Theorem: MeasureTheory.set_integral_gt_gt
File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

tactic: rw [← sub_pos, ← smul_eq_mul, ← set_integral_const, ← integral_sub hfint this,
  set_integral_pos_iff_support_of_nonneg_ae]
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ ENNReal.toReal (↑↑μ {x | R < f x}) * R < ∫ (x : α) in {x | R < f x}, f x ∂μ

tactic: refine' ⟨aestronglyMeasurable_const, lt_of_le_of_lt _ hfint.2⟩
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
⊢ IntegrableOn (fun x => R) {x | R < f x}

tactic: refine'
  set_lintegral_mono (Measurable.nnnorm _).coe_nnreal_ennreal hfm.nnnorm.coe_nnreal_ennreal
    fun x hx => _
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
⊢ (∫⁻ (a : α) in {x | R < f x}, ↑‖(fun x => R) a‖₊ ∂μ) ≤ ∫⁻ (a : α) in {x | R < f x}, ↑‖f a‖₊ ∂μ

tactic: exact measurable_const
state_before: case refine'_1
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
⊢ Measurable fun a => (fun x => R) a

tactic: simp only [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg hR,
  Real.nnnorm_of_nonneg (hR.trans <| le_of_lt hx), Subtype.mk_le_mk]
state_before: case refine'_2
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
x : α
hx : x ∈ {x | R < f x}
⊢ ↑‖(fun x => R) x‖₊ ≤ ↑‖f x‖₊

tactic: exact le_of_lt hx
state_before: case refine'_2
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
x : α
hx : x ∈ {x | R < f x}
⊢ { val := R, property := hR } ≤ { val := f x, property := (_ : 0 ≤ f x) }

tactic: rw [← zero_lt_iff] at hμ
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})

tactic: rwa [Set.inter_eq_self_of_subset_right]
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : 0 < ↑↑μ {x | R < f x}
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ 0 < ↑↑μ ((support fun a => f a - R) ∩ {x | R < f x})

tactic: exact fun x hx => Ne.symm (ne_of_lt <| sub_pos.2 hx)
state_before: α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : 0 < ↑↑μ {x | R < f x}
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ {x | R < f x} ⊆ support fun a => f a - R

tactic: change ∀ᵐ x ∂μ.restrict _, _
state_before: case hf
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ 0 ≤ᵐ[Measure.restrict μ {x | R < f x}] fun a => f a - R

tactic: rw [ae_restrict_iff]
state_before: case hf
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | R < f x}, OfNat.ofNat 0 x ≤ (fun a => f a - R) x

tactic: exact eventually_of_forall fun x hx => sub_nonneg.2 <| le_of_lt hx
state_before: case hf
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | R < f x} → OfNat.ofNat 0 x ≤ (fun a => f a - R) x

tactic: exact measurableSet_le measurable_zero (hfm.sub measurable_const)
state_before: case hf
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ MeasurableSet {x | OfNat.ofNat 0 x ≤ (fun a => f a - R) x}

tactic: exact Integrable.sub hfint this
state_before: case hfi
α : Type u_1
β : Type ?u.264063
E : Type ?u.264066
F : Type ?u.264069
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
f✝ g : α → E
s t : Set α
μ ν : Measure α
l l' : Filter α
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
R : ℝ
f : α → ℝ
hR : 0 ≤ R
hfm : Measurable f
hfint : IntegrableOn f {x | R < f x}
hμ : ↑↑μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x}
⊢ IntegrableOn (fun a => f a - R) {x | R < f x}