mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
by
  by_cases hG : CompleteSpace G; swap
  · simp [integral, hG]
  borelize G
  by_cases hf_int : Integrable f μ
  swap
  · have hf_int_m : ¬Integrable f (μ.trim hm) := fun hf_int_m =>
      hf_int (integrable_of_integrable_trim hm hf_int_m)
    rw [integral_undef hf_int, integral_undef hf_int_m]
  have : SeparableSpace (range f ∪ {0} : Set G) := hf.separableSpace_range_union_singleton
  let f_seq := @SimpleFunc.approxOn G β _ _ _ m _ hf.measurable (range f ∪ {0}) 0 (by simp) _
  have hf_seq_meas : ∀ n, StronglyMeasurable[m] (f_seq n) := fun n =>
    @SimpleFunc.stronglyMeasurable β G m _ (f_seq n)
  have hf_seq_int : ∀ n, Integrable (f_seq n) μ :=
    SimpleFunc.integrable_approxOn_range (hf.mono hm).measurable hf_int
  have hf_seq_int_m : ∀ n, Integrable (f_seq n) (μ.trim hm) := fun n =>
    (hf_seq_int n).trim hm (hf_seq_meas n)
  have hf_seq_eq : ∀ n, ∫ x, f_seq n x ∂μ = ∫ x, f_seq n x ∂μ.trim hm := fun n =>
    integral_trim_simpleFunc hm (f_seq n) (hf_seq_int n)
  have h_lim_1 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)) := by
    refine tendsto_integral_of_L1 f hf_int.1 (Eventually.of_forall hf_seq_int) ?_
    exact SimpleFunc.tendsto_approxOn_range_L1_enorm (hf.mono hm).measurable hf_int
  have h_lim_2 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ.trim hm)) := by
    simp_rw [hf_seq_eq]
    refine @tendsto_integral_of_L1 β G _ _ m (μ.trim hm) _ f (hf_int.trim hm hf).1 _ _
      (Eventually.of_forall hf_seq_int_m) ?_
    exact @SimpleFunc.tendsto_approxOn_range_L1_enorm β G m _ _ _ f _ _ hf.measurable
      (hf_int.trim hm hf)
  exact tendsto_nhds_unique h_lim_1 h_lim_2

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal ≤ (c' * eLpNorm (↑↑g - ↑↑f) 1 μ).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : AEStronglyMeasurable f (μ.trim hm)
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ c'.toReal * (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal ≤ c'.toReal * (ε / 2 / c'.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ c'.toReal * (ε / 2 / c'.toReal) = ε / 2

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : r = 0
this : f =ᵐ[μ] 0
⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ↑r

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
⊢ Continuous fun f ↦ Integrable.toL1 ↑↑f ⋯

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
⊢ ∫ (x : α), -f x ∂μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop (𝓝 0)

Tactic:
rw [Metric.tendsto_nhds] at δseq_lim

Goal state after:
case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : ∀ ε > 0, ∀ᶠ (x : ℕ) in atTop, dist (δseq x) 0 < ε
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ μ' s ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε
n : ℕ
n_large : n ≥ N
⊢ x ∉ thickening ε E

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ thickenedIndicatorAux (δseq n) E) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ (thickenedIndicatorAux δ E x).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ ∃ C, ∀ (x y : α), dist (thickenedIndicatorAux δ E x).toNNReal (thickenedIndicatorAux δ E y).toNNReal ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ thickenedIndicatorAux δ E x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (thickenedIndicator δ_pos E) x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (thickenedIndicator δ_pos E) x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: X : Type u_2
inst✝ : PseudoEMetricSpace X
δ : ℝ
δ_pos : 0 < δ
F : Set X
x : X
hxF : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F
⊢ 1 ≤ (thickenedIndicator δ_pos F) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
x_out : x ∉ thickening δ E
⊢ (thickenedIndicator δ_pos E) x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ (E.indicator fun x ↦ 1) ≤ ⇑(thickenedIndicator δ_pos E)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
x : α
⊢ thickenedIndicatorAux δ₁ E x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
x : α
⊢ thickenedIndicatorAux δ₂ E x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
⊢ ⇑(thickenedIndicator δ₁_pos E) ≤ ⇑(thickenedIndicator δ₂_pos E)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E₁ E₂ : Set α
subset : E₁ ⊆ E₂
x : α
⊢ thickenedIndicatorAux δ E₁ x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E₁ E₂ : Set α
subset : E₁ ⊆ E₂
x : α
⊢ thickenedIndicatorAux δ E₂ x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ (thickenedIndicator δ_pos E) y ≤ (thickenedIndicator δ_pos E) x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
⊢ μ univ = ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
⊢ (fun f ↦ setToFun μ' T hT' ↑↑f) = fun f ↦ setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ ⇑(thickenedIndicator ⋯ E)) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ thickenedIndicatorAux δ E x ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ δ.toNNReal ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ 1 - infEDist x E / ENNReal.ofReal δ - (1 - infEDist y E / ENNReal.ofReal δ) ≤ ↑δ.toNNReal⁻¹ * edist x y

Tactic:
ENNReal.coe_inv (by simp [δ_pos]),

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ 1 - infEDist x E / ENNReal.ofReal δ - (1 - infEDist y E / ENNReal.ofReal δ) ≤ (↑δ.toNNReal)⁻¹ * edist x y
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
⊢ setToFun μ T hT f = setToFun μ' T hT' f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 1 ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 1 - infEDist x E * (↑δ.toNNReal)⁻¹ - (1 - infEDist y E * (↑δ.toNNReal)⁻¹) ≤
    infEDist y E * (↑δ.toNNReal)⁻¹ - infEDist x E * (↑δ.toNNReal)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 0 < infEDist x E → infEDist x E < infEDist y E → (↑δ.toNNReal)⁻¹ ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
⊢ ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
⊢ setToFun μ T hT f = setToFun μ' T hT' f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ - infEDist x E * (↑δ.toNNReal)⁻¹ ≤ (↑δ.toNNReal)⁻¹ * edist x y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
hT_add : DominatedFinMeasAdditive (μ + μ') T C'
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : Integrable f (μ + μ')
⊢ setToFun (μ + μ') T hT_add f = setToFun μ T hT f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
hposbdd : ∫ (ω : α), max (f ω) 0 ∂μ ≤ μ.real univ • ↑r
⊢ 2 * μ univ * ↑r ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ ≤ edist x y * (↑δ.toNNReal)⁻¹ + infEDist x E * (↑δ.toNNReal)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
hT_add : DominatedFinMeasAdditive (μ + μ') T C'
hT : DominatedFinMeasAdditive μ' T C
f : α → E
hf : Integrable f (μ + μ')
⊢ setToFun (μ + μ') T hT_add f = setToFun μ' T hT f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ LipschitzWith δ.toNNReal⁻¹ ⇑(thickenedIndicator δ_pos E)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F
C C' : ℝ
f : α → E
ν : Measure α
hTμ : DominatedFinMeasAdditive μ T C
hTν : DominatedFinMeasAdditive ν T' C'
hμ : Integrable f μ
hν : Integrable f ν
hTμ_add : DominatedFinMeasAdditive (μ + ν) T (max C 0)
hTν_add : DominatedFinMeasAdditive (μ + ν) T' (max C' 0)
⊢ setToFun (μ + ν) T hTμ_add f + setToFun (μ + ν) T' hTν_add f = setToFun μ T hTμ f + setToFun ν T' hTν f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F
C C' : ℝ
f : α → E
ν : Measure α
hTμ : DominatedFinMeasAdditive μ T C
hTν : DominatedFinMeasAdditive ν T' C'
hμ : Integrable f μ
hν : Integrable f ν
⊢ setToFun (μ + ν) (T - T') ⋯ f = setToFun μ T hTμ f - setToFun ν T' hTν f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α → E
ι : Type u_7
s : Finset ι
hs : s.Nonempty
μ : ι → Measure α
T : ι → Set α → E →L[ℝ] F
C : ι → ℝ
hTs : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)
hf : ∀ i ∈ s, Integrable f (μ i)
⊢ setToFun (∑ i ∈ s, μ i) (∑ i ∈ s, T i) ⋯ f = ∑ i ∈ s, setToFun (μ i) (T i) ⋯ f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive (∞ • μ) T C
f : α → E
⊢ setToFun (∞ • μ) T hT f = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
c : ℝ≥0
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
⊢ ↑c ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
c : ℝ≥0
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
I' : DominatedFinMeasAdditive (μ + μ') T' C'
A' : setToFun (μ + μ') T' I' f = setToFun μ' T' hT' f
⊢ setToFun μ'' T'' hT'' f = setToFun μ T hT f + setToFun μ' T' hT' f

Tactic:
have I'' : DominatedFinMeasAdditive (μ + μ') T'' C'' := .of_measure_le hμ hT'' hC''

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
I' : DominatedFinMeasAdditive (μ + μ') T' C'
A' : setToFun (μ + μ') T' I' f = setToFun μ' T' hT' f
I'' : DominatedFinMeasAdditive (μ + μ') T'' C''
⊢ setToFun μ'' T'' hT'' f = setToFun μ T hT f + setToFun μ' T' hT' f