Loogle!

Result

Found 24 declarations mentioning MeasurableSpace.map.

• MeasurableSpace.map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) (m : MeasurableSpace α) : MeasurableSpace β
• MeasurableSpace.map_id 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {m : MeasurableSpace α} : MeasurableSpace.map id m = m
• MeasurableSpace.comap_map_le 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : α → β} : MeasurableSpace.comap f (MeasurableSpace.map f m) ≤ m
• MeasurableSpace.le_map_comap 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {g : β → α} : m ≤ MeasurableSpace.map g (MeasurableSpace.comap g m)
• MeasurableSpace.map_const 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} (b : β) : MeasurableSpace.map (fun _a => b) m = ⊤
• MeasurableSpace.monotone_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} : Monotone (MeasurableSpace.map f)
• Measurable.le_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β} : Measurable f → m₂ ≤ MeasurableSpace.map f m₁
• Measurable.of_le_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β} : m₂ ≤ MeasurableSpace.map f m₁ → Measurable f
• MeasurableSpace.map_def 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : α → β} {s : Set β} : MeasurableSet s ↔ MeasurableSet (f ⁻¹' s)
• measurable_iff_le_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m₁ : MeasurableSpace α} {m₂ : MeasurableSpace β} {f : α → β} : Measurable f ↔ m₂ ≤ MeasurableSpace.map f m₁
• MeasurableSpace.gc_comap_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) : GaloisConnection (MeasurableSpace.comap f) (MeasurableSpace.map f)
• MeasurableSpace.map_top 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {f : α → β} : MeasurableSpace.map f ⊤ = ⊤
• MeasurableSpace.map_mono 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {f : α → β} (h : m₁ ≤ m₂) : MeasurableSpace.map f m₁ ≤ MeasurableSpace.map f m₂
• MeasurableSpace.comap_le_iff_le_map 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {m' : MeasurableSpace β} {f : α → β} : MeasurableSpace.comap f m' ≤ m ↔ m' ≤ MeasurableSpace.map f m
• MeasurableSpace.map_comp 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {f : α → β} {g : β → γ} : MeasurableSpace.map g (MeasurableSpace.map f m) = MeasurableSpace.map (g ∘ f) m
• MeasurableSpace.map_iInf 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort uι} {f : α → β} {m : ι → MeasurableSpace α} : MeasurableSpace.map f (⨅ i, m i) = ⨅ i, MeasurableSpace.map f (m i)
• MeasurableSpace.map_inf 📋 Mathlib.MeasureTheory.MeasurableSpace.Basic
  {α : Type u_1} {β : Type u_2} {m₁ m₂ : MeasurableSpace α} {f : α → β} : MeasurableSpace.map f (m₁ ⊓ m₂) = MeasurableSpace.map f m₁ ⊓ MeasurableSpace.map f m₂
• MeasurableEquiv.map_eq 📋 Mathlib.MeasureTheory.MeasurableSpace.Embedding
  {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) : MeasurableSpace.map (⇑e) inst✝ = inst✝¹
• AddOpposite.instMeasurableSpace.eq_1 📋 Mathlib.MeasureTheory.Group.Arithmetic
  {α : Type u_2} [h : MeasurableSpace α] : AddOpposite.instMeasurableSpace = MeasurableSpace.map AddOpposite.op h
• MulOpposite.instMeasurableSpace.eq_1 📋 Mathlib.MeasureTheory.Group.Arithmetic
  {α : Type u_2} [h : MeasurableSpace α] : MulOpposite.instMeasurableSpace = MeasurableSpace.map MulOpposite.op h
• Measurable.map_measurableSpace_eq 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Z : Type u_5} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace Z] [MeasurableSpace.CountablySeparated Z] {f : X → Z} (hf : Measurable f) (hsurj : Function.Surjective f) : MeasurableSpace.map f inst✝ = inst✝¹
• Continuous.map_borel_eq 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f) : MeasurableSpace.map f (borel X) = borel Y
• Continuous.map_eq_borel 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f) : MeasurableSpace.map f inst✝ = borel Y
• Measurable.map_measurableSpace_eq_borel 📋 Mathlib.MeasureTheory.Constructions.Polish.Basic
  {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [T0Space Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f) (hsurj : Function.Surjective f) : MeasurableSpace.map f inst✝ = borel Y

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 40fea08