. rw [integrableOn_Icc_iff_integrableOn_Ioc]
  exact hg_int.left
case refine_1
a b : ℝ
hab : a ≤ b
f g : ℝ → ℝ
μ : Measure ℝ
inst✝¹ : NoAtoms μ
inst✝ : IsLocallyFiniteMeasure μ
hf_cont : ContinuousOn f (Set.Icc a b)
hg_int : IntervalIntegrable g μ a b
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Icc a b)] g
h_int : IntervalIntegrable (fun x => f x * g x) μ a b
⊢ IntegrableOn (fun x => g x) (Set.Icc a b)

case refine_2
a b : ℝ
hab : a ≤ b
f g : ℝ → ℝ
μ : Measure ℝ
inst✝¹ : NoAtoms μ
inst✝ : IsLocallyFiniteMeasure μ
hf_cont : ContinuousOn f (Set.Icc a b)
hg_int : IntervalIntegrable g μ a b
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Icc a b)] g
h_int : IntervalIntegrable (fun x => f x * g x) μ a b
⊢ IntegrableOn (fun x => f x * g x) (Set.Icc a b)
case refine_2
a b : ℝ
hab : a ≤ b
f g : ℝ → ℝ
μ : Measure ℝ
inst✝¹ : NoAtoms μ
inst✝ : IsLocallyFiniteMeasure μ
hf_cont : ContinuousOn f (Set.Icc a b)
hg_int : IntervalIntegrable g μ a b
hg_nonneg : 0 ≤ᵐ[Measure.restrict μ (Set.Icc a b)] g
h_int : IntervalIntegrable (fun x => f x * g x) μ a b
⊢ IntegrableOn (fun x => f x * g x) (Set.Icc a b)