mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
x : ((i : ι) → β (some i)) × β none
⊢ (∀ (i : Option ι), e_meas x i ∈ s i) → (∀ (i : ι), x.1 i ∈ s (some i)) ∧ x.2 ∈ s none

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
c : ε
⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ ≤ ‖c‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
⊢ ⇑e_meas ⁻¹' univ.pi s = (univ.pi fun i ↦ s (some i)) ×ˢ s none

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
⊢ map (⇑(MeasurableEquiv.piOptionEquivProd β).symm) ((Measure.pi fun i ↦ μ (some i)).prod (μ none)) = Measure.pi μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
⊢ map (⇑(MeasurableEquiv.piOptionEquivProd β).symm) ((Measure.pi fun i ↦ μ (some i)).prod (μ none)) = Measure.pi μ

Tactic:
by
  refine pi_eq (fun s _ ↦ ?_) |>.symm
  let e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) :=
    MeasurableEquiv.piOptionEquivProd β |>.symm
  have me := MeasurableEquiv.measurableEmbedding e_meas
  have : e_meas ⁻¹' pi univ s = (pi univ (fun i ↦ s (some i))) ×ˢ (s none) := by
    ext x
    simp only [mem_preimage, Set.mem_pi, mem_univ, forall_true_left, mem_prod]
    refine ⟨by tauto, fun _ i ↦ ?_⟩
    rcases i <;> tauto
  simp only [e_meas, me.map_apply, univ_option, Finset.prod_insertNone, this,
    prod_prod, pi_pi, mul_comm]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f : (i : ι) → α i
⊢ (univ.pi fun i ↦ Iio (f i)) =ᵐ[Measure.pi μ] Iic f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
c : ε
hμs : μ s ≠ 0
⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ = ‖c‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ (∫⁻ (x : α), ‖s.indicator (fun x ↦ c) x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) =
    (∫⁻ (x : α), s.indicator (fun x ↦ ‖c‖ₑ ^ p.toReal) x ∂μ) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f : (i : ι) → α i
⊢ (univ.pi fun i ↦ Ioi (f i)) =ᵐ[Measure.pi μ] Ici f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ (∫⁻ (x : α), s.indicator (fun x ↦ ‖c‖ₑ ^ p.toReal) x ∂μ) ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i ↦ Ioo (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i ↦ Ioc (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ 0
hp : p ≠ 0
⊢ eLpNorm (s.indicator fun x ↦ c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i ↦ Ico (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), IsLocallyFiniteMeasure (μ i)
⊢ IsLocallyFiniteMeasure (Measure.pi μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
p : ℝ≥0∞
hp : p ≠ 0
h'p : p ≠ ∞
t : Set α := toMeasurable μ s
⊢ ‖c‖ₑ * μ t ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
p : ℝ≥0∞
⊢ eLpNorm (s.indicator fun x ↦ c) p μ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
hs : MeasurableSet s
hf : MemLp f p μ
⊢ eLpNorm f p μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
hs : MeasurableSet s
⊢ MemLp (s.indicator f) p μ ↔ MemLp f p (μ.restrict s)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
c : E
hμsc : c = 0 ∨ μ s ≠ ∞
⊢ MemLp (s.indicator fun x ↦ c) p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ eLpNormEssSup (s.piecewise f g) μ = max (eLpNormEssSup f (μ.restrict s)) (eLpNormEssSup g (μ.restrict sᶜ))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ
hp : 0 < p
⊢ eLpNorm' f p μ ≤ c • eLpNorm' g p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι✝ : Type u_1
ι' : Type u_2
α : ι✝ → Type u_3
inst✝⁷ : Fintype ι✝
m : (i : ι✝) → OuterMeasure (α i)
inst✝⁶ : (i : ι✝) → MeasurableSpace (α i)
μ✝ : (i : ι✝) → Measure (α i)
inst✝⁵ : ∀ (i : ι✝), SigmaFinite (μ✝ i)
ι : Type u_4
inst✝⁴ : Fintype ι
X : ι → Type u_5
inst✝³ : (i : ι) → PseudoMetricSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : ∀ (i : ι), IsUnifLocDoublingMeasure (μ i)
⊢ IsUnifLocDoublingMeasure (Measure.pi μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: case neg.inr
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
⊢ eLpNorm (s.piecewise f g) p μ < ∞

Tactic:
eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top hp_zero hp_top,

Goal state after:
case neg.inr
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
⊢ ∫⁻ (a : α), ‖s.piecewise f g a‖ₑ ^ p.toReal ∂μ < ∞
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableMul (α i)
inst✝ : ∀ (i : ι), (μ i).IsMulLeftInvariant
⊢ (Measure.pi μ).IsMulLeftInvariant

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
x : α
⊢ ¬p < 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
x : α
⊢ ↑c ^ p * ‖g x‖ₑ ^ p = (↑c * ‖g x‖ₑ) ^ p

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableMul (α i)
inst✝ : ∀ (i : ι), (μ i).IsMulRightInvariant
⊢ (Measure.pi μ).IsMulRightInvariant

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
⊢ eLpNorm' f p μ ≤ c • eLpNorm' g p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
⊢ ¬p < 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableInv (α i)
inst✝ : ∀ (i : ι), (μ i).IsInvInvariant
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ Inv.inv ⁻¹' univ.pi s = univ.pi fun i ↦ Inv.inv ⁻¹' s i

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
x : α
hx : x ∈ s
⊢ ‖s.piecewise f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : c = ∞
hg' : ∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ = 0
⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ ^ p) μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
x : α
hx : x ∈ sᶜ
⊢ ‖s.piecewise f g x‖ₑ ^ p.toReal = ‖g x‖ₑ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : c = ∞
hg' : (fun a ↦ ‖g a‖ₑ ^ p) =ᵐ[μ] 0
x : α
hx : (fun a ↦ ‖g a‖ₑ ^ p) x = 0 x
⊢ ‖g x‖ₑ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : c = ∞
hg' : eLpNorm' g p μ = 0
⊢ ∀ᵐ (x : α) ∂μ, ‖g x‖ₑ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h✝ : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : c = ∞
hg' : eLpNorm' g p μ = 0
this : ∀ᵐ (x : α) ∂μ, ‖g x‖ₑ = 0
x : α
x✝ : ‖g x‖ₑ = 0 ∧ ‖f x‖ₑ ≤ c * ‖g x‖ₑ
h : ‖g x‖ₑ = 0
h' : ‖f x‖ₑ ≤ c * ‖g x‖ₑ
⊢ ‖f x‖ₑ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁵ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝³ : ∀ (i : ι), SigmaFinite (μ i)
inst✝² : (i : ι) → Group (α i)
inst✝¹ : ∀ (i : ι), MeasurableInv (α i)
inst✝ : ∀ (i : ι), (μ i).IsInvInvariant
⊢ (Measure.pi μ).IsInvInvariant

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
⊢ MemLp (s.piecewise f g) p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), (μ i).IsOpenPosMeasure
⊢ (Measure.pi μ).IsOpenPosMeasure

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : ¬c = ∞
⊢ c ^ p ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hq0_lt : 0 < q
hfq : eLpNorm' f q μ < ∞
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), IsFiniteMeasureOnCompacts (μ i)
K : Set ((i : ι) → α i)
hK : IsCompact K
this : (Measure.pi μ) (univ.pi fun j ↦ eval j '' K) < ∞
⊢ (Measure.pi μ) K < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hfp : eLpNorm f p μ < ∞
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), IsFiniteMeasureOnCompacts (μ i)
⊢ IsFiniteMeasureOnCompacts (Measure.pi μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
s : Set α
β : Type u_9
inst✝ : NormedAddCommGroup β
μ : Measure α
hp' : p ≠ ∞
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
hp : ¬p = 0
⊢ ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x ↦ c) x‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞ → eLpNorm f p μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
⊢ eLpNorm' f 0 μ = 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
f : α → ε
⊢ eLpNorm f 0 μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : ENorm ε
inst✝ : TopologicalSpace ε
f : α → ε
⊢ MemLp f 0 μ ↔ AEStronglyMeasurable f μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
hp0_lt : 0 < q
⊢ eLpNorm' 0 q μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
s : Set α
β : Type u_9
inst✝ : NormedAddCommGroup β
μ : Measure α
hp' : p ≠ ∞
hs : MeasurableSet s
f g : α → β
c : ℝ
hc : 0 ≤ c
hf : ∀ x ∈ s, dist (f x) (g x) ≤ c
⊢ eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
hq0_ne : q ≠ 0
hμ : μ ≠ 0
⊢ eLpNorm' 0 q μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
⊢ eLpNormEssSup 0 μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
p : ι → Prop
inst✝ : DecidablePred p
⊢ MeasurePreserving (⇑(MeasurableEquiv.piEquivPiSubtypeProd α p)) (Measure.pi μ)
    ((Measure.pi fun i ↦ μ ↑i).prod (Measure.pi fun i ↦ μ ↑i))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε' : Type u_6
inst✝³ : TopologicalSpace ε'
inst✝² : ContinuousENorm ε'
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
g : α → ε'
p : ℝ
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ
hp : 0 < p
hp' : ¬p < 0
hc : ¬c = ∞
this : c ^ p ≠ ∞
x : α
⊢ c ^ p * ‖g x‖ₑ ^ p = (c * ‖g x‖ₑ) ^ p

Tactic:
by

Goal state after: