hoelzl@35582
   972
hoelzl@35582
   973
      have "real (natfloor (f t * 2^n)) \<le> f t * 2^n"
hoelzl@35582
   974
        using nonneg[OF t] by (auto intro!: real_natfloor_le mult_nonneg_nonneg)
hoelzl@35582
   975
      hence less: "real (natfloor (f t * 2^n)) / 2^n \<le> f t" unfolding divide_le_eq by auto
hoelzl@35582
   976
hoelzl@35582
   977
      have "f t * 2 ^ n - 1 < real (natfloor (f t * 2^n))" using real_natfloor_gt_diff_one .
hoelzl@35582
   978
      hence "f t - real (natfloor (f t * 2^n)) / 2^n < 1 / 2^n"
hoelzl@35582
   979
        by (auto simp: less_divide_eq divide_less_eq algebra_simps)
hoelzl@35582
   980
      also have "... \<le> 1 / 2^?N" using `?N \<le> n`
hoelzl@35582
   981
        by (auto intro!: divide_left_mono mult_pos_pos simp del: power_Suc)
hoelzl@35582
   982
      also have "... < r" using less_r .
hoelzl@35582
   983
      finally show "norm (?w n t - f t) < r" using `f t < real n` less by auto
hoelzl@35582
   984
    qed
hoelzl@35582
   985
  qed
hoelzl@35582
   986
hoelzl@35582
   987
  fix n
hoelzl@35582
   988
  show "?x n \<in> psfis (?u n)"
hoelzl@35582
   989
  proof (rule psfis_intro)
hoelzl@35582
   990
    show "?A n ` ?I n \<subseteq> sets M"
hoelzl@35582
   991
    proof safe
hoelzl@35582
   992
      fix i :: nat
hoelzl@35582
   993
      from Int[OF
hoelzl@35582
   994
        f_borel[unfolded borel_measurable_less_iff, rule_format, of "real (i+1) / 2^n"]
hoelzl@35582
   995
        f_borel[unfolded borel_measurable_ge_iff, rule_format, of "real i / 2^n"]]
hoelzl@35582
   996
      show "?A n i \<in> sets M"
hoelzl@35582
   997
        by (metis Collect_conj_eq Int_commute Int_left_absorb Int_left_commute)
hoelzl@35582
   998
    qed
hoelzl@35582
   999
    show "nonneg (\<lambda>i :: nat. real i / 2 ^ n)"
hoelzl@35582
  1000
      unfolding nonneg_def by (auto intro!: divide_nonneg_pos)
hoelzl@35582
  1001
  qed auto
hoelzl@35582
  1002
qed
hoelzl@35582
  1003
hoelzl@35582
  1004
lemma nnfis_dom_conv:
hoelzl@35582
  1005
  assumes borel: "f \<in> borel_measurable M"
hoelzl@35582
  1006
  and nnfis: "b \<in> nnfis g"
hoelzl@35582
  1007
  and ord: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
hoelzl@35582
  1008
  and nonneg: "\<And>t. t \<in> space M \<Longrightarrow> 0 \<le> f t"
hoelzl@35582
  1009
  shows "\<exists>a. a \<in> nnfis f \<and> a \<le> b"
hoelzl@35582
  1010
proof -
hoelzl@35582
  1011
  obtain u x where mc_f: "mono_convergent u f (space M)" and
hoelzl@35582
  1012
    psfis: "\<And>n. x n \<in> psfis (u n)"
hoelzl@35582
  1013
    using borel_measurable_mon_conv_psfis[OF borel nonneg] by auto
hoelzl@35582
  1014
hoelzl@35582
  1015
  { fix n
hoelzl@35582
  1016
    { fix t assume t: "t \<in> space M"
hoelzl@35582
  1017
      note mono_convergent_le[OF mc_f this, of n]
hoelzl@35582
  1018
      also note ord[OF t]
hoelzl@35582
  1019
      finally have "u n t \<le> g t" . }
hoelzl@35582
  1020
    from nnfis_mono[OF psfis_nnfis[OF psfis] nnfis this]
hoelzl@35582
  1021
    have "x n \<le> b" by simp }
hoelzl@35582
  1022
  note x_less_b = this
hoelzl@35582
  1023
hoelzl@35582
  1024
  have "convergent x"
hoelzl@35582
  1025
  proof (safe intro!: Bseq_mono_convergent)
hoelzl@35582
  1026
    from x_less_b pos_psfis[OF psfis]
hoelzl@35582
  1027
    show "Bseq x" by (auto intro!: BseqI'[of _ b])
hoelzl@35582
  1028
hoelzl@35582
  1029
    fix n m :: nat assume *: "n \<le> m"
hoelzl@35582
  1030
    show "x n \<le> x m"
hoelzl@35582
  1031
    proof (rule psfis_mono[OF `x n \<in> psfis (u n)` `x m \<in> psfis (u m)`])
hoelzl@35582
  1032
      fix t assume "t \<in> space M"
hoelzl@35582
  1033
      from mc_f[THEN mono_convergentD(1), unfolded incseq_def, OF this]
hoelzl@35582
  1034
      show "u n t \<le> u m t" using * by auto
hoelzl@35582
  1035
    qed
hoelzl@35582
  1036
  qed
hoelzl@35582
  1037
  then obtain a where "x ----> a" unfolding convergent_def by auto
hoelzl@35582
  1038
  with LIMSEQ_le_const2[OF `x ----> a`] x_less_b mc_f psfis
hoelzl@35582
  1039
  show ?thesis unfolding nnfis_def by auto
hoelzl@35582
  1040
qed
hoelzl@35582
  1041
hoelzl@35582
  1042
lemma the_nnfis[simp]: "a \<in> nnfis f \<Longrightarrow> (THE a. a \<in> nnfis f) = a"
hoelzl@35582
  1043
  by (auto intro: the_equality nnfis_unique)
hoelzl@35582
  1044
hoelzl@35582
  1045
lemma nnfis_cong:
hoelzl@35582
  1046
  assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@35582
  1047
  shows "nnfis f = nnfis g"
hoelzl@35582
  1048
proof -
hoelzl@35582
  1049
  { fix f g :: "'a \<Rightarrow> real" assume cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@35582
  1050
    fix x assume "x \<in> nnfis f"
hoelzl@35582
  1051
    then guess u y unfolding nnfis_def by safe note x = this
hoelzl@35582
  1052
    hence "mono_convergent u g (space M)"
hoelzl@35582
  1053
      unfolding mono_convergent_def using cong by auto
hoelzl@35582
  1054
    with x(2,3) have "x \<in> nnfis g" unfolding nnfis_def by auto }
hoelzl@35582
  1055
  from this[OF cong] this[OF cong[symmetric]]
hoelzl@35582
  1056
  show ?thesis by safe
hoelzl@35582
  1057
qed
hoelzl@35582
  1058
hoelzl@35692
  1059
section "Lebesgue Integral"
hoelzl@35692
  1060
hoelzl@35692
  1061
definition
hoelzl@35692
  1062
  "integrable f \<longleftrightarrow> (\<exists>x. x \<in> nnfis (pos_part f)) \<and> (\<exists>y. y \<in> nnfis (neg_part f))"
hoelzl@35692
  1063
hoelzl@35692
  1064
definition
hoelzl@35692
  1065
  "integral f = (THE i :: real. i \<in> nnfis (pos_part f)) - (THE j. j \<in> nnfis (neg_part f))"
hoelzl@35692
  1066
hoelzl@35582
  1067
lemma integral_cong:
hoelzl@35582
  1068
  assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@35582
  1069
  shows "integral f = integral g"
hoelzl@35582
  1070
proof -
hoelzl@35582
  1071
  have "nnfis (pos_part f) = nnfis (pos_part g)"
hoelzl@35582
  1072
    using cong by (auto simp: pos_part_def intro!: nnfis_cong)
hoelzl@35582
  1073
  moreover
hoelzl@35582
  1074
  have "nnfis (neg_part f) = nnfis (neg_part g)"
hoelzl@35582
  1075
    using cong by (auto simp: neg_part_def intro!: nnfis_cong)
hoelzl@35582
  1076
  ultimately show ?thesis
hoelzl@35582
  1077
    unfolding integral_def by auto
hoelzl@35582
  1078
qed
hoelzl@35582
  1079
hoelzl@35582
  1080
lemma nnfis_integral:
hoelzl@35582
  1081
  assumes "a \<in> nnfis f"
hoelzl@35582
  1082
  shows "integrable f" and "integral f = a"
hoelzl@35582
  1083
proof -
hoelzl@35582
  1084
  have a: "a \<in> nnfis (pos_part f)"
hoelzl@35582
  1085
    using assms nnfis_pos_on_mspace[OF assms]
hoelzl@35582
  1086
    by (auto intro!: nnfis_mon_conv[of "\<lambda>i. f" _ "\<lambda>i. a"]
hoelzl@35582
  1087
      LIMSEQ_const simp: mono_convergent_def pos_part_def incseq_def max_def)
hoelzl@35582
  1088
hoelzl@35582
  1089
  have "\<And>t. t \<in> space M \<Longrightarrow> neg_part f t = 0"
hoelzl@35582
  1090
    unfolding neg_part_def min_def
hoelzl@35582
  1091
    using nnfis_pos_on_mspace[OF assms] by auto
hoelzl@35582
  1092
  hence 0: "0 \<in> nnfis (neg_part f)"
hoelzl@35582
  1093
    by (auto simp: nnfis_def mono_convergent_def psfis_0 incseq_def
hoelzl@35582
  1094
          intro!: LIMSEQ_const exI[of _ "\<lambda> x n. 0"] exI[of _ "\<lambda> n. 0"])
hoelzl@35582
  1095
hoelzl@35582
  1096
  from 0 a show "integrable f" "integral f = a"
hoelzl@35582
  1097
    unfolding integrable_def integral_def by auto
hoelzl@35582
  1098
qed
hoelzl@35582
  1099
hoelzl@35582
  1100
lemma nnfis_minus_nnfis_integral:
hoelzl@35582
  1101
  assumes "a \<in> nnfis f" and "b \<in> nnfis g"
hoelzl@35582
  1102
  shows "integrable (\<lambda>t. f t - g t)" and "integral (\<lambda>t. f t - g t) = a - b"
hoelzl@35582
  1103
proof -
hoelzl@35582
  1104
  have borel: "(\<lambda>t. f t - g t) \<in> borel_measurable M" using assms
hoelzl@35582
  1105
    by (blast intro:
hoelzl@35582
  1106
      borel_measurable_diff_borel_measurable nnfis_borel_measurable)
hoelzl@35582
  1107
hoelzl@35582
  1108
  have "\<exists>x. x \<in> nnfis (pos_part (\<lambda>t. f t - g t)) \<and> x \<le> a + b"
hoelzl@35582
  1109
    (is "\<exists>x. x \<in> nnfis ?pp \<and> _")
hoelzl@35582
  1110
  proof (rule nnfis_dom_conv)
hoelzl@35582
  1111
    show "?pp \<in> borel_measurable M"
hoelzl@35692
  1112
      using borel by (rule pos_part_borel_measurable neg_part_borel_measurable)
hoelzl@35582
  1113
    show "a + b \<in> nnfis (\<lambda>t. f t + g t)" using assms by (rule nnfis_add)
hoelzl@35582
  1114
    fix t assume "t \<in> space M"
hoelzl@35582
  1115
    with assms nnfis_add assms[THEN nnfis_pos_on_mspace[OF _ this]]
hoelzl@35582
  1116
    show "?pp t \<le> f t + g t" unfolding pos_part_def by auto
hoelzl@35582
  1117
    show "0 \<le> ?pp t" using nonneg_pos_part[of "\<lambda>t. f t - g t"]
hoelzl@35582
  1118
      unfolding nonneg_def by auto
hoelzl@35582
  1119
  qed
hoelzl@35582
  1120
  then obtain x where x: "x \<in> nnfis ?pp" by auto
hoelzl@35582
  1121
  moreover
hoelzl@35582
  1122
  have "\<exists>x. x \<in> nnfis (neg_part (\<lambda>t. f t - g t)) \<and> x \<le> a + b"
hoelzl@35582
  1123
    (is "\<exists>x. x \<in> nnfis ?np \<and> _")
hoelzl@35582
  1124
  proof (rule nnfis_dom_conv)
hoelzl@35582
  1125
    show "?np \<in> borel_measurable M"
hoelzl@35692
  1126
      using borel by (rule pos_part_borel_measurable neg_part_borel_measurable)
hoelzl@35582
  1127
    show "a + b \<in> nnfis (\<lambda>t. f t + g t)" using assms by (rule nnfis_add)
hoelzl@35582
  1128
    fix t assume "t \<in> space M"
hoelzl@35582
  1129
    with assms nnfis_add assms[THEN nnfis_pos_on_mspace[OF _ this]]
hoelzl@35582
  1130
    show "?np t \<le> f t + g t" unfolding neg_part_def by auto
hoelzl@35582
  1131
    show "0 \<le> ?np t" using nonneg_neg_part[of "\<lambda>t. f t - g t"]
hoelzl@35582
  1132
      unfolding nonneg_def by auto
hoelzl@35582
  1133
  qed
hoelzl@35582
  1134
  then obtain y where y: "y \<in> nnfis ?np" by auto
hoelzl@35582
  1135
  ultimately show "integrable (\<lambda>t. f t - g t)"
hoelzl@35582
  1136
    unfolding integrable_def by auto
hoelzl@35582
  1137
hoelzl@35582
  1138
  from x and y
hoelzl@35582
  1139
  have "a + y \<in> nnfis (\<lambda>t. f t + ?np t)"
hoelzl@35582
  1140
    and "b + x \<in> nnfis (\<lambda>t. g t + ?pp t)" using assms by (auto intro: nnfis_add)
hoelzl@35582
  1141
  moreover
hoelzl@35582
  1142
  have "\<And>t. f t + ?np t = g t + ?pp t"
hoelzl@35582
  1143
    unfolding pos_part_def neg_part_def by auto
hoelzl@35582
  1144
  ultimately have "a - b = x - y"
hoelzl@35582
  1145
    using nnfis_unique by (auto simp: algebra_simps)
hoelzl@35582
  1146
  thus "integral (\<lambda>t. f t - g t) = a - b"
hoelzl@35582
  1147
    unfolding integral_def
hoelzl@35582
  1148
    using the_nnfis[OF x] the_nnfis[OF y] by simp
hoelzl@35582
  1149
qed
hoelzl@35582
  1150
hoelzl@35582
  1151
lemma integral_borel_measurable:
hoelzl@35582
  1152
  "integrable f \<Longrightarrow> f \<in> borel_measurable M"
hoelzl@35582
  1153
  unfolding integrable_def
hoelzl@35582
  1154
  by (subst pos_part_neg_part_borel_measurable_iff)
hoelzl@35582
  1155
   (auto intro: nnfis_borel_measurable)
hoelzl@35582
  1156
hoelzl@35582
  1157
lemma integral_indicator_fn:
hoelzl@35582
  1158
  assumes "a \<in> sets M"
hoelzl@35582
  1159
  shows "integral (indicator_fn a) = measure M a"
hoelzl@35582
  1160
  and "integrable (indicator_fn a)"
hoelzl@35582
  1161
  using psfis_indicator[OF assms, THEN psfis_nnfis]
hoelzl@35582
  1162
  by (auto intro!: nnfis_integral)
hoelzl@35582
  1163
hoelzl@35582
  1164
lemma integral_add:
hoelzl@35582
  1165
  assumes "integrable f" and "integrable g"
hoelzl@35582
  1166
  shows "integrable (\<lambda>t. f t + g t)"
hoelzl@35582
  1167
  and "integral (\<lambda>t. f t + g t) = integral f + integral g"
hoelzl@35582
  1168
proof -
hoelzl@35582
  1169
  { fix t
hoelzl@35582
  1170
    have "pos_part f t + pos_part g t - (neg_part f t + neg_part g t) =
hoelzl@35582
  1171
      f t + g t"
hoelzl@35582
  1172
      unfolding pos_part_def neg_part_def by auto }
hoelzl@35582
  1173
  note part_sum = this
hoelzl@35582
  1174
hoelzl@35582
  1175
  from assms obtain a b c d where
hoelzl@35582
  1176
    a: "a \<in> nnfis (pos_part f)" and b: "b \<in> nnfis (neg_part f)" and
hoelzl@35582
  1177
    c: "c \<in> nnfis (pos_part g)" and d: "d \<in> nnfis (neg_part g)"
hoelzl@35582
  1178
    unfolding integrable_def by auto
hoelzl@35582
  1179
  note sums = nnfis_add[OF a c] nnfis_add[OF b d]
hoelzl@35582
  1180
  note int = nnfis_minus_nnfis_integral[OF sums, unfolded part_sum]
hoelzl@35582
  1181
hoelzl@35582
  1182
  show "integrable (\<lambda>t. f t + g t)" using int(1) .
hoelzl@35582
  1183
hoelzl@35582
  1184
  show "integral (\<lambda>t. f t + g t) = integral f + integral g"
hoelzl@35582
  1185
    using int(2) sums a b c d by (simp add: the_nnfis integral_def)
hoelzl@35582
  1186
qed
hoelzl@35582
  1187
hoelzl@35582
  1188
lemma integral_mono:
hoelzl@35582
  1189
  assumes "integrable f" and "integrable g"
hoelzl@35582
  1190
  and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
hoelzl@35582
  1191
  shows "integral f \<le> integral g"
hoelzl@35582
  1192
proof -
hoelzl@35582
  1193
  from assms obtain a b c d where
hoelzl@35582
  1194
    a: "a \<in> nnfis (pos_part f)" and b: "b \<in> nnfis (neg_part f)" and
hoelzl@35582
  1195
    c: "c \<in> nnfis (pos_part g)" and d: "d \<in> nnfis (neg_part g)"
hoelzl@35582
  1196
    unfolding integrable_def by auto
hoelzl@35582
  1197
hoelzl@35582
  1198
  have "a \<le> c"
hoelzl@35582
  1199
  proof (rule nnfis_mono[OF a c])
hoelzl@35582
  1200
    fix t assume "t \<in> space M"
hoelzl@35582
  1201
    from mono[OF this] show "pos_part f t \<le> pos_part g t"
hoelzl@35582
  1202
      unfolding pos_part_def by auto
hoelzl@35582
  1203
  qed
hoelzl@35582
  1204
  moreover have "d \<le> b"
hoelzl@35582
  1205
  proof (rule nnfis_mono[OF d b])
hoelzl@35582
  1206
    fix t assume "t \<in> space M"
hoelzl@35582
  1207
    from mono[OF this] show "neg_part g t \<le> neg_part f t"
hoelzl@35582
  1208
      unfolding neg_part_def by auto
hoelzl@35582
  1209
  qed
hoelzl@35582
  1210
  ultimately have "a - b \<le> c - d" by auto
hoelzl@35582
  1211
  thus ?thesis unfolding integral_def
hoelzl@35582
  1212
    using a b c d by (simp add: the_nnfis)
hoelzl@35582
  1213
qed
hoelzl@35582
  1214
hoelzl@35582
  1215
lemma integral_uminus:
hoelzl@35582
  1216
  assumes "integrable f"
hoelzl@35582
  1217
  shows "integrable (\<lambda>t. - f t)"
hoelzl@35582
  1218
  and "integral (\<lambda>t. - f t) = - integral f"
hoelzl@35582
  1219
proof -
hoelzl@35582
  1220
  have "pos_part f = neg_part (\<lambda>t.-f t)" and "neg_part f = pos_part (\<lambda>t.-f t)"
hoelzl@35582
  1221
    unfolding pos_part_def neg_part_def by (auto intro!: ext)
hoelzl@35582
  1222
  with assms show "integrable (\<lambda>t.-f t)" and "integral (\<lambda>t.-f t) = -integral f"
hoelzl@35582
  1223
    unfolding integrable_def integral_def by simp_all
hoelzl@35582
  1224
qed
hoelzl@35582
  1225
hoelzl@35582
  1226
lemma integral_times_const:
hoelzl@35582
  1227
  assumes "integrable f"
hoelzl@35582
  1228
  shows "integrable (\<lambda>t. a * f t)" (is "?P a")
hoelzl@35582
  1229
  and "integral (\<lambda>t. a * f t) = a * integral f" (is "?I a")
hoelzl@35582
  1230
proof -
hoelzl@35582
  1231
  { fix a :: real assume "0 \<le> a"
hoelzl@35582
  1232
    hence "pos_part (\<lambda>t. a * f t) = (\<lambda>t. a * pos_part f t)"
hoelzl@35582
  1233
      and "neg_part (\<lambda>t. a * f t) = (\<lambda>t. a * neg_part f t)"
hoelzl@35582
  1234
      unfolding pos_part_def neg_part_def max_def min_def
hoelzl@35582
  1235
      by (auto intro!: ext simp: zero_le_mult_iff)
hoelzl@35582
  1236
    moreover
hoelzl@35582
  1237
    obtain x y where x: "x \<in> nnfis (pos_part f)" and y: "y \<in> nnfis (neg_part f)"
hoelzl@35582
  1238
      using assms unfolding integrable_def by auto
hoelzl@35582
  1239
    ultimately
hoelzl@35582
  1240
    have "a * x \<in> nnfis (pos_part (\<lambda>t. a * f t))" and
hoelzl@35582
  1241
      "a * y \<in> nnfis (neg_part (\<lambda>t. a * f t))"
hoelzl@35582
  1242
      using nnfis_times[OF _ `0 \<le> a`] by auto
hoelzl@35582
  1243
    with x y have "?P a \<and> ?I a"
hoelzl@35582
  1244
      unfolding integrable_def integral_def by (auto simp: algebra_simps) }
hoelzl@35582
  1245
  note int = this
hoelzl@35582
  1246
hoelzl@35582
  1247
  have "?P a \<and> ?I a"
hoelzl@35582
  1248
  proof (cases "0 \<le> a")
hoelzl@35582
  1249
    case True from int[OF this] show ?thesis .
hoelzl@35582
  1250
  next
hoelzl@35582
  1251
    case False with int[of "- a"] integral_uminus[of "\<lambda>t. - a * f t"]
hoelzl@35582
  1252
    show ?thesis by auto
hoelzl@35582
  1253
  qed
hoelzl@35582
  1254
  thus "integrable (\<lambda>t. a * f t)"
hoelzl@35582
  1255
    and "integral (\<lambda>t. a * f t) = a * integral f" by simp_all
hoelzl@35582
  1256
qed
hoelzl@35582
  1257
hoelzl@35582
  1258
lemma integral_cmul_indicator:
hoelzl@35582
  1259
  assumes "s \<in> sets M"
hoelzl@35582
  1260
  shows "integral (\<lambda>x. c * indicator_fn s x) = c * (measure M s)"
hoelzl@35582
  1261
  and "integrable (\<lambda>x. c * indicator_fn s x)"
hoelzl@35582
  1262
using assms integral_times_const integral_indicator_fn by auto
hoelzl@35582
  1263
hoelzl@35582
  1264
lemma integral_zero:
hoelzl@35582
  1265
  shows "integral (\<lambda>x. 0) = 0"
hoelzl@35582
  1266
  and "integrable (\<lambda>x. 0)"
hoelzl@35582
  1267
  using integral_cmul_indicator[OF empty_sets, of 0]
hoelzl@35582
  1268
  unfolding indicator_fn_def by auto
hoelzl@35582
  1269
hoelzl@35582
  1270
lemma integral_setsum:
hoelzl@35582
  1271
  assumes "finite S"
hoelzl@35582
  1272
  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
hoelzl@35582
  1273
  shows "integral (\<lambda>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
hoelzl@35582
  1274
    and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I s")
hoelzl@35582
  1275
proof -
hoelzl@35582
  1276
  from assms have "?int S \<and> ?I S"
hoelzl@35582
  1277
  proof (induct S)
hoelzl@35582
  1278
    case empty thus ?case by (simp add: integral_zero)
hoelzl@35582
  1279
  next
hoelzl@35582
  1280
    case (insert i S)
hoelzl@35582
  1281
    thus ?case
hoelzl@35582
  1282
      apply simp
hoelzl@35582
  1283
      apply (subst integral_add)
hoelzl@35582
  1284
      using assms apply auto
hoelzl@35582
  1285
      apply (subst integral_add)
hoelzl@35582
  1286
      using assms by auto
hoelzl@35582
  1287
  qed
hoelzl@35582
  1288
  thus "?int S" and "?I S" by auto
hoelzl@35582
  1289
qed
hoelzl@35582
  1290
hoelzl@36624
  1291
lemma (in measure_space) integrable_abs:
hoelzl@36624
  1292
  assumes "integrable f"
hoelzl@36624
  1293
  shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
hoelzl@36624
  1294
using assms
hoelzl@36624
  1295
proof -
hoelzl@36624
  1296
  from assms obtain p q where pq: "p \<in> nnfis (pos_part f)" "q \<in> nnfis (neg_part f)"
hoelzl@36624
  1297
    unfolding integrable_def by auto
hoelzl@36624
  1298
  hence "p + q \<in> nnfis (\<lambda> x. pos_part f x + neg_part f x)"
hoelzl@36624
  1299
    using nnfis_add by auto
hoelzl@36624
  1300
  hence "p + q \<in> nnfis (\<lambda> x. \<bar>f x\<bar>)" using pos_neg_part_abs[of f] by simp
hoelzl@36624
  1301
  thus ?thesis unfolding integrable_def
hoelzl@36624
  1302
    using ext[OF pos_part_abs[of f], of "\<lambda> y. y"]
hoelzl@36624
  1303
      ext[OF neg_part_abs[of f], of "\<lambda> y. y"]
hoelzl@36624
  1304
    using nnfis_0 by auto
hoelzl@36624
  1305
qed
hoelzl@36624
  1306
hoelzl@35582
  1307
lemma markov_ineq:
hoelzl@35582
  1308
  assumes "integrable f" "0 < a" "integrable (\<lambda>x. \<bar>f x\<bar>^n)"
hoelzl@35582
  1309
  shows "measure M (f -` {a ..} \<inter> space M) \<le> integral (\<lambda>x. \<bar>f x\<bar>^n) / a^n"
hoelzl@35582
  1310
using assms
hoelzl@35582
  1311
proof -
hoelzl@35582
  1312
  from assms have "0 < a ^ n" using real_root_pow_pos by auto
hoelzl@35582
  1313
  from assms have "f \<in> borel_measurable M"
hoelzl@35582
  1314
    using integral_borel_measurable[OF `integrable f`] by auto
hoelzl@35582
  1315
  hence w: "{w . w \<in> space M \<and> a \<le> f w} \<in> sets M"
hoelzl@35582
  1316
    using borel_measurable_ge_iff by auto
hoelzl@35582
  1317
  have i: "integrable (indicator_fn {w . w \<in> space M \<and> a \<le> f w})"
hoelzl@35582
  1318
    using integral_indicator_fn[OF w] by simp
hoelzl@35582
  1319
  have v1: "\<And> t. a ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t 
hoelzl@35582
  1320
            \<le> (f t) ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t"
hoelzl@35582
  1321
    unfolding indicator_fn_def
hoelzl@35582
  1322
    using `0 < a` power_mono[of a] assms by auto
hoelzl@35582
  1323
  have v2: "\<And> t. (f t) ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t \<le> \<bar>f t\<bar> ^ n"
hoelzl@35582
  1324
    unfolding indicator_fn_def 
hoelzl@35582
  1325
    using power_mono[of a _ n] abs_ge_self `a > 0` 
hoelzl@35582
  1326
    by auto
hoelzl@35582
  1327
  have "{w \<in> space M. a \<le> f w} \<inter> space M = {w . a \<le> f w} \<inter> space M"
hoelzl@35582
  1328
    using Collect_eq by auto
hoelzl@35582
  1329
  from Int_absorb2[OF sets_into_space[OF w]] `0 < a ^ n` sets_into_space[OF w] w this
hoelzl@35582
  1330
  have "(a ^ n) * (measure M ((f -` {y . a \<le> y}) \<inter> space M)) =
hoelzl@35582
  1331
        (a ^ n) * measure M {w . w \<in> space M \<and> a \<le> f w}"
hoelzl@35582
  1332
    unfolding vimage_Collect_eq[of f] by simp
hoelzl@35582
  1333
  also have "\<dots> = integral (\<lambda> t. a ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t)"
hoelzl@35582
  1334
    using integral_cmul_indicator[OF w] i by auto
hoelzl@35582
  1335
  also have "\<dots> \<le> integral (\<lambda> t. \<bar> f t \<bar> ^ n)"
hoelzl@35582
  1336
    apply (rule integral_mono)
hoelzl@35582
  1337
    using integral_cmul_indicator[OF w]
hoelzl@35582
  1338
      `integrable (\<lambda> x. \<bar>f x\<bar> ^ n)` real_le_trans[OF v1 v2] by auto
hoelzl@35582
  1339
  finally show "measure M (f -` {a ..} \<inter> space M) \<le> integral (\<lambda>x. \<bar>f x\<bar>^n) / a^n"
hoelzl@35582
  1340
    unfolding atLeast_def
hoelzl@35582
  1341
    by (auto intro!: mult_imp_le_div_pos[OF `0 < a ^ n`], simp add: real_mult_commute)
hoelzl@35582
  1342
qed
hoelzl@35582
  1343
hoelzl@36624
  1344
lemma (in measure_space) integral_0:
hoelzl@36624
  1345
  fixes f :: "'a \<Rightarrow> real"
hoelzl@36624
  1346
  assumes "integrable f" "integral f = 0" "nonneg f" and borel: "f \<in> borel_measurable M"
hoelzl@36624
  1347
  shows "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0"
hoelzl@36624
  1348
proof -
hoelzl@36624
  1349
  have "{x. f x \<noteq> 0} = {x. \<bar>f x\<bar> > 0}" by auto
hoelzl@36624
  1350
  moreover
hoelzl@36624
  1351
  { fix y assume "y \<in> {x. \<bar> f x \<bar> > 0}"
hoelzl@36624
  1352
    hence "\<bar> f y \<bar> > 0" by auto
hoelzl@36624
  1353
    hence "\<exists> n. \<bar>f y\<bar> \<ge> inverse (real (Suc n))"
hoelzl@36624
  1354
      using ex_inverse_of_nat_Suc_less[of "\<bar>f y\<bar>"] less_imp_le unfolding real_of_nat_def by auto
hoelzl@36624
  1355
    hence "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
hoelzl@36624
  1356
      by auto }
hoelzl@36624
  1357
  moreover
hoelzl@36624
  1358
  { fix y assume "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
hoelzl@36624
  1359
    then obtain n where n: "y \<in> {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}" by auto
hoelzl@36624
  1360
    hence "\<bar>f y\<bar> \<ge> inverse (real (Suc n))" by auto
hoelzl@36624
  1361
    hence "\<bar>f y\<bar> > 0"
hoelzl@36624
  1362
      using real_of_nat_Suc_gt_zero
hoelzl@36624
  1363
        positive_imp_inverse_positive[of "real_of_nat (Suc n)"] by fastsimp
hoelzl@36624
  1364
    hence "y \<in> {x. \<bar>f x\<bar> > 0}" by auto }
hoelzl@36624
  1365
  ultimately have fneq0_UN: "{x. f x \<noteq> 0} = (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
hoelzl@36624
  1366
    by blast
hoelzl@36624
  1367
  { fix n
hoelzl@36624
  1368
    have int_one: "integrable (\<lambda> x. \<bar>f x\<bar> ^ 1)" using integrable_abs assms by auto
hoelzl@36624
  1369
    have "measure M (f -` {inverse (real (Suc n))..} \<inter> space M)
hoelzl@36624
  1370
           \<le> integral (\<lambda> x. \<bar>f x\<bar> ^ 1) / (inverse (real (Suc n)) ^ 1)"
hoelzl@36624
  1371
      using markov_ineq[OF `integrable f` _ int_one] real_of_nat_Suc_gt_zero by auto
hoelzl@36624
  1372
    hence le0: "measure M (f -` {inverse (real (Suc n))..} \<inter> space M) \<le> 0"
hoelzl@36624
  1373
      using assms unfolding nonneg_def by auto
hoelzl@36624
  1374
    have "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
hoelzl@36624
  1375
      apply (subst Int_commute) unfolding Int_def
hoelzl@36624
  1376
      using borel[unfolded borel_measurable_ge_iff] by simp
hoelzl@36624
  1377
    hence m0: "measure M ({x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0 \<and>
hoelzl@36624
  1378
      {x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
hoelzl@36624
  1379
      using positive le0 unfolding atLeast_def by fastsimp }
hoelzl@36624
  1380
  moreover hence "range (\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) \<subseteq> sets M"
hoelzl@36624
  1381
    by auto
hoelzl@36624
  1382
  moreover
hoelzl@36624
  1383
  { fix n
hoelzl@36624
  1384
    have "inverse (real (Suc n)) \<ge> inverse (real (Suc (Suc n)))"
hoelzl@36624
  1385
      using less_imp_inverse_less real_of_nat_Suc_gt_zero[of n] by fastsimp
hoelzl@36624
  1386
    hence "\<And> x. f x \<ge> inverse (real (Suc n)) \<Longrightarrow> f x \<ge> inverse (real (Suc (Suc n)))" by (rule order_trans)
hoelzl@36624
  1387
    hence "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M
hoelzl@36624
  1388
         \<subseteq> {x. f x \<ge> inverse (real (Suc (Suc n)))} \<inter> space M" by auto }
hoelzl@36624
  1389
  ultimately have "(\<lambda> x. 0) ----> measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M)"
hoelzl@36624
  1390
    using monotone_convergence[of "\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M"]
hoelzl@36624
  1391
    unfolding o_def by (simp del: of_nat_Suc)
hoelzl@36624
  1392
  hence "measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0"
hoelzl@36624
  1393
    using LIMSEQ_const[of 0] LIMSEQ_unique by simp
hoelzl@36624
  1394
  hence "measure M ((\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}) \<inter> space M) = 0"
hoelzl@36624
  1395
    using assms unfolding nonneg_def by auto
hoelzl@36624
  1396
  thus "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0" using fneq0_UN by simp
hoelzl@36624
  1397
qed
hoelzl@36624
  1398
hoelzl@35748
  1399
section "Lebesgue integration on countable spaces"
hoelzl@35748
  1400
hoelzl@35748
  1401
lemma nnfis_on_countable:
hoelzl@35748
  1402
  assumes borel: "f \<in> borel_measurable M"
hoelzl@35748
  1403
  and bij: "bij_betw enum S (f ` space M - {0})"
hoelzl@35748
  1404
  and enum_zero: "enum ` (-S) \<subseteq> {0}"
hoelzl@35748
  1405
  and nn_enum: "\<And>n. 0 \<le> enum n"
hoelzl@35748
  1406
  and sums: "(\<lambda>r. enum r * measure M (f -` {enum r} \<inter> space M)) sums x" (is "?sum sums x")
hoelzl@35748
  1407
  shows "x \<in> nnfis f"
hoelzl@35748
  1408
proof -
hoelzl@35748
  1409
  have inj_enum: "inj_on enum S"
hoelzl@35748
  1410
    and range_enum: "enum ` S = f ` space M - {0}"
hoelzl@35748
  1411
    using bij by (auto simp: bij_betw_def)
hoelzl@35748
  1412
hoelzl@35748
  1413
  let "?x n z" = "\<Sum>i = 0..<n. enum i * indicator_fn (f -` {enum i} \<inter> space M) z"
hoelzl@35748
  1414
hoelzl@35748
  1415
  show ?thesis
hoelzl@35748
  1416
  proof (rule nnfis_mon_conv)
hoelzl@35748
  1417
    show "(\<lambda>n. \<Sum>i = 0..<n. ?sum i) ----> x" using sums unfolding sums_def .
hoelzl@35748
  1418
  next
hoelzl@35748
  1419
    fix n
hoelzl@35748
  1420
    show "(\<Sum>i = 0..<n. ?sum i) \<in> nnfis (?x n)"
hoelzl@35748
  1421
    proof (induct n)
hoelzl@35748
  1422
      case 0 thus ?case by (simp add: nnfis_0)
hoelzl@35748
  1423
    next
hoelzl@35748
  1424
      case (Suc n) thus ?case using nn_enum
hoelzl@35748
  1425
        by (auto intro!: nnfis_add nnfis_times psfis_nnfis[OF psfis_indicator] borel_measurable_vimage[OF borel])
hoelzl@35748
  1426
    qed
hoelzl@35748
  1427
  next
hoelzl@35748
  1428
    show "mono_convergent ?x f (space M)"
hoelzl@35748
  1429
    proof (rule mono_convergentI)
hoelzl@35748
  1430
      fix x
hoelzl@35748
  1431
      show "incseq (\<lambda>n. ?x n x)"
hoelzl@35748
  1432
        by (rule incseq_SucI, auto simp: indicator_fn_def nn_enum)
hoelzl@35748
  1433
hoelzl@35748
  1434
      have fin: "\<And>n. finite (enum ` ({0..<n} \<inter> S))" by auto
hoelzl@35748
  1435
hoelzl@35748
  1436
      assume "x \<in> space M"
hoelzl@35748
  1437
      hence "f x \<in> enum ` S \<or> f x = 0" using range_enum by auto
hoelzl@35748
  1438
      thus "(\<lambda>n. ?x n x) ----> f x"
hoelzl@35748
  1439
      proof (rule disjE)
hoelzl@35748
  1440
        assume "f x \<in> enum ` S"
hoelzl@35748
  1441
        then obtain i where "i \<in> S" and "f x = enum i" by auto
hoelzl@35748
  1442
hoelzl@35748
  1443
        { fix n
hoelzl@35748
  1444
          have sum_ranges:
hoelzl@35748
  1445
            "i < n \<Longrightarrow> enum`({0..<n} \<inter> S) \<inter> {z. enum i = z \<and> x\<in>space M} = {enum i}"
hoelzl@35748
  1446
            "\<not> i < n \<Longrightarrow> enum`({0..<n} \<inter> S) \<inter> {z. enum i = z \<and> x\<in>space M} = {}"
hoelzl@35748
  1447
            using `x \<in> space M` `i \<in> S` inj_enum[THEN inj_on_iff] by auto
hoelzl@35748
  1448
          have "?x n x =
hoelzl@35748
  1449
            (\<Sum>i \<in> {0..<n} \<inter> S. enum i * indicator_fn (f -` {enum i} \<inter> space M) x)"
hoelzl@35748
  1450
            using enum_zero by (auto intro!: setsum_mono_zero_cong_right)
hoelzl@35748
  1451
          also have "... =
hoelzl@35748
  1452
            (\<Sum>z \<in> enum`({0..<n} \<inter> S). z * indicator_fn (f -` {z} \<inter> space M) x)"
hoelzl@35748
  1453
            using inj_enum[THEN subset_inj_on] by (auto simp: setsum_reindex)
hoelzl@35748
  1454
          also have "... = (if i < n then f x else 0)"
hoelzl@35748
  1455
            unfolding indicator_fn_def if_distrib[where x=1 and y=0]
hoelzl@35748
  1456
              setsum_cases[OF fin]
hoelzl@35748
  1457
            using sum_ranges `f x = enum i`
hoelzl@35748
  1458
            by auto
hoelzl@35748
  1459
          finally have "?x n x = (if i < n then f x else 0)" . }
hoelzl@35748
  1460
        note sum_equals_if = this
hoelzl@35748
  1461
hoelzl@35748
  1462
        show ?thesis unfolding sum_equals_if
hoelzl@35748
  1463
          by (rule LIMSEQ_offset[where k="i + 1"]) (auto intro!: LIMSEQ_const)
hoelzl@35748
  1464
      next
hoelzl@35748
  1465
        assume "f x = 0"
hoelzl@35748
  1466
        { fix n have "?x n x = 0"
hoelzl@35748
  1467
            unfolding indicator_fn_def if_distrib[where x=1 and y=0]
hoelzl@35748
  1468
              setsum_cases[OF finite_atLeastLessThan]
hoelzl@35748
  1469
            using `f x = 0` `x \<in> space M`
hoelzl@35748
  1470
            by (auto split: split_if) }
hoelzl@35748
  1471
        thus ?thesis using `f x = 0` by (auto intro!: LIMSEQ_const)
hoelzl@35748
  1472
      qed
hoelzl@35748
  1473
    qed
hoelzl@35748
  1474
  qed
hoelzl@35748
  1475
qed
hoelzl@35748
  1476
hoelzl@35748
  1477
lemma integral_on_countable:
hoelzl@35833
  1478
  fixes enum :: "nat \<Rightarrow> real"
hoelzl@35748
  1479
  assumes borel: "f \<in> borel_measurable M"
hoelzl@35748
  1480
  and bij: "bij_betw enum S (f ` space M)"
hoelzl@35748
  1481
  and enum_zero: "enum ` (-S) \<subseteq> {0}"
hoelzl@35748
  1482
  and abs_summable: "summable (\<lambda>r. \<bar>enum r * measure M (f -` {enum r} \<inter> space M)\<bar>)"
hoelzl@35748
  1483
  shows "integrable f"
hoelzl@35748
  1484
  and "integral f = (\<Sum>r. enum r * measure M (f -` {enum r} \<inter> space M))" (is "_ = suminf (?sum f enum)")
hoelzl@35748
  1485
proof -
hoelzl@35748
  1486
  { fix f enum
hoelzl@35748
  1487
    assume borel: "f \<in> borel_measurable M"
hoelzl@35748
  1488
      and bij: "bij_betw enum S (f ` space M)"
hoelzl@35748
  1489
      and enum_zero: "enum ` (-S) \<subseteq> {0}"
hoelzl@35748
  1490
      and abs_summable: "summable (\<lambda>r. \<bar>enum r * measure M (f -` {enum r} \<inter> space M)\<bar>)"
hoelzl@35748
  1491
    have inj_enum: "inj_on enum S" and range_enum: "f ` space M = enum ` S"
hoelzl@35748
  1492
      using bij unfolding bij_betw_def by auto
hoelzl@35748
  1493
hoelzl@35748
  1494
    have [simp, intro]: "\<And>X. 0 \<le> measure M (f -` {X} \<inter> space M)"
hoelzl@35748
  1495
      by (rule positive, rule borel_measurable_vimage[OF borel])
hoelzl@35748
  1496
hoelzl@35748
  1497
    have "(\<Sum>r. ?sum (pos_part f) (pos_part enum) r) \<in> nnfis (pos_part f) \<and>
hoelzl@35748
  1498
          summable (\<lambda>r. ?sum (pos_part f) (pos_part enum) r)"
hoelzl@35748
  1499
    proof (rule conjI, rule nnfis_on_countable)
hoelzl@35748
  1500
      have pos_f_image: "pos_part f ` space M - {0} = f ` space M \<inter> {0<..}"
hoelzl@35748
  1501
        unfolding pos_part_def max_def by auto
hoelzl@35748
  1502
hoelzl@35748
  1503
      show "bij_betw (pos_part enum) {x \<in> S. 0 < enum x} (pos_part f ` space M - {0})"
hoelzl@35748
  1504
        unfolding bij_betw_def pos_f_image
hoelzl@35748
  1505
        unfolding pos_part_def max_def range_enum
hoelzl@35748
  1506
        by (auto intro!: inj_onI simp: inj_enum[THEN inj_on_eq_iff])
hoelzl@35748
  1507
hoelzl@35748
  1508
      show "\<And>n. 0 \<le> pos_part enum n" unfolding pos_part_def max_def by auto
hoelzl@35748
  1509
hoelzl@35748
  1510
      show "pos_part f \<in> borel_measurable M"
hoelzl@35748
  1511
        by (rule pos_part_borel_measurable[OF borel])
hoelzl@35748
  1512
hoelzl@35748
  1513
      show "pos_part enum ` (- {x \<in> S. 0 < enum x}) \<subseteq> {0}"
hoelzl@35748
  1514
        unfolding pos_part_def max_def using enum_zero by auto
hoelzl@35748
  1515
hoelzl@35748
  1516
      show "summable (\<lambda>r. ?sum (pos_part f) (pos_part enum) r)"
hoelzl@35748
  1517
      proof (rule summable_comparison_test[OF _ abs_summable], safe intro!: exI[of _ 0])
hoelzl@35748
  1518
        fix n :: nat
hoelzl@35748
  1519
        have "pos_part enum n \<noteq> 0 \<Longrightarrow> (pos_part f -` {enum n} \<inter> space M) =
hoelzl@35748
  1520
          (if 0 < enum n then (f -` {enum n} \<inter> space M) else {})"
hoelzl@35748
  1521
          unfolding pos_part_def max_def by (auto split: split_if_asm)
hoelzl@35748
  1522
        thus "norm (?sum (pos_part f) (pos_part enum) n) \<le> \<bar>?sum f enum n \<bar>"
hoelzl@35748
  1523
          by (cases "pos_part enum n = 0",
hoelzl@35748
  1524
            auto simp: pos_part_def max_def abs_mult not_le split: split_if_asm intro!: mult_nonpos_nonneg)
hoelzl@35748
  1525
      qed
hoelzl@35748
  1526
      thus "(\<lambda>r. ?sum (pos_part f) (pos_part enum) r) sums (\<Sum>r. ?sum (pos_part f) (pos_part enum) r)"
hoelzl@35748
  1527
        by (rule summable_sums)
hoelzl@35748
  1528
    qed }
hoelzl@35748
  1529
  note pos = this
hoelzl@35748
  1530
hoelzl@35748
  1531
  note pos_part = pos[OF assms(1-4)]
hoelzl@35748
  1532
  moreover
hoelzl@35748
  1533
  have neg_part_to_pos_part:
hoelzl@35748
  1534
    "\<And>f :: _ \<Rightarrow> real. neg_part f = pos_part (uminus \<circ> f)"
hoelzl@35748
  1535
    by (auto simp: pos_part_def neg_part_def min_def max_def expand_fun_eq)
hoelzl@35748
  1536
hoelzl@35748
  1537
  have neg_part: "(\<Sum>r. ?sum (neg_part f) (neg_part enum) r) \<in> nnfis (neg_part f) \<and>
hoelzl@35748
  1538
    summable (\<lambda>r. ?sum (neg_part f) (neg_part enum) r)"
hoelzl@35748
  1539
    unfolding neg_part_to_pos_part
hoelzl@35748
  1540
  proof (rule pos)
hoelzl@35748
  1541
    show "uminus \<circ> f \<in> borel_measurable M" unfolding comp_def
hoelzl@35748
  1542
      by (rule borel_measurable_uminus_borel_measurable[OF borel])
hoelzl@35748
  1543
hoelzl@35748
  1544
    show "bij_betw (uminus \<circ> enum) S ((uminus \<circ> f) ` space M)"
hoelzl@35748
  1545
      using bij unfolding bij_betw_def
hoelzl@35748
  1546
      by (auto intro!: comp_inj_on simp: image_compose)
hoelzl@35748
  1547
hoelzl@35748
  1548
    show "(uminus \<circ> enum) ` (- S) \<subseteq> {0}"
hoelzl@35748
  1549
      using enum_zero by auto
hoelzl@35748
  1550
hoelzl@35748
  1551
    have minus_image: "\<And>r. (uminus \<circ> f) -` {(uminus \<circ> enum) r} \<inter> space M = f -` {enum r} \<inter> space M"
hoelzl@35748
  1552
      by auto
hoelzl@35748
  1553
    show "summable (\<lambda>r. \<bar>(uminus \<circ> enum) r * measure_space.measure M ((uminus \<circ> f) -` {(uminus \<circ> enum) r} \<inter> space M)\<bar>)"
hoelzl@35748
  1554
      unfolding minus_image using abs_summable by simp
hoelzl@35748
  1555
  qed
hoelzl@35748
  1556
  ultimately show "integrable f" unfolding integrable_def by auto
hoelzl@35748
  1557
hoelzl@35748
  1558
  { fix r
hoelzl@35748
  1559
    have "?sum (pos_part f) (pos_part enum) r - ?sum (neg_part f) (neg_part enum) r = ?sum f enum r"
hoelzl@35748
  1560
    proof (cases rule: linorder_cases)
hoelzl@35748
  1561
      assume "0 < enum r"
hoelzl@35748
  1562
      hence "pos_part f -` {enum r} \<inter> space M = f -` {enum r} \<inter> space M"
hoelzl@35748
  1563
        unfolding pos_part_def max_def by (auto split: split_if_asm)
hoelzl@35748
  1564
      with `0 < enum r` show ?thesis unfolding pos_part_def neg_part_def min_def max_def expand_fun_eq
hoelzl@35748
  1565
        by auto
hoelzl@35748
  1566
    next
hoelzl@35748
  1567
      assume "enum r < 0"
hoelzl@35748
  1568
      hence "neg_part f -` {- enum r} \<inter> space M = f -` {enum r} \<inter> space M"
hoelzl@35748
  1569
        unfolding neg_part_def min_def by (auto split: split_if_asm)
hoelzl@35748
  1570
      with `enum r < 0` show ?thesis unfolding pos_part_def neg_part_def min_def max_def expand_fun_eq
hoelzl@35748
  1571
        by auto
hoelzl@35748
  1572
    qed (simp add: neg_part_def pos_part_def) }
hoelzl@35748
  1573
  note sum_diff_eq_sum = this
hoelzl@35748
  1574
hoelzl@35748
  1575
  have "(\<Sum>r. ?sum (pos_part f) (pos_part enum) r) - (\<Sum>r. ?sum (neg_part f) (neg_part enum) r)
hoelzl@35748
  1576
    = (\<Sum>r. ?sum (pos_part f) (pos_part enum) r - ?sum (neg_part f) (neg_part enum) r)"
hoelzl@35748
  1577
    using neg_part pos_part by (auto intro: suminf_diff)
hoelzl@35748
  1578
  also have "... = (\<Sum>r. ?sum f enum r)" unfolding sum_diff_eq_sum ..
hoelzl@35748
  1579
  finally show "integral f = suminf (?sum f enum)"
hoelzl@35748
  1580
    unfolding integral_def using pos_part neg_part
hoelzl@35748
  1581
    by (auto dest: the_nnfis)
hoelzl@35748
  1582
qed
hoelzl@35748
  1583
hoelzl@35692
  1584
section "Lebesgue integration on finite space"
hoelzl@35692
  1585
hoelzl@35582
  1586
lemma integral_finite_on_sets:
hoelzl@35582
  1587
  assumes "f \<in> borel_measurable M" and "finite (space M)" and "a \<in> sets M"
hoelzl@35582
  1588
  shows "integral (\<lambda>x. f x * indicator_fn a x) =
hoelzl@35582
  1589
    (\<Sum> r \<in> f`a. r * measure M (f -` {r} \<inter> a))" (is "integral ?f = _")
hoelzl@35582
  1590
proof -
hoelzl@35582
  1591
  { fix x assume "x \<in> a"
hoelzl@35582
  1592
    with assms have "f -` {f x} \<inter> space M \<in> sets M"
hoelzl@35582
  1593
      by (subst Int_commute)
hoelzl@35582
  1594
         (auto simp: vimage_def Int_def
hoelzl@35582
  1595
               intro!: borel_measurable_const
hoelzl@35582
  1596
                      borel_measurable_eq_borel_measurable)
hoelzl@35582
  1597
    from Int[OF this assms(3)]
hoelzl@35582
  1598
         sets_into_space[OF assms(3), THEN Int_absorb1]
hoelzl@35582
  1599
    have "f -` {f x} \<inter> a \<in> sets M" by (simp add: Int_assoc) }
hoelzl@35582
  1600
  note vimage_f = this
hoelzl@35582
  1601
hoelzl@35582
  1602
  have "finite a"
hoelzl@35582
  1603
    using assms(2,3) sets_into_space
hoelzl@35582
  1604
    by (auto intro: finite_subset)
hoelzl@35582
  1605
hoelzl@35582
  1606
  have "integral (\<lambda>x. f x * indicator_fn a x) =
hoelzl@35582
  1607
    integral (\<lambda>x. \<Sum>i\<in>f ` a. i * indicator_fn (f -` {i} \<inter> a) x)"
hoelzl@35582
  1608
    (is "_ = integral (\<lambda>x. setsum (?f x) _)")
hoelzl@35582
  1609
    unfolding indicator_fn_def if_distrib
hoelzl@35582
  1610
    using `finite a` by (auto simp: setsum_cases intro!: integral_cong)
hoelzl@35582
  1611
  also have "\<dots> = (\<Sum>i\<in>f`a. integral (\<lambda>x. ?f x i))"
hoelzl@35582
  1612
  proof (rule integral_setsum, safe)
hoelzl@35582
  1613
    fix n x assume "x \<in> a"
hoelzl@35582
  1614
    thus "integrable (\<lambda>y. ?f y (f x))"
hoelzl@35582
  1615
      using integral_indicator_fn(2)[OF vimage_f]
hoelzl@35582
  1616
      by (auto intro!: integral_times_const)
hoelzl@35582
  1617
  qed (simp add: `finite a`)
hoelzl@35582
  1618
  also have "\<dots> = (\<Sum>i\<in>f`a. i * measure M (f -` {i} \<inter> a))"
hoelzl@35582
  1619
    using integral_cmul_indicator[OF vimage_f]
hoelzl@35582
  1620
    by (auto intro!: setsum_cong)
hoelzl@35582
  1621
  finally show ?thesis .
hoelzl@35582
  1622
qed
hoelzl@35582
  1623
hoelzl@35582
  1624
lemma integral_finite:
hoelzl@35582
  1625
  assumes "f \<in> borel_measurable M" and "finite (space M)"
hoelzl@35582
  1626
  shows "integral f = (\<Sum> r \<in> f ` space M. r * measure M (f -` {r} \<inter> space M))"
hoelzl@35582
  1627
  using integral_finite_on_sets[OF assms top]
hoelzl@35582
  1628
    integral_cong[of "\<lambda>x. f x * indicator_fn (space M) x" f]
hoelzl@35582
  1629
  by (auto simp add: indicator_fn_def)
hoelzl@35582
  1630
hoelzl@35692
  1631
section "Radon–Nikodym derivative"
hoelzl@35582
  1632
hoelzl@35692
  1633
definition
hoelzl@35692
  1634
  "RN_deriv v \<equiv> SOME f. measure_space (M\<lparr>measure := v\<rparr>) \<and>
hoelzl@35692
  1635
    f \<in> borel_measurable M \<and>
hoelzl@35692
  1636
    (\<forall>a \<in> sets M. (integral (\<lambda>x. f x * indicator_fn a x) = v a))"
hoelzl@35582
  1637
hoelzl@35977
  1638
end
hoelzl@35977
  1639
hoelzl@35977
  1640
lemma sigma_algebra_cong:
hoelzl@35977
  1641
  fixes M :: "('a, 'b) algebra_scheme" and M' :: "('a, 'c) algebra_scheme"
hoelzl@35977
  1642
  assumes *: "sigma_algebra M"
hoelzl@35977
  1643
  and cong: "space M = space M'" "sets M = sets M'"
hoelzl@35977
  1644
  shows "sigma_algebra M'"
hoelzl@35977
  1645
using * unfolding sigma_algebra_def algebra_def sigma_algebra_axioms_def unfolding cong .
hoelzl@35977
  1646
hoelzl@35977
  1647
lemma finite_Pow_additivity_sufficient:
hoelzl@35977
  1648
  assumes "finite (space M)" and "sets M = Pow (space M)"
hoelzl@35977
  1649
  and "positive M (measure M)" and "additive M (measure M)"
hoelzl@35977
  1650
  shows "finite_measure_space M"
hoelzl@35977
  1651
proof -
hoelzl@35977
  1652
  have "sigma_algebra M"
hoelzl@35977
  1653
    using assms by (auto intro!: sigma_algebra_cong[OF sigma_algebra_Pow])
hoelzl@35977
  1654
hoelzl@35977
  1655
  have "measure_space M"
hoelzl@35977
  1656
    by (rule Measure.finite_additivity_sufficient) (fact+)
hoelzl@35977
  1657
  thus ?thesis
hoelzl@35977
  1658
    unfolding finite_measure_space_def finite_measure_space_axioms_def
hoelzl@35977
  1659
    using assms by simp
hoelzl@35977
  1660
qed
hoelzl@35977
  1661
hoelzl@35977
  1662
lemma finite_measure_spaceI:
hoelzl@35977
  1663
  assumes "measure_space M" and "finite (space M)" and "sets M = Pow (space M)"
hoelzl@35977
  1664
  shows "finite_measure_space M"
hoelzl@35977
  1665
  unfolding finite_measure_space_def finite_measure_space_axioms_def
hoelzl@35977
  1666
  using assms by simp
hoelzl@35977
  1667
hoelzl@35977
  1668
lemma (in finite_measure_space) integral_finite_singleton:
hoelzl@35977
  1669
  "integral f = (\<Sum>x \<in> space M. f x * measure M {x})"
hoelzl@35977
  1670
proof -
hoelzl@35977
  1671
  have "f \<in> borel_measurable M"
hoelzl@35977
  1672
    unfolding borel_measurable_le_iff
hoelzl@35977
  1673
    using sets_eq_Pow by auto
hoelzl@35977
  1674
  { fix r let ?x = "f -` {r} \<inter> space M"
hoelzl@35977
  1675
    have "?x \<subseteq> space M" by auto
hoelzl@35977
  1676
    with finite_space sets_eq_Pow have "measure M ?x = (\<Sum>i \<in> ?x. measure M {i})"
hoelzl@35977
  1677
      by (auto intro!: measure_real_sum_image) }
hoelzl@35977
  1678
  note measure_eq_setsum = this
hoelzl@35977
  1679
  show ?thesis
hoelzl@35977
  1680
    unfolding integral_finite[OF `f \<in> borel_measurable M` finite_space]
hoelzl@35977
  1681
      measure_eq_setsum setsum_right_distrib
hoelzl@35977
  1682
    apply (subst setsum_Sigma)
hoelzl@35977
  1683
    apply (simp add: finite_space)
hoelzl@35977
  1684
    apply (simp add: finite_space)
hoelzl@35977
  1685
  proof (rule setsum_reindex_cong[symmetric])
hoelzl@35977
  1686
    fix a assume "a \<in> Sigma (f ` space M) (\<lambda>x. f -` {x} \<inter> space M)"
hoelzl@35977
  1687
    thus "(\<lambda>(x, y). x * measure M {y}) a = f (snd a) * measure_space.measure M {snd a}"
hoelzl@35977
  1688
      by auto
hoelzl@35977
  1689
  qed (auto intro!: image_eqI inj_onI)
hoelzl@35977
  1690
qed
hoelzl@35977
  1691
hoelzl@35977
  1692
lemma (in finite_measure_space) RN_deriv_finite_singleton:
hoelzl@35582
  1693
  fixes v :: "'a set \<Rightarrow> real"
hoelzl@35977
  1694
  assumes ms_v: "measure_space (M\<lparr>measure := v\<rparr>)"
hoelzl@36624
  1695
  and eq_0: "\<And>x. \<lbrakk> x \<in> space M ; measure M {x} = 0 \<rbrakk> \<Longrightarrow> v {x} = 0"
hoelzl@35582
  1696
  and "x \<in> space M" and "measure M {x} \<noteq> 0"
hoelzl@35582
  1697
  shows "RN_deriv v x = v {x} / (measure M {x})" (is "_ = ?v x")
hoelzl@35582
  1698
  unfolding RN_deriv_def
hoelzl@35582
  1699
proof (rule someI2_ex[where Q = "\<lambda>f. f x = ?v x"], rule exI[where x = ?v], safe)
hoelzl@35582
  1700
  show "(\<lambda>a. v {a} / measure_space.measure M {a}) \<in> borel_measurable M"
hoelzl@35977
  1701
    unfolding borel_measurable_le_iff using sets_eq_Pow by auto
hoelzl@35582
  1702
next
hoelzl@35582
  1703
  fix a assume "a \<in> sets M"
hoelzl@35582
  1704
  hence "a \<subseteq> space M" and "finite a"
hoelzl@35977
  1705
    using sets_into_space finite_space by (auto intro: finite_subset)
hoelzl@36624
  1706
  have *: "\<And>x a. x \<in> space M \<Longrightarrow> (if measure M {x} = 0 then 0 else v {x} * indicator_fn a x) =
hoelzl@35582
  1707
    v {x} * indicator_fn a x" using eq_0 by auto
hoelzl@35582
  1708
hoelzl@35582
  1709
  from measure_space.measure_real_sum_image[OF ms_v, of a]
hoelzl@35977
  1710
    sets_eq_Pow `a \<in> sets M` sets_into_space `finite a`
hoelzl@35582
  1711
  have "v a = (\<Sum>x\<in>a. v {x})" by auto
hoelzl@35582
  1712
  thus "integral (\<lambda>x. v {x} / measure_space.measure M {x} * indicator_fn a x) = v a"
hoelzl@35977
  1713
    apply (simp add: eq_0 integral_finite_singleton)
hoelzl@35582
  1714
    apply (unfold divide_1)
hoelzl@35977
  1715
    by (simp add: * indicator_fn_def if_distrib setsum_cases finite_space `a \<subseteq> space M` Int_absorb1)
hoelzl@35582
  1716
next
hoelzl@35582
  1717
  fix w assume "w \<in> borel_measurable M"
hoelzl@35582
  1718
  assume int_eq_v: "\<forall>a\<in>sets M. integral (\<lambda>x. w x * indicator_fn a x) = v a"
hoelzl@35977
  1719
  have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
hoelzl@35582
  1720
hoelzl@35582
  1721
  have "w x * measure M {x} =
hoelzl@35582
  1722
    (\<Sum>y\<in>space M. w y * indicator_fn {x} y * measure M {y})"
hoelzl@35582
  1723
    apply (subst (3) mult_commute)
hoelzl@35977
  1724
    unfolding indicator_fn_def if_distrib setsum_cases[OF finite_space]
hoelzl@35582
  1725
    using `x \<in> space M` by simp
hoelzl@35582
  1726
  also have "... = v {x}"
hoelzl@35582
  1727
    using int_eq_v[rule_format, OF `{x} \<in> sets M`]
hoelzl@35977
  1728
    by (simp add: integral_finite_singleton)
hoelzl@35582
  1729
  finally show "w x = v {x} / measure M {x}"
hoelzl@35582
  1730
    using `measure M {x} \<noteq> 0` by (simp add: eq_divide_eq)
hoelzl@35582
  1731
qed fact
hoelzl@35582
  1732
hoelzl@35748
  1733
end