Theorem: MeasureTheory.integral_eq_zero_iff_of_nonneg_ae
File: Mathlib/MeasureTheory/Integral/Bochner.lean

tactic: simp_rw [integral_eq_lintegral_of_nonneg_ae hf hfi.1, ENNReal.toReal_eq_zero_iff,
  ← ENNReal.not_lt_top, ← hasFiniteIntegral_iff_ofReal hf, hfi.2, not_true, or_false_iff]
state_before: α : Type u_1
E : Type ?u.1106298
F : Type ?u.1106301
𝕜 : Type ?u.1106304
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1108995
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f
⊢ (∫ (x : α), f x ∂μ) = 0 ↔ f =ᵐ[μ] 0

tactic: rw [lintegral_eq_zero_iff']
state_before: α : Type u_1
E : Type ?u.1106298
F : Type ?u.1106301
𝕜 : Type ?u.1106304
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1108995
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f
⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ) = 0 ↔ f =ᵐ[μ] 0

tactic: rw [← hf.le_iff_eq, Filter.EventuallyEq, Filter.EventuallyLE]
state_before: α : Type u_1
E : Type ?u.1106298
F : Type ?u.1106301
𝕜 : Type ?u.1106304
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1108995
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f
⊢ (fun a => ENNReal.ofReal (f a)) =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0

tactic: simp only [Pi.zero_apply, ofReal_eq_zero]
state_before: α : Type u_1
E : Type ?u.1106298
F : Type ?u.1106301
𝕜 : Type ?u.1106304
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1108995
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f
⊢ (∀ᵐ (x : α) ∂μ, ENNReal.ofReal (f x) = OfNat.ofNat 0 x) ↔ ∀ᵐ (x : α) ∂μ, f x ≤ OfNat.ofNat 0 x

tactic: exact (ENNReal.measurable_ofReal.comp_aemeasurable hfi.1.aemeasurable)
state_before: α : Type u_1
E : Type ?u.1106298
F : Type ?u.1106301
𝕜 : Type ?u.1106304
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1108995
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f
⊢ AEMeasurable fun a => ENNReal.ofReal (f a)