Documentation

Mathlib.MeasureTheory.Measure.Count

Counting measure #

In this file we define the counting measure MeasurTheory.Measure.count as MeasureTheory.Measure.sum MeasureTheory.Measure.dirac and prove basic properties of this measure.

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def MeasureTheory.Measure.count {α : Type u_1} [MeasurableSpace α] :
MeasureTheory.Measure α

Counting measure on any measurable space.

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Instances For
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    theorem MeasureTheory.Measure.le_count_apply {α : Type u_1} [MeasurableSpace α] {s : Set α} :
    ∑' (x : ↑s), 1 ≤ ↑↑MeasureTheory.Measure.count s
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    theorem MeasureTheory.Measure.count_apply {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s = ∑' (i : ↑s), 1
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    theorem MeasureTheory.Measure.count_empty {α : Type u_1} [MeasurableSpace α] :
    ↑↑MeasureTheory.Measure.count ∅ = 0
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    @[simp]
    theorem MeasureTheory.Measure.count_apply_finset' {α : Type u_1} [MeasurableSpace α] {s : Finset α} (s_mble : MeasurableSet ↑s) :
    ↑↑MeasureTheory.Measure.count ↑s = ↑s.card
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    @[simp]
    theorem MeasureTheory.Measure.count_apply_finset {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (s : Finset α) :
    ↑↑MeasureTheory.Measure.count ↑s = ↑s.card
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    theorem MeasureTheory.Measure.count_apply_finite' {α : Type u_1} [MeasurableSpace α] {s : Set α} (s_fin : Set.Finite s) (s_mble : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s = ↑(Set.Finite.toFinset s_fin).card
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    theorem MeasureTheory.Measure.count_apply_finite {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (s : Set α) (hs : Set.Finite s) :
    ↑↑MeasureTheory.Measure.count s = ↑(Set.Finite.toFinset hs).card
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    theorem MeasureTheory.Measure.count_apply_infinite {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : Set.Infinite s) :
    ↑↑MeasureTheory.Measure.count s = ⊤

    count measure evaluates to infinity at infinite sets.

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    @[simp]
    theorem MeasureTheory.Measure.count_apply_eq_top' {α : Type u_1} [MeasurableSpace α] {s : Set α} (s_mble : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s = ⊤ ↔ Set.Infinite s
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    @[simp]
    theorem MeasureTheory.Measure.count_apply_eq_top {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] :
    ↑↑MeasureTheory.Measure.count s = ⊤ ↔ Set.Infinite s
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    @[simp]
    theorem MeasureTheory.Measure.count_apply_lt_top' {α : Type u_1} [MeasurableSpace α] {s : Set α} (s_mble : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s < ⊤ ↔ Set.Finite s
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    @[simp]
    theorem MeasureTheory.Measure.count_apply_lt_top {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] :
    ↑↑MeasureTheory.Measure.count s < ⊤ ↔ Set.Finite s
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    theorem MeasureTheory.Measure.empty_of_count_eq_zero' {α : Type u_1} [MeasurableSpace α] {s : Set α} (s_mble : MeasurableSet s) (hsc : ↑↑MeasureTheory.Measure.count s = 0) :
    s = ∅
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    theorem MeasureTheory.Measure.empty_of_count_eq_zero {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] (hsc : ↑↑MeasureTheory.Measure.count s = 0) :
    s = ∅
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    @[simp]
    theorem MeasureTheory.Measure.count_eq_zero_iff' {α : Type u_1} [MeasurableSpace α] {s : Set α} (s_mble : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s = 0 ↔ s = ∅
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    @[simp]
    theorem MeasureTheory.Measure.count_eq_zero_iff {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] :
    ↑↑MeasureTheory.Measure.count s = 0 ↔ s = ∅
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    theorem MeasureTheory.Measure.count_ne_zero' {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs' : Set.Nonempty s) (s_mble : MeasurableSet s) :
    ↑↑MeasureTheory.Measure.count s ≠ 0
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    theorem MeasureTheory.Measure.count_ne_zero {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] (hs' : Set.Nonempty s) :
    ↑↑MeasureTheory.Measure.count s ≠ 0
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    @[simp]
    theorem MeasureTheory.Measure.count_singleton' {α : Type u_1} [MeasurableSpace α] {a : α} (ha : MeasurableSet {a}) :
    ↑↑MeasureTheory.Measure.count {a} = 1
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    theorem MeasureTheory.Measure.count_singleton {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) :
    ↑↑MeasureTheory.Measure.count {a} = 1
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    theorem MeasureTheory.Measure.count_injective_image' {α : Type u_2} {β : Type u_1} [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Function.Injective f) {s : Set β} (s_mble : MeasurableSet s) (fs_mble : MeasurableSet (f '' s)) :
    ↑↑MeasureTheory.Measure.count (f '' s) = ↑↑MeasureTheory.Measure.count s
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    theorem MeasureTheory.Measure.count_injective_image {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] [MeasurableSingletonClass β] {f : β → α} (hf : Function.Injective f) (s : Set β) :
    ↑↑MeasureTheory.Measure.count (f '' s) = ↑↑MeasureTheory.Measure.count s
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    instance MeasureTheory.Measure.count.isFiniteMeasure {α : Type u_1} [MeasurableSpace α] [Finite α] :
    MeasureTheory.IsFiniteMeasure MeasureTheory.Measure.count
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    • ⋯ = ⋯