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

Goal state before:
pp: case pos
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
⊢ ∫ (a : α), ↑(f a).toNNReal ∂μ + ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ (2 * μ univ * ↑r).toReal

pp: case pos
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
⊢ Integrable (fun a ↦ ↑(-f a).toNNReal) μ

Tactic:
]

Goal state after:
case pos
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
⊢ ∫ (a : α), ↑(f a).toNNReal ∂μ + ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ (2 * μ univ * ↑r).toReal

case pos
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
⊢ Integrable (fun a ↦ ↑(-f a).toNNReal) μ
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
hc0 : c = 0
hT_smul : DominatedFinMeasAdditive 0 T C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
⊢ 0 = 0

pp: case pos.h
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
hc0 : c = 0
hT_smul : DominatedFinMeasAdditive 0 T C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
⊢ c • μ = 0

Tactic:
]

Goal state after:
case pos.h
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
hc0 : c = 0
hT_smul : DominatedFinMeasAdditive 0 T C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
⊢ c • μ = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg.refine_2
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
hc0 : ¬c = 0
⊢ μ = μ

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), max (-f a) 0 ∂μ ≤ ∫ (a : α), max (f a) 0 ∂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
⊢ ∫ (a : α), max (f a) 0 ∂μ + ∫ (a : α), max (f a) 0 ∂μ ≤ 2 * (μ univ).toReal * ↑r

Tactic:
]

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), max (-f a) 0 ∂μ ≤ ∫ (a : α), max (f a) 0 ∂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
⊢ ∫ (a : α), max (f a) 0 ∂μ + ∫ (a : α), max (f a) 0 ∂μ ≤ 2 * (μ univ).toReal * ↑r
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), max (-f a) 0 ∂μ ≤ ∫ (a : α), max (f a) 0 ∂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
⊢ ∫ (a : α), max (f a) 0 ∂μ ≤ (μ univ).toReal * ↑r

Tactic:
]

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), max (-f a) 0 ∂μ ≤ ∫ (a : α), max (f a) 0 ∂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
⊢ ∫ (a : α), max (f a) 0 ∂μ ≤ (μ univ).toReal * ↑r
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
obs : ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ thickening δ E
δ : ℝ
x_notin_thE : x ∉ thickening δ E
a✝ : δ ∈ Ioi 0
⊢ (thickening δ E).mulIndicator f x = (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
simp only [x_notin_thE, not_false_eq_true, mulIndicator_of_notMem, x_mem_closure]

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
I'' : DominatedFinMeasAdditive (μ + μ') T'' C''
A'' : setToFun (μ + μ') T'' I'' f = setToFun μ'' T'' hT'' f
⊢ setToFun (μ + μ') T'' I'' f = setToFun (μ + μ') T I f + setToFun (μ + μ') T' I' f

Tactic:
]

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''
A'' : setToFun (μ + μ') T'' I'' f = setToFun μ'' T'' hT'' f
⊢ setToFun (μ + μ') T'' I'' f = setToFun (μ + μ') T I f + setToFun (μ + μ') T' I' f
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
δ : ℝ
a✝ : δ ∈ univ
obs : x ∈ cthickening δ E
⊢ f x = f x

Tactic:
]

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

Goal state before:
pp: case pos
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 : ↥(Lp E 1 μ)
hC : 0 ≤ C
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) f‖ ≤ C * ‖f‖

Tactic:
]

Goal state after:
case pos
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 : ↥(Lp E 1 μ)
hC : 0 ≤ C
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) f‖ ≤ C * ‖f‖
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (thickening i E).mulIndicator f x) (𝓝[>] 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (thickening i E).mulIndicator f x) (𝓝[>] 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
x : α
⊢ Tendsto (fun x_1 ↦ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x) (𝓝[>] 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
x : α
⊢ Tendsto (fun x_1 ↦ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x) (𝓝[>] 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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 : ↥(Lp E 1 μ)
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) f‖ ≤ max C 0 * ‖f‖

Tactic:
]

Goal state after:
case pos
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 : ↥(Lp E 1 μ)
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) f‖ ≤ max C 0 * ‖f‖
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (cthickening i E).mulIndicator f x) (𝓝 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (cthickening i E).mulIndicator f x) (𝓝 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
x : α
⊢ Tendsto (fun x_1 ↦ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x) (𝓝 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
x : α
⊢ Tendsto (fun x_1 ↦ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x) (𝓝 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hC : 0 ≤ C
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) (Integrable.toL1 f hf)‖ ≤ C * ‖Integrable.toL1 f hf‖

Tactic:
]

Goal state after:
case pos
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hC : 0 ≤ C
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) (Integrable.toL1 f hf)‖ ≤ C * ‖Integrable.toL1 f hf‖
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hst : Disjoint s t
x : (i : δ) → X i
e : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i) ≃ᵐ ((i : ↥(s ∪ t)) → X ↑i) := MeasurableEquiv.piFinsetUnion X hst
⊢ ∫⁻ (a : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i)),
      f
        (updateFinset x (s ∪ t)
          ((MeasurableEquiv.piFinsetUnion X hst) a)) ∂(Measure.pi fun i ↦ μ ↑i).prod (Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (y : ((i : ↥s) → X ↑i) × ((j : ↥t) → X ↑j)),
      f (updateFinset x (s ∪ t) (?m.100 y)) ∂(Measure.pi fun i ↦ μ ↑i).prod (Measure.pi fun j ↦ μ ↑j)

Tactic:
]

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

Goal state before:
pp: case pos
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) (Integrable.toL1 f hf)‖ ≤ max C 0 * ‖Integrable.toL1 f hf‖

Tactic:
]

Goal state after:
case pos
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hF : CompleteSpace F
⊢ ‖(L1.setToL1 hT) (Integrable.toL1 f hf)‖ ≤ max C 0 * ‖Integrable.toL1 f hf‖
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
s t : Finset δ
hst : Disjoint s t
⊢ ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ

Tactic:
]

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
x : (i : δ) → X i
⊢ (fun x ↦ ∫⁻ (xᵢ : X i), (∫⋯∫⁻_s, f ∂μ) (update x i xᵢ) ∂μ i) x = ∫⁻ (xᵢ : X i), (∫⋯∫⁻_s, f ∂μ) (update x i xᵢ) ∂μ i

Tactic:
]

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

Goal state before:
pp: case hbc
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ∂μ).toReal = (∫⁻ (a : α), ‖f a‖ₑ ∂μ).toReal

Tactic:
]

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
⊢ ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ = ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ

Tactic:
]

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

Goal state before:
pp: case h₂
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ∂μ).toReal ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
]

Goal state after:
case h₂
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
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : Integrable f μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ∂μ).toReal ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: case insert
δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
a✝ : δ'
s✝ : Finset δ'
hi : a✝ ∉ s✝
ih :
  ∀ (x : (i : δ) → X i),
    (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
      (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) (update x (e a✝) xᵢ) ∂μ (e a✝) =
    ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) (update ((fun {x_1} ↦ x) ∘' e) a✝ xᵢ) ∂((fun {x} ↦ μ) ∘' e) a✝

Tactic:
]

Goal state after:
case insert
δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
a✝ : δ'
s✝ : Finset δ'
hi : a✝ ∉ s✝
ih :
  ∀ (x : (i : δ) → X i),
    (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
      (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) (update x (e a✝) xᵢ) ∂μ (e a✝) =
    ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) (update ((fun {x_1} ↦ x) ∘' e) a✝ xᵢ) ∂((fun {x} ↦ μ) ∘' e) a✝
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: case insert
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s✝ : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
i : δ
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
y : X i
i' : δ
s : Finset δ
hi' : i' ∉ s
ih : i ∉ s → ∀ (x : (i : δ) → X i), (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) x
hi : i ∉ insert i' s
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∂μ) (update (update x i y) i' xᵢ) ∂μ i' =
    ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) (update x i' xᵢ) ∂μ i'

Tactic:
]

Goal state after:
case insert
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s✝ : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
i : δ
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
y : X i
i' : δ
s : Finset δ
hi' : i' ∉ s
ih : i ∉ s → ∀ (x : (i : δ) → X i), (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) x
hi : i ∉ insert i' s
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∂μ) (update (update x i y) i' xᵢ) ∂μ i' =
    ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) (update x i' xᵢ) ∂μ i'
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
f_int : Integrable f μ
⊢ Tendsto (fun e ↦ ‖Integrable.toL1 (fs e) ⋯ - Integrable.toL1 f f_int‖) atTop (𝓝 0)

Tactic:
]

Goal state after:
case pos
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
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
f_int : Integrable f μ
⊢ Tendsto (fun e ↦ ‖Integrable.toL1 (fs e) ⋯ - Integrable.toL1 f f_int‖) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, g ∂μ ∂μ = ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, g ∂μ ∂μ

Tactic:
]

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, f ∂μ ∂μ ≤ ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, g ∂μ ∂μ

Tactic:
]

Goal state after:
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, f ∂μ ∂μ ≤ ∫⋯∫⁻_t \ s, ∫⋯∫⁻_s, g ∂μ ∂μ