File: Mathlib/MeasureTheory/OuterMeasure/Basic.lean

full_name: MeasureTheory.measure_empty
code: @[simp]
theorem measure_empty : μ ∅ = 0 
goal: μ ∅ = 0

full_name: MeasureTheory.measure_mono
code: @[mono, gcongr]
theorem measure_mono (h : s ⊆ t) : μ s ≤ μ t 
goal: μ s ≤ μ t

full_name: MeasureTheory.measure_mono_null
code: theorem measure_mono_null (h : s ⊆ t) (ht : μ t = 0) : μ s = 0 
goal: μ s = 0

full_name: MeasureTheory.measure_pos_of_superset
code: theorem measure_pos_of_superset (h : s ⊆ t) (hs : μ s ≠ 0) : 0 < μ t 
goal: 0 < μ t

full_name: MeasureTheory.measure_iUnion_le
code: theorem measure_iUnion_le [Countable ι] (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) 
goal: μ (⋃ i, s i) ≤ ∑' (i : ι), μ (s i)

full_name: MeasureTheory.measure_biUnion_le
code: theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
    μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) 
goal: μ (⋃ i ∈ I, s i) ≤ ∑' (i : ↑I), μ (s ↑i)

full_name: MeasureTheory.measure_biUnion_finset_le
code: theorem measure_biUnion_finset_le (I : Finset ι) (s : ι → Set α) :
    μ (⋃ i ∈ I, s i) ≤ ∑ i ∈ I, μ (s i) 
goal: μ (⋃ i ∈ I, s i) ≤ ∑ i ∈ I, μ (s i)

full_name: MeasureTheory.measure_iUnion_fintype_le
code: theorem measure_iUnion_fintype_le [Fintype ι] (μ : F) (s : ι → Set α) :
    μ (⋃ i, s i) ≤ ∑ i, μ (s i) 
goal: μ (⋃ i, s i) ≤ ∑ i : ι, μ (s i)

full_name: MeasureTheory.measure_union_le
code: theorem measure_union_le (s t : Set α) : μ (s ∪ t) ≤ μ s + μ t 
goal: μ (s ∪ t) ≤ μ s + μ t

full_name: MeasureTheory.measure_le_inter_add_diff
code: theorem measure_le_inter_add_diff (μ : F) (s t : Set α) : μ s ≤ μ (s ∩ t) + μ (s \ t) 
goal: μ s ≤ μ (s ∩ t) + μ (s \ t)

full_name: MeasureTheory.measure_diff_null
code: theorem measure_diff_null (ht : μ t = 0) : μ (s \ t) = μ s 
goal: μ (s \ t) = μ s

full_name: MeasureTheory.measure_biUnion_null_iff
code: theorem measure_biUnion_null_iff {I : Set ι} (hI : I.Countable) {s : ι → Set α} :
    μ (⋃ i ∈ I, s i) = 0 ↔ ∀ i ∈ I, μ (s i) = 0 
goal: μ (⋃ i ∈ I, s i) = 0 ↔ ∀ i ∈ I, μ (s i) = 0

full_name: MeasureTheory.measure_sUnion_null_iff
code: theorem measure_sUnion_null_iff {S : Set (Set α)} (hS : S.Countable) :
    μ (⋃₀ S) = 0 ↔ ∀ s ∈ S, μ s = 0 
goal: μ (⋃₀ S) = 0 ↔ ∀ s ∈ S, μ s = 0

full_name: MeasureTheory.measure_iUnion_null_iff
code: @[simp]
theorem measure_iUnion_null_iff {ι : Sort*} [Countable ι] {s : ι → Set α} :
    μ (⋃ i, s i) = 0 ↔ ∀ i, μ (s i) = 0 
goal: μ (⋃ i, s i) = 0 ↔ ∀ (i : ι), μ (s i) = 0

full_name: MeasureTheory.measure_union_null_iff
code: @[simp]
theorem measure_union_null_iff : μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0 
goal: μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0

full_name: MeasureTheory.measure_union_null
code: theorem measure_union_null (hs : μ s = 0) (ht : μ t = 0) : μ (s ∪ t) = 0 
goal: μ (s ∪ t) = 0

full_name: MeasureTheory.measure_null_iff_singleton
code: lemma measure_null_iff_singleton (hs : s.Countable) : μ s = 0 ↔ ∀ x ∈ s, μ {x} = 0 
goal: μ s = 0 ↔ ∀ x ∈ s, μ {x} = 0

full_name: MeasureTheory.measure_iUnion_of_tendsto_zero
code: /-- Let `μ` be an (outer) measure; let `s : ι → Set α` be a sequence of sets, `S = ⋃ n, s n`.
If `μ (S \ s n)` tends to zero along some nontrivial filter (usually `Filter.atTop` on `ι = ℕ`),
then `μ S = ⨆ n, μ (s n)`. -/
theorem measure_iUnion_of_tendsto_zero {ι} (μ : F) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => μ ((⋃ n, s n) \ s k)) l (𝓝 0)) : μ (⋃ n, s n) = ⨆ n, μ (s n) 
goal: μ (⋃ n, s n) = ⨆ n, μ (s n)

full_name: MeasureTheory.measure_null_of_locally_null
code: /-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem measure_null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α]
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, μ u = 0) : μ s = 0 
goal: μ s = 0

full_name: MeasureTheory.exists_mem_forall_mem_nhdsWithin_pos_measure
code: /-- If `m s ≠ 0`, then for some point `x ∈ s` and any `t ∈ 𝓝[s] x` we have `0 < m t`. -/
theorem exists_mem_forall_mem_nhdsWithin_pos_measure [TopologicalSpace α]
    [SecondCountableTopology α] {s : Set α} (hs : μ s ≠ 0) :
    ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t 
goal: ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t

full_name: MeasureTheory.OuterMeasure.empty'
code: @[deprecated measure_empty (since := "2024-05-14")]
theorem empty' (m : OuterMeasure α) : m ∅ = 0 
goal: m ∅ = 0

full_name: MeasureTheory.OuterMeasure.mono'
code: @[deprecated measure_mono (since := "2024-05-14")]
theorem mono' (m : OuterMeasure α) {s₁ s₂} (h : s₁ ⊆ s₂) : m s₁ ≤ m s₂ 
goal: m s₁ ≤ m s₂

full_name: MeasureTheory.OuterMeasure.mono_null
code: @[deprecated measure_mono_null (since := "2024-05-14")]
theorem mono_null (m : OuterMeasure α) {s t} (h : s ⊆ t) (ht : m t = 0) : m s = 0 
goal: m s = 0

full_name: MeasureTheory.OuterMeasure.pos_of_subset_ne_zero
code: @[deprecated measure_pos_of_superset (since := "2024-05-14")]
theorem pos_of_subset_ne_zero (m : OuterMeasure α) {a b : Set α} (hs : a ⊆ b) (hnz : m a ≠ 0) :
    0 < m b 
goal: 0 < m b

full_name: MeasureTheory.OuterMeasure.iUnion
code: @[deprecated measure_iUnion_le (since := "2024-05-14")]
protected theorem iUnion (m : OuterMeasure α) {β} [Countable β] (s : β → Set α) :
    m (⋃ i, s i) ≤ ∑' i, m (s i) 
goal: m (⋃ i, s i) ≤ ∑' (i : β), m (s i)

full_name: MeasureTheory.OuterMeasure.biUnion_null_iff
code: @[deprecated measure_biUnion_null_iff (since := "2024-05-14")]
theorem biUnion_null_iff (m : OuterMeasure α) {s : Set β} (hs : s.Countable) {t : β → Set α} :
    m (⋃ i ∈ s, t i) = 0 ↔ ∀ i ∈ s, m (t i) = 0 
goal: m (⋃ i ∈ s, t i) = 0 ↔ ∀ i ∈ s, m (t i) = 0

full_name: MeasureTheory.OuterMeasure.sUnion_null_iff
code: @[deprecated measure_sUnion_null_iff (since := "2024-05-14")]
theorem sUnion_null_iff (m : OuterMeasure α) {S : Set (Set α)} (hS : S.Countable) :
    m (⋃₀ S) = 0 ↔ ∀ s ∈ S, m s = 0 
goal: m (⋃₀ S) = 0 ↔ ∀ s ∈ S, m s = 0

full_name: MeasureTheory.OuterMeasure.iUnion_null_iff
code: @[deprecated measure_iUnion_null_iff (since := "2024-05-14")]
theorem iUnion_null_iff {ι : Sort*} [Countable ι] (m : OuterMeasure α) {s : ι → Set α} :
    m (⋃ i, s i) = 0 ↔ ∀ i, m (s i) = 0 
goal: m (⋃ i, s i) = 0 ↔ ∀ (i : ι), m (s i) = 0

full_name: MeasureTheory.OuterMeasure.iUnion_null_iff'
code: @[deprecated (since := "2024-01-14")]
theorem iUnion_null_iff' (m : OuterMeasure α) {ι : Prop} {s : ι → Set α} :
    m (⋃ i, s i) = 0 ↔ ∀ i, m (s i) = 0 
goal: m (⋃ (i : ι), s i) = 0 ↔ ∀ (i : ι), m (s i) = 0

full_name: MeasureTheory.OuterMeasure.iUnion_finset
code: @[deprecated measure_biUnion_finset_le (since := "2024-05-14")]
protected theorem iUnion_finset (m : OuterMeasure α) (s : β → Set α) (t : Finset β) :
    m (⋃ i ∈ t, s i) ≤ ∑ i ∈ t, m (s i) 
goal: m (⋃ i ∈ t, s i) ≤ ∑ i ∈ t, m (s i)

full_name: MeasureTheory.OuterMeasure.union
code: @[deprecated measure_union_le (since := "2024-05-14")]
protected theorem union (m : OuterMeasure α) (s₁ s₂ : Set α) : m (s₁ ∪ s₂) ≤ m s₁ + m s₂ 
goal: m (s₁ ∪ s₂) ≤ m s₁ + m s₂

full_name: MeasureTheory.OuterMeasure.null_of_locally_null
code: /-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
@[deprecated measure_null_of_locally_null (since := "2024-05-14")]
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 
goal: m s = 0

full_name: MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos
code: /-- If `m s ≠ 0`, then for some point `x ∈ s` and any `t ∈ 𝓝[s] x` we have `0 < m t`. -/
@[deprecated exists_mem_forall_mem_nhdsWithin_pos_measure (since := "2024-05-14")]
theorem exists_mem_forall_mem_nhds_within_pos [TopologicalSpace α] [SecondCountableTopology α]
    (m : OuterMeasure α) {s : Set α} (hs : m s ≠ 0) : ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t 
goal: ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t

full_name: MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero
code: /-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) 
goal: m (⋃ n, s n) = ⨆ n, m (s n)

full_name: MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top
code: /-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞) :
    m (⋃ n, s n) = ⨆ n, m (s n) 
goal: m (⋃ n, s n) = ⨆ n, m (s n)

full_name: MeasureTheory.OuterMeasure.le_inter_add_diff
code: @[deprecated measure_le_inter_add_diff (since := "2024-05-14")]
theorem le_inter_add_diff {m : OuterMeasure α} {t : Set α} (s : Set α) :
    m t ≤ m (t ∩ s) + m (t \ s) 
goal: m t ≤ m (t ∩ s) + m (t \ s)

full_name: MeasureTheory.OuterMeasure.diff_null
code: @[deprecated measure_diff_null (since := "2024-05-14")]
theorem diff_null (m : OuterMeasure α) (s : Set α) {t : Set α} (ht : m t = 0) : m (s \ t) = m s 
goal: m (s \ t) = m s

full_name: MeasureTheory.OuterMeasure.union_null
code: @[deprecated measure_union_null (since := "2024-05-14")]
theorem union_null (m : OuterMeasure α) {s₁ s₂ : Set α} (h₁ : m s₁ = 0) (h₂ : m s₂ = 0) :
    m (s₁ ∪ s₂) = 0 
goal: m (s₁ ∪ s₂) = 0

full_name: MeasureTheory.OuterMeasure.coe_fn_injective
code: theorem coe_fn_injective : Injective fun (μ : OuterMeasure α) (s : Set α) => μ s 
goal: Function.Injective fun μ s => μ s

full_name: MeasureTheory.OuterMeasure.ext
code: @[ext]
theorem ext {μ₁ μ₂ : OuterMeasure α} (h : ∀ s, μ₁ s = μ₂ s) : μ₁ = μ₂ 
goal: μ₁ = μ₂

full_name: MeasureTheory.OuterMeasure.ext_nonempty
code: /-- A version of `MeasureTheory.OuterMeasure.ext` that assumes `μ₁ s = μ₂ s` on all *nonempty*
sets `s`, and gets `μ₁ ∅ = μ₂ ∅` from `MeasureTheory.OuterMeasure.empty'`. -/
theorem ext_nonempty {μ₁ μ₂ : OuterMeasure α} (h : ∀ s : Set α, s.Nonempty → μ₁ s = μ₂ s) :
    μ₁ = μ₂ 
goal: μ₁ = μ₂