File: Mathlib/MeasureTheory/Function/EssSup.lean

full_name: essSup
code: /-- Essential supremum of `f` with respect to measure `μ`: the smallest `c : β` such that
`f x ≤ c` a.e. -/
def essSup {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) 
goal: {x : MeasurableSpace α} → (α → β) → MeasureTheory.Measure α → β

full_name: essInf
code: /-- Essential infimum of `f` with respect to measure `μ`: the greatest `c : β` such that
`c ≤ f x` a.e. -/
def essInf {_ : MeasurableSpace α} (f : α → β) (μ : Measure α) 
goal: {x : MeasurableSpace α} → (α → β) → MeasureTheory.Measure α → β

full_name: essSup_congr_ae
code: theorem essSup_congr_ae {f g : α → β} (hfg : f =ᵐ[μ] g) : essSup f μ = essSup g μ 
goal: essSup f μ = essSup g μ

full_name: essInf_congr_ae
code: theorem essInf_congr_ae {f g : α → β} (hfg : f =ᵐ[μ] g) : essInf f μ = essInf g μ 
goal: essInf f μ = essInf g μ

full_name: essSup_const'
code: @[simp]
theorem essSup_const' [NeZero μ] (c : β) : essSup (fun _ : α => c) μ = c 
goal: essSup (fun x => c) μ = c

full_name: essInf_const'
code: @[simp]
theorem essInf_const' [NeZero μ] (c : β) : essInf (fun _ : α => c) μ = c 
goal: essInf (fun x => c) μ = c

full_name: essSup_const
code: theorem essSup_const (c : β) (hμ : μ ≠ 0) : essSup (fun _ : α => c) μ = c 
goal: essSup (fun x => c) μ = c

full_name: essInf_const
code: theorem essInf_const (c : β) (hμ : μ ≠ 0) : essInf (fun _ : α => c) μ = c 
goal: essInf (fun x => c) μ = c

full_name: essSup_eq_sInf
code: theorem essSup_eq_sInf {m : MeasurableSpace α} (μ : Measure α) (f : α → β) :
    essSup f μ = sInf { a | μ { x | a < f x } = 0 } 
goal: essSup f μ = sInf {a | μ {x | a < f x} = 0}

full_name: essInf_eq_sSup
code: theorem essInf_eq_sSup {m : MeasurableSpace α} (μ : Measure α) (f : α → β) :
    essInf f μ = sSup { a | μ { x | f x < a } = 0 } 
goal: essInf f μ = sSup {a | μ {x | f x < a} = 0}

full_name: ae_lt_of_essSup_lt
code: theorem ae_lt_of_essSup_lt (hx : essSup f μ < x)
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, f y < x 
goal: ∀ᵐ (y : α) ∂μ, f y < x

full_name: ae_lt_of_lt_essInf
code: theorem ae_lt_of_lt_essInf (hx : x < essInf f μ)
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, x < f y 
goal: ∀ᵐ (y : α) ∂μ, x < f y

full_name: ae_le_essSup
code: theorem ae_le_essSup
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, f y ≤ essSup f μ 
goal: ∀ᵐ (y : α) ∂μ, f y ≤ essSup f μ

full_name: ae_essInf_le
code: theorem ae_essInf_le
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, essInf f μ ≤ f y 
goal: ∀ᵐ (y : α) ∂μ, essInf f μ ≤ f y

full_name: meas_essSup_lt
code: theorem meas_essSup_lt
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    μ { y | essSup f μ < f y } = 0 
goal: μ {y | essSup f μ < f y} = 0

full_name: meas_lt_essInf
code: theorem meas_lt_essInf
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    μ { y | f y < essInf f μ } = 0 
goal: μ {y | f y < essInf f μ} = 0

full_name: essSup_measure_zero
code: @[simp]
theorem essSup_measure_zero {m : MeasurableSpace α} {f : α → β} : essSup f (0 : Measure α) = ⊥ 
goal: essSup f 0 = ⊥

full_name: essInf_measure_zero
code: @[simp]
theorem essInf_measure_zero {_ : MeasurableSpace α} {f : α → β} : essInf f (0 : Measure α) = ⊤ 
goal: ∀ {x : MeasurableSpace α} {f : α → β}, essInf f 0 = ⊤

full_name: essSup_mono_ae
code: theorem essSup_mono_ae {f g : α → β} (hfg : f ≤ᵐ[μ] g) : essSup f μ ≤ essSup g μ 
goal: essSup f μ ≤ essSup g μ

full_name: essInf_mono_ae
code: theorem essInf_mono_ae {f g : α → β} (hfg : f ≤ᵐ[μ] g) : essInf f μ ≤ essInf g μ 
goal: essInf f μ ≤ essInf g μ

full_name: essSup_le_of_ae_le
code: theorem essSup_le_of_ae_le {f : α → β} (c : β) (hf : f ≤ᵐ[μ] fun _ => c) : essSup f μ ≤ c 
goal: essSup f μ ≤ c

full_name: le_essInf_of_ae_le
code: theorem le_essInf_of_ae_le {f : α → β} (c : β) (hf : (fun _ => c) ≤ᵐ[μ] f) : c ≤ essInf f μ 
goal: c ≤ essInf f μ

full_name: essSup_const_bot
code: theorem essSup_const_bot : essSup (fun _ : α => (⊥ : β)) μ = (⊥ : β) 
goal: essSup (fun x => ⊥) μ = ⊥

full_name: essInf_const_top
code: theorem essInf_const_top : essInf (fun _ : α => (⊤ : β)) μ = (⊤ : β) 
goal: essInf (fun x => ⊤) μ = ⊤

full_name: OrderIso.essSup_apply
code: theorem OrderIso.essSup_apply {m : MeasurableSpace α} {γ} [CompleteLattice γ] (f : α → β)
    (μ : Measure α) (g : β ≃o γ) : g (essSup f μ) = essSup (fun x => g (f x)) μ 
goal: g (essSup f μ) = essSup (fun x => g (f x)) μ

full_name: OrderIso.essInf_apply
code: theorem OrderIso.essInf_apply {_ : MeasurableSpace α} {γ} [CompleteLattice γ] (f : α → β)
    (μ : Measure α) (g : β ≃o γ) : g (essInf f μ) = essInf (fun x => g (f x)) μ 
goal: ∀ {x : MeasurableSpace α} {γ : Type u_3} [inst : CompleteLattice γ] (f : α → β) (μ : MeasureTheory.Measure α)
  (g : β ≃o γ), g (essInf f μ) = essInf (fun x => g (f x)) μ

full_name: essSup_mono_measure
code: theorem essSup_mono_measure {f : α → β} (hμν : ν ≪ μ) : essSup f ν ≤ essSup f μ 
goal: essSup f ν ≤ essSup f μ

full_name: essSup_mono_measure'
code: theorem essSup_mono_measure' {α : Type*} {β : Type*} {_ : MeasurableSpace α}
    {μ ν : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (hμν : ν ≤ μ) :
    essSup f ν ≤ essSup f μ 
goal: ∀ {x : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [inst : CompleteLattice β] {f : α → β},
  ν ≤ μ → essSup f ν ≤ essSup f μ

full_name: essInf_antitone_measure
code: theorem essInf_antitone_measure {f : α → β} (hμν : μ ≪ ν) : essInf f ν ≤ essInf f μ 
goal: essInf f ν ≤ essInf f μ

full_name: essSup_smul_measure
code: theorem essSup_smul_measure {f : α → β} {c : ℝ≥0∞} (hc : c ≠ 0) :
    essSup f (c • μ) = essSup f μ 
goal: essSup f (c • μ) = essSup f μ

full_name: essSup_comp_le_essSup_map_measure
code: theorem essSup_comp_le_essSup_map_measure (hf : AEMeasurable f μ) :
    essSup (g ∘ f) μ ≤ essSup g (Measure.map f μ) 
goal: essSup (g ∘ f) μ ≤ essSup g (MeasureTheory.Measure.map f μ)

full_name: MeasurableEmbedding.essSup_map_measure
code: theorem MeasurableEmbedding.essSup_map_measure (hf : MeasurableEmbedding f) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ

full_name: essSup_map_measure_of_measurable
code: theorem essSup_map_measure_of_measurable (hg : Measurable g) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ

full_name: essSup_map_measure
code: theorem essSup_map_measure (hg : AEMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ

full_name: ENNReal.essSup_piecewise
code: lemma essSup_piecewise {s : Set α} [DecidablePred (· ∈ s)] {g} (hs : MeasurableSet s) :
    essSup (s.piecewise f g) μ = max (essSup f (μ.restrict s)) (essSup g (μ.restrict sᶜ)) 
goal: essSup (s.piecewise f g) μ = max (essSup f (μ.restrict s)) (essSup g (μ.restrict sᶜ))

full_name: ENNReal.essSup_indicator_eq_essSup_restrict
code: theorem essSup_indicator_eq_essSup_restrict {s : Set α} {f : α → ℝ≥0∞} (hs : MeasurableSet s) :
    essSup (s.indicator f) μ = essSup f (μ.restrict s) 
goal: essSup (s.indicator f) μ = essSup f (μ.restrict s)

full_name: ENNReal.ae_le_essSup
code: theorem ae_le_essSup (f : α → ℝ≥0∞) : ∀ᵐ y ∂μ, f y ≤ essSup f μ 
goal: ∀ᵐ (y : α) ∂μ, f y ≤ essSup f μ

full_name: ENNReal.essSup_eq_zero_iff
code: @[simp]
theorem essSup_eq_zero_iff : essSup f μ = 0 ↔ f =ᵐ[μ] 0 
goal: essSup f μ = 0 ↔ f =ᵐ[μ] 0

full_name: ENNReal.essSup_const_mul
code: theorem essSup_const_mul {a : ℝ≥0∞} : essSup (fun x : α => a * f x) μ = a * essSup f μ 
goal: essSup (fun x => a * f x) μ = a * essSup f μ

full_name: ENNReal.essSup_mul_le
code: theorem essSup_mul_le (f g : α → ℝ≥0∞) : essSup (f * g) μ ≤ essSup f μ * essSup g μ 
goal: essSup (f * g) μ ≤ essSup f μ * essSup g μ

full_name: ENNReal.essSup_add_le
code: theorem essSup_add_le (f g : α → ℝ≥0∞) : essSup (f + g) μ ≤ essSup f μ + essSup g μ 
goal: essSup (f + g) μ ≤ essSup f μ + essSup g μ

full_name: ENNReal.essSup_liminf_le
code: theorem essSup_liminf_le {ι} [Countable ι] [LinearOrder ι] (f : ι → α → ℝ≥0∞) :
    essSup (fun x => atTop.liminf fun n => f n x) μ ≤
      atTop.liminf fun n => essSup (fun x => f n x) μ 
goal: essSup (fun x => Filter.liminf (fun n => f n x) Filter.atTop) μ ≤
  Filter.liminf (fun n => essSup (fun x => f n x) μ) Filter.atTop

full_name: ENNReal.coe_essSup
code: theorem coe_essSup {f : α → ℝ≥0} (hf : IsBoundedUnder (· ≤ ·) (ae μ) f) :
    ((essSup f μ : ℝ≥0) : ℝ≥0∞) = essSup (fun x => (f x : ℝ≥0∞)) μ 
goal: ↑(essSup f μ) = essSup (fun x => ↑(f x)) μ

full_name: ENNReal.essSup_restrict_eq_of_support_subset
code: lemma essSup_restrict_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
    essSup f (μ.restrict s) = essSup f μ 
goal: essSup f (μ.restrict s) = essSup f μ