@@ -505,7 +505,7 @@ theorem integral_eq_lintegral_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f
505505theorem integral_norm_eq_lintegral_enorm {P : Type *} [NormedAddCommGroup P] {f : α → P}
506506 (hf : AEStronglyMeasurable f μ) : ∫ x, ‖f x‖ ∂μ = (∫⁻ x, ‖f x‖ₑ ∂μ).toReal := by
507507 rw [integral_eq_lintegral_of_nonneg_ae _ hf.norm]
508- · simp_rw [ofReal_norm_eq_enorm ]
508+ · simp_rw [ofReal_norm ]
509509 · filter_upwards; simp_rw [Pi.zero_apply, norm_nonneg, imp_true_iff]
510510
511511theorem ofReal_integral_norm_eq_lintegral_enorm {P : Type *} [NormedAddCommGroup P] {f : α → P}
@@ -689,7 +689,7 @@ theorem ofReal_integral_eq_lintegral_ofReal {f : α → ℝ} (hfi : Integrable f
689689 ENNReal.ofReal (∫ x, f x ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
690690 have : f =ᵐ[μ] (‖f ·‖) := f_nn.mono fun _x hx ↦ (abs_of_nonneg hx).symm
691691 simp_rw [integral_congr_ae this, ofReal_integral_norm_eq_lintegral_enorm hfi,
692- ← ofReal_norm_eq_enorm ]
692+ ← ofReal_norm ]
693693 exact lintegral_congr_ae (this.symm.fun_comp ENNReal.ofReal)
694694
695695theorem integral_toReal {f : α → ℝ≥0 ∞} (hfm : AEMeasurable f μ) (hf : ∀ᵐ x ∂μ, f x < ∞) :
@@ -907,7 +907,7 @@ theorem MemLp.eLpNorm_eq_integral_rpow_norm {f : α → H} {p : ℝ≥0∞} (hp1
907907 (hf : MemLp f p μ) :
908908 eLpNorm f p μ = ENNReal.ofReal ((∫ a, ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹) := by
909909 have A : ∫⁻ a : α, ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ a : α, ‖f a‖ₑ ^ p.toReal ∂μ := by
910- simp_rw [← ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm_eq_enorm ]
910+ simp_rw [← ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm ]
911911 simp only [eLpNorm_eq_lintegral_rpow_enorm_toReal hp1 hp2, one_div]
912912 rw [integral_eq_lintegral_of_nonneg_ae]; rotate_left
913913 · exact ae_of_all _ fun x => by positivity
@@ -1145,13 +1145,13 @@ theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α →
11451145 -- replace norms by nnnorm
11461146 have h_left : ∫⁻ a, ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ =
11471147 ∫⁻ a, ((‖f ·‖ₑ) * (‖g ·‖ₑ)) a ∂μ := by
1148- simp_rw [Pi.mul_apply, ← ofReal_norm_eq_enorm , ENNReal.ofReal_mul (norm_nonneg _)]
1148+ simp_rw [Pi.mul_apply, ← ofReal_norm , ENNReal.ofReal_mul (norm_nonneg _)]
11491149 have h_right_f : ∫⁻ a, .ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ a, ‖f a‖ₑ ^ p ∂μ := by
11501150 refine lintegral_congr fun x => ?_
1151- rw [← ofReal_norm_eq_enorm , ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg]
1151+ rw [← ofReal_norm , ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg]
11521152 have h_right_g : ∫⁻ a, .ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ a, ‖g a‖ₑ ^ q ∂μ := by
11531153 refine lintegral_congr fun x => ?_
1154- rw [← ofReal_norm_eq_enorm , ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg]
1154+ rw [← ofReal_norm , ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg]
11551155 rw [h_left, h_right_f, h_right_g]
11561156 -- we can now apply `ENNReal.lintegral_mul_le_Lp_mul_Lq` (up to the `toReal` application)
11571157 refine ENNReal.toReal_mono ?_ ?_
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