feat(measure_theory/decomposition/jordan): the Jordan decomposition theorem for signed measures (#8645)

This PR proves the Jordan decomposition theorem for signed measures, that is, given a signed measure `s`, there exists a unique pair of mutually singular measures `μ` and `ν`, such that `s = μ - ν`.

Author: kex-y

src/measure_theory/decomposition/jordan.lean

@@ -0,0 +1,381 @@
+/-
+Copyright (c) 2021 Kexing Ying. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kexing Ying
+-/
+import measure_theory.decomposition.signed_hahn
+
+/-!
+# Jordan decomposition
+
+This file proves the existence and uniqueness of the Jordan decomposition for signed measures.
+The Jordan decomposition theorem states that, given a signed measure `s`, there exists a
+unique pair of mutually singular measures `μ` and `ν`, such that `s = μ - ν`.
+
+The Jordan decomposition theorem for measures is a corollary of the Hahn decomposition theorem and
+is useful for the Lebesgue decomposition theorem.
+
+## Main definitions
+
+* `measure_theory.jordan_decomposition`: a Jordan decomposition of a measurable space is a
+  pair of mutually singular finite measures. We say `j` is a Jordan decomposition of a signed
+  meausre `s` if `s = j.pos_part - j.neg_part`.
+* `measure_theory.signed_measure.to_jordan_decomposition`: the Jordan decomposition of a
+  signed measure.
+* `measure_theory.signed_measure.to_jordan_decomposition_equiv`: is the `equiv` between
+  `measure_theory.signed_measure` and `measure_theory.jordan_decomposition` formed by
+  `measure_theory.signed_measure.to_jordan_decomposition`.
+
+## Main results
+
+* `measure_theory.signed_measure.to_signed_measure_to_jordan_decomposition` : the Jordan
+  decomposition theorem.
+* `measure_thoery.signed_measure.to_signed_measure_injective` : the Jordan decomposition of a
+  signed measure is unique.
+
+## Tags
+
+Jordan decomposition theorem
+-/
+
+noncomputable theory
+open_locale classical measure_theory ennreal
+
+variables {α β : Type*} [measurable_space α]
+
+namespace measure_theory
+
+/-- A Jordan decomposition of a measurable space is a pair of mutually singular,
+finite measures. -/
+@[ext] structure jordan_decomposition (α : Type*) [measurable_space α] :=
+(pos_part neg_part : measure α)
+[pos_part_finite : finite_measure pos_part]
+[neg_part_finite : finite_measure neg_part]
+(mutually_singular : pos_part ⊥ₘ neg_part)
+
+attribute [instance] jordan_decomposition.pos_part_finite
+attribute [instance] jordan_decomposition.neg_part_finite
+
+namespace jordan_decomposition
+
+open measure vector_measure
+
+variable (j : jordan_decomposition α)
+
+instance : has_zero (jordan_decomposition α) :=
+{ zero := ⟨0, 0, mutually_singular.zero⟩ }
+
+instance : inhabited (jordan_decomposition α) :=
+{ default := 0 }
+
+instance : has_neg (jordan_decomposition α) :=
+{ neg := λ j, ⟨j.neg_part, j.pos_part, j.mutually_singular.symm⟩ }
+
+@[simp] lemma zero_pos_part : (0 : jordan_decomposition α).pos_part = 0 := rfl
+@[simp] lemma zero_neg_part : (0 : jordan_decomposition α).neg_part = 0 := rfl
+
+@[simp] lemma neg_pos_part : (-j).pos_part = j.neg_part := rfl
+@[simp] lemma neg_neg_part : (-j).neg_part = j.pos_part := rfl
+
+/-- The signed measure associated with a Jordan decomposition. -/
+def to_signed_measure : signed_measure α :=
+j.pos_part.to_signed_measure - j.neg_part.to_signed_measure
+
+lemma to_signed_measure_zero : (0 : jordan_decomposition α).to_signed_measure = 0 :=
+begin
+  ext1 i hi,
+  erw [to_signed_measure, to_signed_measure_sub_apply hi, sub_self, zero_apply],
+end
+
+lemma to_signed_measure_neg : (-j).to_signed_measure = -j.to_signed_measure :=
+begin
+  ext1 i hi,
+  rw [neg_apply, to_signed_measure, to_signed_measure,
+      to_signed_measure_sub_apply hi, to_signed_measure_sub_apply hi, neg_sub],
+  refl,
+end
+
+/-- A Jordan decomposition provides a Hahn decomposition. -/
+lemma exists_compl_positive_negative :
+  ∃ S : set α, measurable_set S ∧
+    j.to_signed_measure ≤[S] 0 ∧ 0 ≤[Sᶜ] j.to_signed_measure ∧
+    j.pos_part S = 0 ∧ j.neg_part Sᶜ = 0 :=
+begin
+  obtain ⟨S, hS₁, hS₂, hS₃⟩ := j.mutually_singular,
+  refine ⟨S, hS₁, _, _, hS₂, hS₃⟩,
+  { refine restrict_le_restrict_of_subset_le _ _ (λ A hA hA₁, _),
+    rw [to_signed_measure, to_signed_measure_sub_apply hA,
+        show j.pos_part A = 0, by exact nonpos_iff_eq_zero.1 (hS₂ ▸ measure_mono hA₁),
+        ennreal.zero_to_real, zero_sub, neg_le, zero_apply, neg_zero],
+    exact ennreal.to_real_nonneg },
+  { refine restrict_le_restrict_of_subset_le _ _ (λ A hA hA₁, _),
+    rw [to_signed_measure, to_signed_measure_sub_apply hA,
+        show j.neg_part A = 0, by exact nonpos_iff_eq_zero.1 (hS₃ ▸ measure_mono hA₁),
+        ennreal.zero_to_real, sub_zero],
+    exact ennreal.to_real_nonneg },
+end
+
+end jordan_decomposition
+
+namespace signed_measure
+
+open measure vector_measure jordan_decomposition classical
+
+variables {s : signed_measure α} {μ ν : measure α} [finite_measure μ] [finite_measure ν]
+
+/-- Given a signed measure `s`, `s.to_jordan_decomposition` is the Jordan decomposition `j`,
+such that `s = j.to_signed_measure`. This property is known as the Jordan decomposition
+theorem, and is shown by
+`measure_theory.signed_measure.to_signed_measure_to_jordan_decomposition`. -/
+def to_jordan_decomposition (s : signed_measure α) : jordan_decomposition α :=
+let i := some s.exists_compl_positive_negative in
+let hi := some_spec s.exists_compl_positive_negative in
+{ pos_part := s.to_measure_of_zero_le i hi.1 hi.2.1,
+  neg_part := s.to_measure_of_le_zero iᶜ hi.1.compl hi.2.2,
+  pos_part_finite := infer_instance,
+  neg_part_finite := infer_instance,
+  mutually_singular :=
+  begin
+    refine ⟨iᶜ, hi.1.compl, _, _⟩,
+    { rw [to_measure_of_zero_le_apply _ _ hi.1 hi.1.compl], simpa },
+    { rw [to_measure_of_le_zero_apply _ _ hi.1.compl hi.1.compl.compl], simpa }
+  end }
+
+lemma to_jordan_decomposition_spec (s : signed_measure α) :
+  ∃ (i : set α) (hi₁ : measurable_set i) (hi₂ : 0 ≤[i] s) (hi₃ : s ≤[iᶜ] 0),
+    s.to_jordan_decomposition.pos_part = s.to_measure_of_zero_le i hi₁ hi₂ ∧
+    s.to_jordan_decomposition.neg_part = s.to_measure_of_le_zero iᶜ hi₁.compl hi₃ :=
+begin
+  set i := some s.exists_compl_positive_negative,
+  obtain ⟨hi₁, hi₂, hi₃⟩ := some_spec s.exists_compl_positive_negative,
+  exact ⟨i, hi₁, hi₂, hi₃, rfl, rfl⟩,
+end
+
+/-- **The Jordan decomposition theorem**: Given a signed measure `s`, there exists a pair of
+mutually singular measures `μ` and `ν` such that `s = μ - ν`. In this case, the measures `μ`
+and `ν` are given by `s.to_jordan_decomposition.pos_part` and
+`s.to_jordan_decomposition.neg_part` respectively.
+
+Note that we use `measure_theory.jordan_decomposition.to_signed_measure` to represent the
+signed measure corresponding to
+`s.to_jordan_decomposition.pos_part - s.to_jordan_decomposition.neg_part`. -/
+@[simp] lemma to_signed_measure_to_jordan_decomposition (s : signed_measure α) :
+  s.to_jordan_decomposition.to_signed_measure = s :=
+begin
+  obtain ⟨i, hi₁, hi₂, hi₃, hμ, hν⟩ := s.to_jordan_decomposition_spec,
+  simp only [jordan_decomposition.to_signed_measure, hμ, hν],
+  ext k hk,
+  rw [to_signed_measure_sub_apply hk, to_measure_of_zero_le_apply _ hi₂ hi₁ hk,
+      to_measure_of_le_zero_apply _ hi₃ hi₁.compl hk],
+  simp only [ennreal.coe_to_real, subtype.coe_mk, ennreal.some_eq_coe, sub_neg_eq_add],
+  rw [← of_union _ (measurable_set.inter hi₁ hk) (measurable_set.inter hi₁.compl hk),
+      set.inter_comm i, set.inter_comm iᶜ, set.inter_union_compl _ _],
+  { apply_instance },
+  { rintro x ⟨⟨hx₁, _⟩, hx₂, _⟩,
+    exact false.elim (hx₂ hx₁) }
+end
+
+section
+
+variables {u v w : set α}
+
+/-- A subset `v` of a null-set `w` has zero measure if `w` is a subset of a positive set `u`. -/
+lemma subset_positive_null_set
+  (hu : measurable_set u) (hv : measurable_set v) (hw : measurable_set w)
+  (hsu : 0 ≤[u] s) (hw₁ : s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w) : s v = 0 :=
+begin
+  have : s v + s (w \ v) = 0,
+  { rw [← hw₁, ← of_union set.disjoint_diff hv (hw.diff hv),
+        set.union_diff_self, set.union_eq_self_of_subset_left hwt],
+    apply_instance },
+  have h₁ := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu (hwt.trans hw₂)),
+  have h₂ := nonneg_of_zero_le_restrict _
+    (restrict_le_restrict_subset _ _ hu hsu ((w.diff_subset v).trans hw₂)),
+  linarith,
+end
+
+/-- A subset `v` of a null-set `w` has zero measure if `w` is a subset of a negative set `u`. -/
+lemma subset_negative_null_set
+  (hu : measurable_set u) (hv : measurable_set v) (hw : measurable_set w)
+  (hsu : s ≤[u] 0) (hw₁ : s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w) : s v = 0 :=
+begin
+  rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu,
+  have := subset_positive_null_set hu hv hw hsu,
+  simp only [pi.neg_apply, neg_eq_zero, coe_neg] at this,
+  exact this hw₁ hw₂ hwt,
+end
+
+/-- If the symmetric difference of two positive sets is a null-set, then so are the differences
+between the two sets. -/
+lemma of_diff_eq_zero_of_symm_diff_eq_zero_positive
+  (hu : measurable_set u) (hv : measurable_set v)
+  (hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u Δ v) = 0) :
+  s (u \ v) = 0 ∧ s (v \ u) = 0 :=
+begin
+  rw restrict_le_restrict_iff at hsu hsv,
+  have a := hsu (hu.diff hv) (u.diff_subset v),
+  have b := hsv (hv.diff hu) (v.diff_subset u),
+  erw [of_union (set.disjoint_of_subset_left (u.diff_subset v) set.disjoint_diff)
+        (hu.diff hv) (hv.diff hu)] at hs,
+  rw zero_apply at a b,
+  split,
+  all_goals { linarith <|> apply_instance <|> assumption },
+end
+
+/-- If the symmetric difference of two negative sets is a null-set, then so are the differences
+between the two sets. -/
+lemma of_diff_eq_zero_of_symm_diff_eq_zero_negative
+  (hu : measurable_set u) (hv : measurable_set v)
+  (hsu : s ≤[u] 0) (hsv : s ≤[v] 0) (hs : s (u Δ v) = 0) :
+  s (u \ v) = 0 ∧ s (v \ u) = 0 :=
+begin
+  rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu,
+  rw [← s.neg_le_neg_iff _ hv, neg_zero] at hsv,
+  have := of_diff_eq_zero_of_symm_diff_eq_zero_positive hu hv hsu hsv,
+  simp only [pi.neg_apply, neg_eq_zero, coe_neg] at this,
+  exact this hs,
+end
+
+lemma of_inter_eq_of_symm_diff_eq_zero_positive
+  (hu : measurable_set u) (hv : measurable_set v) (hw : measurable_set w)
+  (hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u Δ v) = 0) :
+  s (w ∩ u) = s (w ∩ v) :=
+begin
+  have hwuv : s ((w ∩ u) Δ (w ∩ v)) = 0,
+  { refine subset_positive_null_set (hu.union hv) ((hw.inter hu).symm_diff (hw.inter hv))
+      (hu.symm_diff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs _ _,
+    { exact symm_diff_le_sup u v },
+    { rintro x (⟨⟨hxw, hxu⟩, hx⟩ | ⟨⟨hxw, hxv⟩, hx⟩);
+      rw [set.mem_inter_eq, not_and] at hx,
+      { exact or.inl ⟨hxu, hx hxw⟩ },
+      { exact or.inr ⟨hxv, hx hxw⟩ } } },
+  obtain ⟨huv, hvu⟩ := of_diff_eq_zero_of_symm_diff_eq_zero_positive
+    (hw.inter hu) (hw.inter hv)
+    (restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right u))
+    (restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right v)) hwuv,
+  rw [← of_diff_of_diff_eq_zero (hw.inter hu) (hw.inter hv) hvu, huv, zero_add]
+end
+
+lemma of_inter_eq_of_symm_diff_eq_zero_negative
+  (hu : measurable_set u) (hv : measurable_set v) (hw : measurable_set w)
+  (hsu : s ≤[u] 0) (hsv : s ≤[v] 0) (hs : s (u Δ v) = 0) :
+  s (w ∩ u) = s (w ∩ v) :=
+begin
+  rw [← s.neg_le_neg_iff _ hu, neg_zero] at hsu,
+  rw [← s.neg_le_neg_iff _ hv, neg_zero] at hsv,
+  have := of_inter_eq_of_symm_diff_eq_zero_positive hu hv hw hsu hsv,
+  simp only [pi.neg_apply, neg_inj, neg_eq_zero, coe_neg] at this,
+  exact this hs,
+end
+
+end
+
+end signed_measure
+
+namespace jordan_decomposition
+
+open measure vector_measure signed_measure function
+
+private lemma eq_of_pos_part_eq_pos_part {j₁ j₂ : jordan_decomposition α}
+  (hj : j₁.pos_part = j₂.pos_part) (hj' : j₁.to_signed_measure = j₂.to_signed_measure) :
+  j₁ = j₂ :=
+begin
+  ext1,
+  { exact hj },
+  { rw ← to_signed_measure_eq_to_signed_measure_iff,
+    suffices : j₁.pos_part.to_signed_measure - j₁.neg_part.to_signed_measure =
+      j₁.pos_part.to_signed_measure - j₂.neg_part.to_signed_measure,
+    { exact sub_right_inj.mp this },
+    convert hj' }
+end
+
+/-- The Jordan decomposition of a signed measure is unique. -/
+theorem to_signed_measure_injective :
+  injective $ @jordan_decomposition.to_signed_measure α _ :=
+begin
+  /- The main idea is that two Jordan decompositions of a signed measure provide two
+  Hahn decompositions for that measure. Then, from `of_symm_diff_compl_positive_negative`,
+  the symmetric difference of the two Hahn decompositions has measure zero, thus, allowing us to
+  show the equality of the underlying measures of the Jordan decompositions. -/
+  intros j₁ j₂ hj,
+  -- obtain the two Hahn decompositions from the Jordan decompositions
+  obtain ⟨S, hS₁, hS₂, hS₃, hS₄, hS₅⟩ := j₁.exists_compl_positive_negative,
+  obtain ⟨T, hT₁, hT₂, hT₃, hT₄, hT₅⟩ := j₂.exists_compl_positive_negative,
+  rw ← hj at hT₂ hT₃,
+  -- the symmetric differences of the two Hahn decompositions have measure zero
+  obtain ⟨hST₁, -⟩ := of_symm_diff_compl_positive_negative hS₁.compl hT₁.compl
+    ⟨hS₃, (compl_compl S).symm ▸ hS₂⟩ ⟨hT₃, (compl_compl T).symm ▸ hT₂⟩,
+  -- it suffices to show the Jordan decompositions have the same positive parts
+  refine eq_of_pos_part_eq_pos_part _ hj,
+  ext1 i hi,
+  -- we see that the positive parts of the two Jordan decompositions are equal to their
+  -- associated signed measures restricted on their associated Hahn decompositions
+  have hμ₁ : (j₁.pos_part i).to_real = j₁.to_signed_measure (i ∩ Sᶜ),
+  { rw [to_signed_measure, to_signed_measure_sub_apply (hi.inter hS₁.compl),
+        show j₁.neg_part (i ∩ Sᶜ) = 0, by exact nonpos_iff_eq_zero.1
+          (hS₅ ▸ measure_mono (set.inter_subset_right _ _)),
+        ennreal.zero_to_real, sub_zero],
+    conv_lhs { rw ← set.inter_union_compl i S },
+    rw [measure_union, show j₁.pos_part (i ∩ S) = 0, by exact nonpos_iff_eq_zero.1
+          (hS₄ ▸ measure_mono (set.inter_subset_right _ _)), zero_add],
+    { refine set.disjoint_of_subset_left (set.inter_subset_right _ _)
+        (set.disjoint_of_subset_right (set.inter_subset_right _ _) disjoint_compl_right) },
+    { exact hi.inter hS₁ },
+    { exact hi.inter hS₁.compl } },
+  have hμ₂ : (j₂.pos_part i).to_real = j₂.to_signed_measure (i ∩ Tᶜ),
+  { rw [to_signed_measure, to_signed_measure_sub_apply (hi.inter hT₁.compl),
+        show j₂.neg_part (i ∩ Tᶜ) = 0, by exact nonpos_iff_eq_zero.1
+          (hT₅ ▸ measure_mono (set.inter_subset_right _ _)),
+        ennreal.zero_to_real, sub_zero],
+    conv_lhs { rw ← set.inter_union_compl i T },
+    rw [measure_union, show j₂.pos_part (i ∩ T) = 0, by exact nonpos_iff_eq_zero.1
+          (hT₄ ▸ measure_mono (set.inter_subset_right _ _)), zero_add],
+    { exact set.disjoint_of_subset_left (set.inter_subset_right _ _)
+        (set.disjoint_of_subset_right (set.inter_subset_right _ _) disjoint_compl_right) },
+    { exact hi.inter hT₁ },
+    { exact hi.inter hT₁.compl } },
+  -- since the two signed measures associated with the Jordan decompositions are the same,
+  -- and the symmetric difference of the Hahn decompositions have measure zero, the result follows
+  rw [← ennreal.to_real_eq_to_real (measure_lt_top _ _) (measure_lt_top _ _),
+      hμ₁, hμ₂, ← hj],
+  exact of_inter_eq_of_symm_diff_eq_zero_positive hS₁.compl hT₁.compl hi hS₃ hT₃ hST₁,
+  all_goals { apply_instance },
+end
+
+@[simp]
+lemma to_jordan_decomposition_to_signed_measure (j : jordan_decomposition α) :
+  (j.to_signed_measure).to_jordan_decomposition = j :=
+(@to_signed_measure_injective _ _ j (j.to_signed_measure).to_jordan_decomposition (by simp)).symm
+
+end jordan_decomposition
+
+namespace signed_measure
+
+open jordan_decomposition
+
+/-- `measure_theory.signed_measure.to_jordan_decomposition` and
+`measure_theory.jordan_decomposition.to_signed_measure` form a `equiv`. -/
+@[simps apply symm_apply]
+def to_jordan_decomposition_equiv (α : Type*) [measurable_space α] :
+  signed_measure α ≃ jordan_decomposition α :=
+{ to_fun := to_jordan_decomposition,
+  inv_fun := to_signed_measure,
+  left_inv := to_signed_measure_to_jordan_decomposition,
+  right_inv := to_jordan_decomposition_to_signed_measure }
+
+lemma to_jordan_decomposition_zero : (0 : signed_measure α).to_jordan_decomposition = 0 :=
+begin
+  apply to_signed_measure_injective,
+  simp [to_signed_measure_zero],
+end
+
+lemma to_jordan_decomposition_neg (s : signed_measure α) :
+  (-s).to_jordan_decomposition = -s.to_jordan_decomposition :=
+begin
+  apply to_signed_measure_injective,
+  simp [to_signed_measure_neg],
+end
+
+end signed_measure
+
+end measure_theory