File: Mathlib/MeasureTheory/Measure/MutuallySingular.lean

full_name: MeasureTheory.Measure.MutuallySingular
code: /-- Two measures `μ`, `ν` are said to be mutually singular if there exists a measurable set `s`
such that `μ s = 0` and `ν sᶜ = 0`. -/
def MutuallySingular {_ : MeasurableSpace α} (μ ν : Measure α) : Prop 
goal: {x : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → Prop

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Measure_MutuallySingular___macroRules_MeasureTheory_term_⟂ₘ__1»
code: @[inherit_doc MeasureTheory.Measure.MutuallySingular]
scoped[MeasureTheory] infixl:60 " ⟂ₘ " => MeasureTheory.Measure.MutuallySingular
goal: Lean.Macro

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_Measure_MutuallySingular___unexpand_MeasureTheory_Measure_MutuallySingular_1
code: @[inherit_doc MeasureTheory.Measure.MutuallySingular]
scoped[MeasureTheory] infixl:60 " ⟂ₘ " => MeasureTheory.Measure.MutuallySingular
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.Measure.MutuallySingular.mk
code: theorem mk {s t : Set α} (hs : μ s = 0) (ht : ν t = 0) (hst : univ ⊆ s ∪ t) :
    MutuallySingular μ ν 
goal: μ ⟂ₘ ν

full_name: MeasureTheory.Measure.MutuallySingular.nullSet
code: /-- A set such that `μ h.nullSet = 0` and `ν h.nullSetᶜ = 0`. -/
def nullSet (h : μ ⟂ₘ ν) : Set α 
goal: Set α

full_name: MeasureTheory.Measure.MutuallySingular.measurableSet_nullSet
code: lemma measurableSet_nullSet (h : μ ⟂ₘ ν) : MeasurableSet h.nullSet 
goal: MeasurableSet h.nullSet

full_name: MeasureTheory.Measure.MutuallySingular.measure_nullSet
code: @[simp]
lemma measure_nullSet (h : μ ⟂ₘ ν) : μ h.nullSet = 0 
goal: μ h.nullSet = 0

full_name: MeasureTheory.Measure.MutuallySingular.measure_compl_nullSet
code: @[simp]
lemma measure_compl_nullSet (h : μ ⟂ₘ ν) : ν h.nullSetᶜ = 0 
goal: ν h.nullSetᶜ = 0

full_name: MeasureTheory.Measure.MutuallySingular.restrict_nullSet
code: @[simp]
lemma restrict_nullSet (h : μ ⟂ₘ ν) : μ.restrict h.nullSet = 0 
goal: μ.restrict h.nullSet = 0

full_name: MeasureTheory.Measure.MutuallySingular.restrict_compl_nullSet
code: @[simp]
lemma restrict_compl_nullSet (h : μ ⟂ₘ ν) : ν.restrict h.nullSetᶜ = 0 
goal: ν.restrict h.nullSetᶜ = 0

full_name: MeasureTheory.Measure.MutuallySingular.zero_right
code: @[simp]
theorem zero_right : μ ⟂ₘ 0 
goal: μ ⟂ₘ 0

full_name: MeasureTheory.Measure.MutuallySingular.symm
code: @[symm]
theorem symm (h : ν ⟂ₘ μ) : μ ⟂ₘ ν 
goal: μ ⟂ₘ ν

full_name: MeasureTheory.Measure.MutuallySingular.comm
code: theorem comm : μ ⟂ₘ ν ↔ ν ⟂ₘ μ 
goal: μ ⟂ₘ ν ↔ ν ⟂ₘ μ

full_name: MeasureTheory.Measure.MutuallySingular.zero_left
code: @[simp]
theorem zero_left : 0 ⟂ₘ μ 
goal: 0 ⟂ₘ μ

full_name: MeasureTheory.Measure.MutuallySingular.mono_ac
code: theorem mono_ac (h : μ₁ ⟂ₘ ν₁) (hμ : μ₂ ≪ μ₁) (hν : ν₂ ≪ ν₁) : μ₂ ⟂ₘ ν₂ 
goal: μ₂ ⟂ₘ ν₂

full_name: MeasureTheory.Measure.MutuallySingular.mono
code: theorem mono (h : μ₁ ⟂ₘ ν₁) (hμ : μ₂ ≤ μ₁) (hν : ν₂ ≤ ν₁) : μ₂ ⟂ₘ ν₂ 
goal: μ₂ ⟂ₘ ν₂

full_name: MeasureTheory.Measure.MutuallySingular.self_iff
code: @[simp]
lemma self_iff (μ : Measure α) : μ ⟂ₘ μ ↔ μ = 0 
goal: μ ⟂ₘ μ ↔ μ = 0

full_name: MeasureTheory.Measure.MutuallySingular.sum_left
code: @[simp]
theorem sum_left {ι : Type*} [Countable ι] {μ : ι → Measure α} : sum μ ⟂ₘ ν ↔ ∀ i, μ i ⟂ₘ ν 
goal: MeasureTheory.Measure.sum μ ⟂ₘ ν ↔ ∀ (i : ι), μ i ⟂ₘ ν

full_name: MeasureTheory.Measure.MutuallySingular.sum_right
code: @[simp]
theorem sum_right {ι : Type*} [Countable ι] {ν : ι → Measure α} : μ ⟂ₘ sum ν ↔ ∀ i, μ ⟂ₘ ν i 
goal: μ ⟂ₘ MeasureTheory.Measure.sum ν ↔ ∀ (i : ι), μ ⟂ₘ ν i

full_name: MeasureTheory.Measure.MutuallySingular.add_left_iff
code: @[simp]
theorem add_left_iff : μ₁ + μ₂ ⟂ₘ ν ↔ μ₁ ⟂ₘ ν ∧ μ₂ ⟂ₘ ν 
goal: μ₁ + μ₂ ⟂ₘ ν ↔ μ₁ ⟂ₘ ν ∧ μ₂ ⟂ₘ ν

full_name: MeasureTheory.Measure.MutuallySingular.add_right_iff
code: @[simp]
theorem add_right_iff : μ ⟂ₘ ν₁ + ν₂ ↔ μ ⟂ₘ ν₁ ∧ μ ⟂ₘ ν₂ 
goal: μ ⟂ₘ ν₁ + ν₂ ↔ μ ⟂ₘ ν₁ ∧ μ ⟂ₘ ν₂

full_name: MeasureTheory.Measure.MutuallySingular.add_left
code: theorem add_left (h₁ : ν₁ ⟂ₘ μ) (h₂ : ν₂ ⟂ₘ μ) : ν₁ + ν₂ ⟂ₘ μ 
goal: ν₁ + ν₂ ⟂ₘ μ

full_name: MeasureTheory.Measure.MutuallySingular.add_right
code: theorem add_right (h₁ : μ ⟂ₘ ν₁) (h₂ : μ ⟂ₘ ν₂) : μ ⟂ₘ ν₁ + ν₂ 
goal: μ ⟂ₘ ν₁ + ν₂

full_name: MeasureTheory.Measure.MutuallySingular.smul
code: theorem smul (r : ℝ≥0∞) (h : ν ⟂ₘ μ) : r • ν ⟂ₘ μ 
goal: r • ν ⟂ₘ μ

full_name: MeasureTheory.Measure.MutuallySingular.smul_nnreal
code: theorem smul_nnreal (r : ℝ≥0) (h : ν ⟂ₘ μ) : r • ν ⟂ₘ μ 
goal: r • ν ⟂ₘ μ

full_name: MeasureTheory.Measure.MutuallySingular.restrict
code: lemma restrict (h : μ ⟂ₘ ν) (s : Set α) : μ.restrict s ⟂ₘ ν 
goal: μ.restrict s ⟂ₘ ν

full_name: MeasureTheory.Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular
code: lemma eq_zero_of_absolutelyContinuous_of_mutuallySingular {μ ν : Measure α}
    (h_ac : μ ≪ ν) (h_ms : μ ⟂ₘ ν) :
    μ = 0 
goal: μ = 0

full_name: MeasurableEmbedding.mutuallySingular_map
code: lemma _root_.MeasurableEmbedding.mutuallySingular_map {β : Type*} {_ : MeasurableSpace β}
    {f : α → β} (hf : MeasurableEmbedding f) (hμν : μ ⟂ₘ ν) :
    μ.map f ⟂ₘ ν.map f 
goal: ∀ {x : MeasurableSpace β} {f : α → β},
  MeasurableEmbedding f → μ ⟂ₘ ν → MeasureTheory.Measure.map f μ ⟂ₘ MeasureTheory.Measure.map f ν