Loogle!

Result

Found 26 declarations mentioning MeasureTheory.OuterMeasure.map.

• MeasureTheory.OuterMeasure.map 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β
• MeasureTheory.OuterMeasure.map_id 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} (m : MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map id) m = m
• MeasureTheory.OuterMeasure.map_mono 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) : Monotone ⇑(MeasureTheory.OuterMeasure.map f)
• MeasureTheory.OuterMeasure.map_top_of_surjective 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_2} (f : α → β) (hf : Function.Surjective f) : (MeasureTheory.OuterMeasure.map f) ⊤ = ⊤
• MeasureTheory.OuterMeasure.map_apply 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure α) (s : Set β) : ((MeasureTheory.OuterMeasure.map f) m) s = m (f ⁻¹' s)
• MeasureTheory.OuterMeasure.restrict.eq_1 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} (s : Set α) : MeasureTheory.OuterMeasure.restrict s = MeasureTheory.OuterMeasure.map Subtype.val ∘ₗ MeasureTheory.OuterMeasure.comap Subtype.val
• MeasureTheory.OuterMeasure.comap_map 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} {f : α → β} (hf : Function.Injective f) (m : MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m) = m
• MeasureTheory.OuterMeasure.map_comap_of_surjective 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} {f : α → β} (hf : Function.Surjective f) (m : MeasureTheory.OuterMeasure β) : (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) = m
• MeasureTheory.OuterMeasure.le_comap_map 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure α) : m ≤ (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m)
• MeasureTheory.OuterMeasure.map_comap_le 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure β) : (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) ≤ m
• MeasureTheory.OuterMeasure.map_top 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_2} (f : α → β) : (MeasureTheory.OuterMeasure.map f) ⊤ = (MeasureTheory.OuterMeasure.restrict (Set.range f)) ⊤
• MeasureTheory.OuterMeasure.map_iSup 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} {ι : Sort u_4} (f : α → β) (m : ι → MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map f) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.map f) (m i)
• MeasureTheory.OuterMeasure.map_comap 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure β) : (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) = (MeasureTheory.OuterMeasure.restrict (Set.range f)) m
• MeasureTheory.OuterMeasure.map_map 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} {γ : Type u_4} (f : α → β) (g : β → γ) (m : MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map g) ((MeasureTheory.OuterMeasure.map f) m) = (MeasureTheory.OuterMeasure.map (g ∘ f)) m
• MeasureTheory.OuterMeasure.map_le_restrict_range 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} {ma : MeasureTheory.OuterMeasure α} {mb : MeasureTheory.OuterMeasure β} {f : α → β} : (MeasureTheory.OuterMeasure.map f) ma ≤ (MeasureTheory.OuterMeasure.restrict (Set.range f)) mb ↔ (MeasureTheory.OuterMeasure.map f) ma ≤ mb
• MeasureTheory.OuterMeasure.map_sup 📋 Mathlib.MeasureTheory.OuterMeasure.Operations
  {α : Type u_1} {β : Type u_3} (f : α → β) (m m' : MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map f) (m ⊔ m') = (MeasureTheory.OuterMeasure.map f) m ⊔ (MeasureTheory.OuterMeasure.map f) m'
• MeasureTheory.OuterMeasure.map_ofFunction 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {m : Set α → ENNReal} {m_empty : m ∅ = 0} {β : Type u_2} {f : α → β} (hf : Function.Injective f) : (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.ofFunction m m_empty) = MeasureTheory.OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty
• MeasureTheory.OuterMeasure.map_ofFunction_le 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {m : Set α → ENNReal} {m_empty : m ∅ = 0} {β : Type u_2} (f : α → β) : (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.ofFunction m m_empty) ≤ MeasureTheory.OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty
• MeasureTheory.OuterMeasure.map_iInf_le 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {ι : Sort u_2} {β : Type u_3} (f : α → β) (m : ι → MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map f) (⨅ i, m i) ≤ ⨅ i, (MeasureTheory.OuterMeasure.map f) (m i)
• MeasureTheory.OuterMeasure.map_iInf 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {ι : Sort u_2} {β : Type u_3} {f : α → β} (hf : Function.Injective f) (m : ι → MeasureTheory.OuterMeasure α) : (MeasureTheory.OuterMeasure.map f) (⨅ i, m i) = (MeasureTheory.OuterMeasure.restrict (Set.range f)) (⨅ i, (MeasureTheory.OuterMeasure.map f) (m i))
• MeasureTheory.OuterMeasure.map_iInf_comap 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {ι : Sort u_2} {β : Type u_3} [Nonempty ι] {f : α → β} (m : ι → MeasureTheory.OuterMeasure β) : (MeasureTheory.OuterMeasure.map f) (⨅ i, (MeasureTheory.OuterMeasure.comap f) (m i)) = ⨅ i, (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) (m i))
• MeasureTheory.OuterMeasure.map_biInf_comap 📋 Mathlib.MeasureTheory.OuterMeasure.OfFunction
  {α : Type u_1} {ι : Type u_2} {β : Type u_3} {I : Set ι} (hI : I.Nonempty) {f : α → β} (m : ι → MeasureTheory.OuterMeasure β) : (MeasureTheory.OuterMeasure.map f) (⨅ i ∈ I, (MeasureTheory.OuterMeasure.comap f) (m i)) = ⨅ i ∈ I, (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) (m i))
• MeasureTheory.Measure.map_toOuterMeasure 📋 Mathlib.MeasureTheory.Measure.Map
  {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) : (MeasureTheory.Measure.map f μ).toOuterMeasure = ((MeasureTheory.OuterMeasure.map f) μ.toOuterMeasure).trim
• MeasureTheory.Measure.mapₗ.eq_1 📋 Mathlib.MeasureTheory.Measure.Map
  {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : α → β) : MeasureTheory.Measure.mapₗ f = if hf : Measurable f then MeasureTheory.Measure.liftLinear (MeasureTheory.OuterMeasure.map f) ⋯ else 0
• MeasureTheory.OuterMeasure.isometryEquiv_map_mkMetric 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ENNReal → ENNReal) (f : X ≃ᵢ Y) : (MeasureTheory.OuterMeasure.map ⇑f) (MeasureTheory.OuterMeasure.mkMetric m) = MeasureTheory.OuterMeasure.mkMetric m
• MeasureTheory.OuterMeasure.isometry_map_mkMetric 📋 Mathlib.MeasureTheory.Measure.Hausdorff
  {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ENNReal → ENNReal) {f : X → Y} (hf : Isometry f) (H : Monotone m ∨ Function.Surjective f) : (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.mkMetric m) = (MeasureTheory.OuterMeasure.restrict (Set.range f)) (MeasureTheory.OuterMeasure.mkMetric m)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision bce1d65