Loogle!

Result

Found 8 definitions mentioning MeasurableSet.iUnion.

• MeasurableSet.iUnion 📋 Mathlib.MeasureTheory.MeasurableSpace.Defs
  {α : Type u_1} {ι : Sort u_6} {m : MeasurableSpace α} [Countable ι] ⦃f : ι → Set α⦄ (h : ∀ (b : ι), MeasurableSet (f b)) : MeasurableSet (⋃ b, f b)
• MeasurableSpace.generateFrom_induction 📋 Mathlib.MeasureTheory.MeasurableSpace.Defs
  {α : Type u_1} (C : Set (Set α)) (p : (s : Set α) → MeasurableSet s → Prop) (hC : ∀ t ∈ C, ∀ (ht : MeasurableSet t), p t ht) (empty : p ∅ ⋯) (compl : ∀ (t : Set α) (ht : MeasurableSet t), p t ht → p tᶜ ⋯) (iUnion : ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)), (∀ (n : ℕ), p (s n) ⋯) → p (⋃ i, s i) ⋯) (s : Set α) (hs : MeasurableSet s) : p s hs
• MeasureTheory.inducedOuterMeasure_eq 📋 Mathlib.MeasureTheory.OuterMeasure.Induced
  {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ENNReal} (m0 : m ∅ ⋯ = 0) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s : Set α} (hs : MeasurableSet s) : (MeasureTheory.inducedOuterMeasure m ⋯ m0) s = m s hs
• MeasureTheory.inducedOuterMeasure_eq_extend 📋 Mathlib.MeasureTheory.OuterMeasure.Induced
  {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ENNReal} (m0 : m ∅ ⋯ = 0) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s : Set α} (hs : MeasurableSet s) : (MeasureTheory.inducedOuterMeasure m ⋯ m0) s = MeasureTheory.extend m s
• MeasureTheory.extend_mono 📋 Mathlib.MeasureTheory.OuterMeasure.Induced
  {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ENNReal} (m0 : m ∅ ⋯ = 0) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (hs : s₁ ⊆ s₂) : MeasureTheory.extend m s₁ ≤ MeasureTheory.extend m s₂
• MeasureTheory.extend_iUnion_le_tsum_nat 📋 Mathlib.MeasureTheory.OuterMeasure.Induced
  {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ENNReal} (m0 : m ∅ ⋯ = 0) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) (s : ℕ → Set α) : MeasureTheory.extend m (⋃ i, s i) ≤ ∑' (i : ℕ), MeasureTheory.extend m (s i)
• MeasureTheory.Measure.ofMeasurable 📋 Mathlib.MeasureTheory.Measure.MeasureSpaceDef
  {α : Type u_1} [MeasurableSpace α] (m : (s : Set α) → MeasurableSet s → ENNReal) (m0 : m ∅ ⋯ = 0) (mU : ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) : MeasureTheory.Measure α
• MeasureTheory.Measure.ofMeasurable_apply 📋 Mathlib.MeasureTheory.Measure.MeasureSpaceDef
  {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ENNReal} {m0 : m ∅ ⋯ = 0} {mU : ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯} (s : Set α) (hs : MeasurableSet s) : (MeasureTheory.Measure.ofMeasurable m m0 mU) s = m s hs

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, _ * _, _ ^ _, |- _ < _ → _
woould 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 4e1aab0 serving mathlib revision b513113