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] :
Measure α

Counting measure on any measurable space.

Equations
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@[simp]
theorem MeasureTheory.Measure.count_ne_zero'' {α : Type u_1} [MeasurableSpace α] [Nonempty α] :
count ≠ 0
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theorem MeasureTheory.Measure.le_count_apply {α : Type u_1} [MeasurableSpace α] {s : Set α} :
∑' (x : ↑s), 1 ≤ count s
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theorem MeasureTheory.Measure.count_apply {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s) :
count s = ↑s.encard
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@[deprecated MeasureTheory.measure_empty (since := "2025-02-06")]
theorem MeasureTheory.Measure.count_empty {α : Type u_1} [MeasurableSpace α] :
count ∅ = 0
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@[simp]
theorem MeasureTheory.Measure.count_apply_finset' {α : Type u_1} [MeasurableSpace α] {s : Finset α} (hs : MeasurableSet ↑s) :
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) :
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 : s.Finite) (s_mble : MeasurableSet s) :
count s = ↑s_fin.toFinset.card
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theorem MeasureTheory.Measure.count_apply_finite {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (s : Set α) (hs : s.Finite) :
count s = ↑hs.toFinset.card
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theorem MeasureTheory.Measure.count_apply_infinite {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : s.Infinite) :
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) :
count s = ⊤ ↔ s.Infinite
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@[simp]
theorem MeasureTheory.Measure.count_apply_eq_top {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] :
count s = ⊤ ↔ s.Infinite
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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) :
count s < ⊤ ↔ s.Finite
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@[simp]
theorem MeasureTheory.Measure.count_apply_lt_top {α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] :
count s < ⊤ ↔ s.Finite
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@[simp]
theorem MeasureTheory.Measure.count_eq_zero_iff {α : Type u_1} [MeasurableSpace α] {s : Set α} :
count s = 0 ↔ s = ∅
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theorem MeasureTheory.Measure.count_ne_zero_iff {α : Type u_1} [MeasurableSpace α] {s : Set α} :
count s ≠ 0 ↔ s.Nonempty
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theorem MeasureTheory.Measure.count_ne_zero {α : Type u_1} [MeasurableSpace α] {s : Set α} :
s.Nonempty → count s ≠ 0

Alias of the reverse direction of MeasureTheory.Measure.count_ne_zero_iff.

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@[deprecated MeasureTheory.Measure.count_eq_zero_iff (since := "2024-11-20")]
theorem MeasureTheory.Measure.empty_of_count_eq_zero {α : Type u_1} [MeasurableSpace α] {s : Set α} :
count s = 0 → s = ∅

Alias of the forward direction of MeasureTheory.Measure.count_eq_zero_iff.

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@[deprecated MeasureTheory.Measure.empty_of_count_eq_zero (since := "2024-11-20")]
theorem MeasureTheory.Measure.empty_of_count_eq_zero' {α : Type u_1} [MeasurableSpace α] {s : Set α} :
count s = 0 → s = ∅

Alias of the forward direction of MeasureTheory.Measure.count_eq_zero_iff.

Alias of the forward direction of MeasureTheory.Measure.count_eq_zero_iff.

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@[deprecated MeasureTheory.Measure.count_eq_zero_iff (since := "2024-11-20")]
theorem MeasureTheory.Measure.count_eq_zero_iff' {α : Type u_1} [MeasurableSpace α] {s : Set α} :
count s = 0 ↔ s = ∅

Alias of MeasureTheory.Measure.count_eq_zero_iff.

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@[deprecated MeasureTheory.Measure.count_ne_zero (since := "2024-11-20")]
theorem MeasureTheory.Measure.count_ne_zero' {α : Type u_1} [MeasurableSpace α] {s : Set α} :
s.Nonempty → count s ≠ 0

Alias of the reverse direction of MeasureTheory.Measure.count_ne_zero_iff.

Alias of the reverse direction of MeasureTheory.Measure.count_ne_zero_iff.

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@[simp]
theorem MeasureTheory.Measure.ae_count_iff {α : Type u_1} [MeasurableSpace α] {p : α → Prop} :
(∀ᵐ (x : α) ∂count, p x) ↔ ∀ (x : α), p x
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@[simp]
theorem MeasureTheory.Measure.count_singleton' {α : Type u_1} [MeasurableSpace α] {a : α} (ha : MeasurableSet {a}) :
count {a} = 1
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theorem MeasureTheory.Measure.count_singleton {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) :
count {a} = 1
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@[simp]
theorem MeasureTheory.count_real_singleton' {α : Type u_1} [MeasurableSpace α] {a : α} (ha : MeasurableSet {a}) :
Measure.count.real {a} = 1
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theorem MeasureTheory.count_real_singleton {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) :
Measure.count.real {a} = 1
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theorem MeasureTheory.Measure.count_injective_image' {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : β → α} (hf : Function.Injective f) {s : Set β} (s_mble : MeasurableSet s) (fs_mble : MeasurableSet (f '' s)) :
count (f '' s) = 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) :
count (f '' s) = count s
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instance MeasureTheory.Measure.count.instSigmaFinite {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] [Countable α] :
SigmaFinite count
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instance MeasureTheory.Measure.count.isFiniteMeasure {α : Type u_1} [MeasurableSpace α] [Finite α] :
IsFiniteMeasure count
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@[simp]
theorem MeasureTheory.Measure.count_univ {α : Type u_1} [MeasurableSpace α] :
count Set.univ = ↑(ENat.card α)