File: Mathlib/MeasureTheory/Measure/Count.lean

full_name: MeasureTheory.Measure.count
code: /-- Counting measure on any measurable space. -/
def count : Measure α 
goal: MeasureTheory.Measure α

full_name: MeasureTheory.Measure.le_count_apply
code: theorem le_count_apply : ∑' _ : s, (1 : ℝ≥0∞) ≤ count s 
goal: ∑' (x : ↑s), 1 ≤ MeasureTheory.Measure.count s

full_name: MeasureTheory.Measure.count_apply
code: theorem count_apply (hs : MeasurableSet s) : count s = ∑' i : s, 1 
goal: MeasureTheory.Measure.count s = ∑' (i : ↑s), 1

full_name: MeasureTheory.Measure.count_empty
code: theorem count_empty : count (∅ : Set α) = 0 
goal: MeasureTheory.Measure.count ∅ = 0

full_name: MeasureTheory.Measure.count_apply_finset'
code: @[simp]
theorem count_apply_finset' {s : Finset α} (s_mble : MeasurableSet (s : Set α)) :
    count (↑s : Set α) = s.card 
goal: MeasureTheory.Measure.count ↑s = ↑s.card

full_name: MeasureTheory.Measure.count_apply_finset
code: @[simp]
theorem count_apply_finset [MeasurableSingletonClass α] (s : Finset α) :
    count (↑s : Set α) = s.card 
goal: MeasureTheory.Measure.count ↑s = ↑s.card

full_name: MeasureTheory.Measure.count_apply_finite'
code: theorem count_apply_finite' {s : Set α} (s_fin : s.Finite) (s_mble : MeasurableSet s) :
    count s = s_fin.toFinset.card 
goal: MeasureTheory.Measure.count s = ↑s_fin.toFinset.card

full_name: MeasureTheory.Measure.count_apply_finite
code: theorem count_apply_finite [MeasurableSingletonClass α] (s : Set α) (hs : s.Finite) :
    count s = hs.toFinset.card 
goal: MeasureTheory.Measure.count s = ↑hs.toFinset.card

full_name: MeasureTheory.Measure.count_apply_infinite
code: /-- `count` measure evaluates to infinity at infinite sets. -/
theorem count_apply_infinite (hs : s.Infinite) : count s = ∞ 
goal: MeasureTheory.Measure.count s = ⊤

full_name: MeasureTheory.Measure.count_apply_eq_top'
code: @[simp]
theorem count_apply_eq_top' (s_mble : MeasurableSet s) : count s = ∞ ↔ s.Infinite 
goal: MeasureTheory.Measure.count s = ⊤ ↔ s.Infinite

full_name: MeasureTheory.Measure.count_apply_eq_top
code: @[simp]
theorem count_apply_eq_top [MeasurableSingletonClass α] : count s = ∞ ↔ s.Infinite 
goal: MeasureTheory.Measure.count s = ⊤ ↔ s.Infinite

full_name: MeasureTheory.Measure.count_apply_lt_top'
code: @[simp]
theorem count_apply_lt_top' (s_mble : MeasurableSet s) : count s < ∞ ↔ s.Finite 
goal: MeasureTheory.Measure.count s < ⊤ ↔ s.Finite

full_name: MeasureTheory.Measure.count_apply_lt_top
code: @[simp]
theorem count_apply_lt_top [MeasurableSingletonClass α] : count s < ∞ ↔ s.Finite 
goal: MeasureTheory.Measure.count s < ⊤ ↔ s.Finite

full_name: MeasureTheory.Measure.empty_of_count_eq_zero'
code: theorem empty_of_count_eq_zero' (s_mble : MeasurableSet s) (hsc : count s = 0) : s = ∅ 
goal: s = ∅

full_name: MeasureTheory.Measure.empty_of_count_eq_zero
code: theorem empty_of_count_eq_zero [MeasurableSingletonClass α] (hsc : count s = 0) : s = ∅ 
goal: s = ∅

full_name: MeasureTheory.Measure.count_eq_zero_iff'
code: @[simp]
theorem count_eq_zero_iff' (s_mble : MeasurableSet s) : count s = 0 ↔ s = ∅ 
goal: MeasureTheory.Measure.count s = 0 ↔ s = ∅

full_name: MeasureTheory.Measure.count_eq_zero_iff
code: @[simp]
theorem count_eq_zero_iff [MeasurableSingletonClass α] : count s = 0 ↔ s = ∅ 
goal: MeasureTheory.Measure.count s = 0 ↔ s = ∅

full_name: MeasureTheory.Measure.count_ne_zero'
code: theorem count_ne_zero' (hs' : s.Nonempty) (s_mble : MeasurableSet s) : count s ≠ 0 
goal: MeasureTheory.Measure.count s ≠ 0

full_name: MeasureTheory.Measure.count_ne_zero
code: theorem count_ne_zero [MeasurableSingletonClass α] (hs' : s.Nonempty) : count s ≠ 0 
goal: MeasureTheory.Measure.count s ≠ 0

full_name: MeasureTheory.Measure.count_singleton'
code: @[simp]
theorem count_singleton' {a : α} (ha : MeasurableSet ({a} : Set α)) : count ({a} : Set α) = 1 
goal: MeasureTheory.Measure.count {a} = 1

full_name: MeasureTheory.Measure.count_singleton
code: theorem count_singleton [MeasurableSingletonClass α] (a : α) : count ({a} : Set α) = 1 
goal: MeasureTheory.Measure.count {a} = 1

full_name: MeasureTheory.Measure.count_injective_image'
code: theorem count_injective_image' {f : β → α} (hf : Function.Injective f) {s : Set β}
    (s_mble : MeasurableSet s) (fs_mble : MeasurableSet (f '' s)) : count (f '' s) = count s 
goal: MeasureTheory.Measure.count (f '' s) = MeasureTheory.Measure.count s

full_name: MeasureTheory.Measure.count_injective_image
code: theorem count_injective_image [MeasurableSingletonClass α] [MeasurableSingletonClass β] {f : β → α}
    (hf : Function.Injective f) (s : Set β) : count (f '' s) = count s 
goal: MeasureTheory.Measure.count (f '' s) = MeasureTheory.Measure.count s

full_name: MeasureTheory.Measure.count.isFiniteMeasure
code: instance count.isFiniteMeasure [Finite α] :
    IsFiniteMeasure (Measure.count : Measure α) 
goal: MeasureTheory.IsFiniteMeasure MeasureTheory.Measure.count