[Merged by Bors] - chore(MeasureTheory/Function/ConditionalExpectation): remove unneeded lpMeasSubgroup_coe and lpMeas_coe

Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean

@@ -215,14 +215,6 @@ theorem mem_lpMeas_self {m0 : MeasurableSpace α} (μ : Measure α) (f : Lp F p
     f ∈ lpMeas F 𝕜 m0 p μ :=
   mem_lpMeas_iff_aeStronglyMeasurable.mpr (Lp.aestronglyMeasurable f)
 
-theorem lpMeasSubgroup_coe {m _ : MeasurableSpace α} {μ : Measure α} {f : lpMeasSubgroup F m p μ} :
-    (f : _ → _) = (f : Lp F p μ) :=
-  rfl
-
-theorem lpMeas_coe {m _ : MeasurableSpace α} {μ : Measure α} {f : lpMeas F 𝕜 m p μ} :
-    (f : _ → _) = (f : Lp F p μ) :=
-  rfl
-
 theorem mem_lpMeas_indicatorConstLp {m m0 : MeasurableSpace α} (hm : m ≤ m0) {μ : Measure α}
     {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {c : F} :
     indicatorConstLp p (hm s hs) hμs c ∈ lpMeas F 𝕜 m p μ :=
@@ -334,7 +326,6 @@ theorem lpMeasSubgroupToLpTrim_left_inv (hm : m ≤ m0) :
   intro f
   ext1
   ext1
-  rw [← lpMeasSubgroup_coe]
   exact (lpTrimToLpMeasSubgroup_ae_eq hm _).trans (lpMeasSubgroupToLpTrim_ae_eq hm _)
 
 theorem lpMeasSubgroupToLpTrim_add (hm : m ≤ m0) (f g : lpMeasSubgroup F m p μ) :
@@ -350,7 +341,6 @@ theorem lpMeasSubgroupToLpTrim_add (hm : m ≤ m0) (f g : lpMeasSubgroup F m p 
       (EventuallyEq.add (lpMeasSubgroupToLpTrim_ae_eq hm f).symm
         (lpMeasSubgroupToLpTrim_ae_eq hm g).symm)
   refine (Lp.coeFn_add _ _).trans ?_
-  simp_rw [lpMeasSubgroup_coe]
   filter_upwards with x using rfl
 
 theorem lpMeasSubgroupToLpTrim_neg (hm : m ≤ m0) (f : lpMeasSubgroup F m p μ) :
@@ -360,7 +350,6 @@ theorem lpMeasSubgroupToLpTrim_neg (hm : m ≤ m0) (f : lpMeasSubgroup F m p μ)
   refine (Lp.stronglyMeasurable _).ae_eq_trim_of_stronglyMeasurable hm (Lp.stronglyMeasurable _).neg
     <| (lpMeasSubgroupToLpTrim_ae_eq hm _).trans <|
     ((Lp.coeFn_neg _).trans ?_).trans  (lpMeasSubgroupToLpTrim_ae_eq hm f).symm.neg
-  simp_rw [lpMeasSubgroup_coe]
   exact Eventually.of_forall fun x => by rfl
 
 theorem lpMeasSubgroupToLpTrim_sub (hm : m ≤ m0) (f g : lpMeasSubgroup F m p μ) :
@@ -384,7 +373,7 @@ theorem lpMeasToLpTrim_smul (hm : m ≤ m0) (c : 𝕜) (f : lpMeas F 𝕜 m p μ
 theorem lpMeasSubgroupToLpTrim_norm_map [hp : Fact (1 ≤ p)] (hm : m ≤ m0)
     (f : lpMeasSubgroup F m p μ) : ‖lpMeasSubgroupToLpTrim F p μ hm f‖ = ‖f‖ := by
   rw [Lp.norm_def, eLpNorm_trim hm (Lp.stronglyMeasurable _),
-    eLpNorm_congr_ae (lpMeasSubgroupToLpTrim_ae_eq hm _), lpMeasSubgroup_coe, ← Lp.norm_def]
+    eLpNorm_congr_ae (lpMeasSubgroupToLpTrim_ae_eq hm _), ← Lp.norm_def]
   congr
 
 theorem isometry_lpMeasSubgroupToLpTrim [hp : Fact (1 ≤ p)] (hm : m ≤ m0) :
@@ -470,7 +459,6 @@ theorem lpMeasToLpTrimLie_symm_indicator [one_le_p : Fact (1 ≤ p)] [NormedSpac
     ((lpMeasToLpTrimLie F ℝ p μ hm).symm (indicatorConstLp p hs hμs c) : Lp F p μ) =
       indicatorConstLp p (hm s hs) ((le_trim hm).trans_lt hμs.lt_top).ne c := by
   ext1
-  rw [← lpMeas_coe]
   change
     lpTrimToLpMeas F ℝ p μ hm (indicatorConstLp p hs hμs c) =ᵐ[μ]
       (indicatorConstLp p _ _ c : α → F)
@@ -482,7 +470,6 @@ theorem lpMeasToLpTrimLie_symm_toLp [one_le_p : Fact (1 ≤ p)] [NormedSpace ℝ
     ((lpMeasToLpTrimLie F ℝ p μ hm).symm (hf.toLp f) : Lp F p μ) =
       (memLp_of_memLp_trim hm hf).toLp f := by
   ext1
-  rw [← lpMeas_coe]
   refine (lpTrimToLpMeas_ae_eq hm _).trans ?_
   exact (ae_eq_of_ae_eq_trim (MemLp.coeFn_toLp hf)).trans (MemLp.coeFn_toLp _).symm
 

Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean

@@ -379,7 +379,7 @@ theorem condExpInd_of_measurable (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞)
   refine EventuallyEq.trans ?_ indicatorConstLp_coeFn.symm
   refine (condExpInd_ae_eq_condExpIndSMul hm (hm s hs) hμs c).trans ?_
   refine (condExpIndSMul_ae_eq_smul hm (hm s hs) hμs c).trans ?_
-  rw [lpMeas_coe, condExpL2_indicator_of_measurable hm hs hμs (1 : ℝ)]
+  rw [condExpL2_indicator_of_measurable hm hs hμs (1 : ℝ)]
   refine (@indicatorConstLp_coeFn α _ _ 2 μ _ s (hm s hs) hμs (1 : ℝ)).mono fun x hx => ?_
   dsimp only
   rw [hx]

Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean

@@ -112,15 +112,15 @@ theorem norm_condExpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) : ‖condExpL2 E
 
 theorem eLpNorm_condExpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) :
     eLpNorm (ε := E) (condExpL2 E 𝕜 hm f) 2 μ ≤ eLpNorm f 2 μ := by
-  rw [lpMeas_coe, ← ENNReal.toReal_le_toReal (Lp.eLpNorm_ne_top _) (Lp.eLpNorm_ne_top _), ←
+  rw [← ENNReal.toReal_le_toReal (Lp.eLpNorm_ne_top _) (Lp.eLpNorm_ne_top _), ←
     Lp.norm_def, ← Lp.norm_def, Submodule.norm_coe]
   exact norm_condExpL2_le hm f
 
 @[deprecated (since := "2025-01-21")] alias eLpNorm_condexpL2_le := eLpNorm_condExpL2_le
 
 theorem norm_condExpL2_coe_le (hm : m ≤ m0) (f : α →₂[μ] E) :
     ‖(condExpL2 E 𝕜 hm f : α →₂[μ] E)‖ ≤ ‖f‖ := by
-  rw [Lp.norm_def, Lp.norm_def, ← lpMeas_coe]
+  rw [Lp.norm_def, Lp.norm_def]
   exact ENNReal.toReal_mono (Lp.eLpNorm_ne_top _) (eLpNorm_condExpL2_le hm f)
 
 @[deprecated (since := "2025-01-21")] alias norm_condexpL2_coe_le := norm_condExpL2_coe_le
@@ -209,7 +209,6 @@ theorem condExpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ
       rw [Pi.zero_apply] at hx ⊢
       · rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
     · refine Measurable.coe_nnreal_ennreal (Measurable.nnnorm ?_)
-      rw [lpMeas_coe]
       exact (Lp.stronglyMeasurable _).measurable
   refine le_antisymm ?_ (zero_le _)
   refine (lintegral_nnnorm_condExpL2_le hs hμs f).trans (le_of_eq ?_)
@@ -251,9 +250,8 @@ linear maps. -/
 theorem condExpL2_const_inner (hm : m ≤ m0) (f : Lp E 2 μ) (c : E) :
     condExpL2 𝕜 𝕜 hm (((Lp.memLp f).const_inner c).toLp fun a => ⟪c, f a⟫) =ᵐ[μ]
     fun a => ⟪c, (condExpL2 E 𝕜 hm f : α → E) a⟫ := by
-  rw [lpMeas_coe]
   have h_mem_Lp : MemLp (fun a => ⟪c, (condExpL2 E 𝕜 hm f : α → E) a⟫) 2 μ := by
-    refine MemLp.const_inner _ ?_; rw [lpMeas_coe]; exact Lp.memLp _
+    refine MemLp.const_inner _ ?_; exact Lp.memLp _
   have h_eq : h_mem_Lp.toLp _ =ᵐ[μ] fun a => ⟪c, (condExpL2 E 𝕜 hm f : α → E) a⟫ :=
     h_mem_Lp.coeFn_toLp
   refine EventuallyEq.trans ?_ h_eq
@@ -263,14 +261,14 @@ theorem condExpL2_const_inner (hm : m ≤ m0) (f : Lp E 2 μ) (c : E) :
     rw [IntegrableOn, integrable_congr (ae_restrict_of_ae h_eq)]
     exact (integrableOn_condExpL2_of_measure_ne_top hm hμs.ne _).const_inner _
   · intro s hs hμs
-    rw [← lpMeas_coe, integral_condExpL2_eq_of_fin_meas_real _ hs hμs.ne,
-      integral_congr_ae (ae_restrict_of_ae h_eq), lpMeas_coe, ←
+    rw [integral_condExpL2_eq_of_fin_meas_real _ hs hμs.ne,
+      integral_congr_ae (ae_restrict_of_ae h_eq), ←
       L2.inner_indicatorConstLp_eq_setIntegral_inner 𝕜 (↑(condExpL2 E 𝕜 hm f)) (hm s hs) c hμs.ne,
       ← inner_condExpL2_left_eq_right, condExpL2_indicator_of_measurable _ hs,
       L2.inner_indicatorConstLp_eq_setIntegral_inner 𝕜 f (hm s hs) c hμs.ne,
       setIntegral_congr_ae (hm s hs)
         ((MemLp.coeFn_toLp ((Lp.memLp f).const_inner c)).mono fun x hx _ => hx)]
-  · rw [← lpMeas_coe]; exact lpMeas.aeStronglyMeasurable _
+  · exact lpMeas.aeStronglyMeasurable _
   · refine AEStronglyMeasurable.congr ?_ h_eq.symm
     exact (lpMeas.aeStronglyMeasurable _).const_inner
 
@@ -279,7 +277,7 @@ theorem condExpL2_const_inner (hm : m ≤ m0) (f : Lp E 2 μ) (c : E) :
 /-- `condExpL2` verifies the equality of integrals defining the conditional expectation. -/
 theorem integral_condExpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
     (hμs : μ s ≠ ∞) : ∫ x in s, (condExpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ := by
-  rw [← sub_eq_zero, lpMeas_coe, ←
+  rw [← sub_eq_zero, ←
     integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)
       (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs)]
   refine integral_eq_zero_of_forall_integral_inner_eq_zero 𝕜 _ ?_ ?_
@@ -291,7 +289,7 @@ theorem integral_condExpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableS
       ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)
       ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs).const_inner c)]
   have h_ae_eq_f := MemLp.coeFn_toLp (E := 𝕜) ((Lp.memLp f).const_inner c)
-  rw [← lpMeas_coe, sub_eq_zero, ←
+  rw [sub_eq_zero, ←
     setIntegral_congr_ae (hm s hs) ((condExpL2_const_inner hm f c).mono fun x hx _ => hx), ←
     setIntegral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]
   exact integral_condExpL2_eq_of_fin_meas_real _ hs hμs
@@ -313,11 +311,11 @@ theorem condExpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E''
     rw [T.setIntegral_compLp _ (hm s hs),
       T.integral_comp_comm
         (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim hμs.ne),
-      ← lpMeas_coe, ← lpMeas_coe, integral_condExpL2_eq hm f hs hμs.ne,
+      integral_condExpL2_eq hm f hs hμs.ne,
       integral_condExpL2_eq hm (T.compLp f) hs hμs.ne, T.setIntegral_compLp _ (hm s hs),
       T.integral_comp_comm
         (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs.ne)]
-  · rw [← lpMeas_coe]; exact lpMeas.aeStronglyMeasurable _
+  · exact lpMeas.aeStronglyMeasurable _
   · have h_coe := T.coeFn_compLp (condExpL2 E' 𝕜 hm f : α →₂[μ] E')
     rw [← EventuallyEq] at h_coe
     refine AEStronglyMeasurable.congr ?_ h_coe.symm
@@ -340,7 +338,6 @@ theorem condExpL2_indicator_ae_eq_smul (hm : m ≤ m0) (hs : MeasurableSet s) (h
   have h_comp :=
     condExpL2_comp_continuousLinearMap ℝ 𝕜 hm (toSpanSingleton ℝ x)
       (indicatorConstLp 2 hs hμs (1 : ℝ))
-  rw [← lpMeas_coe] at h_comp
   refine h_comp.trans ?_
   exact (toSpanSingleton ℝ x).coeFn_compLp _
 
@@ -351,7 +348,6 @@ theorem condExpL2_indicator_eq_toSpanSingleton_comp (hm : m ≤ m0) (hs : Measur
     (hμs : μ s ≠ ∞) (x : E') : (condExpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) : α →₂[μ] E') =
     (toSpanSingleton ℝ x).compLp (condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1)) := by
   ext1
-  rw [← lpMeas_coe]
   refine (condExpL2_indicator_ae_eq_smul 𝕜 hm hs hμs x).trans ?_
   have h_comp := (toSpanSingleton ℝ x).coeFn_compLp
     (condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α →₂[μ] ℝ)
@@ -375,7 +371,7 @@ theorem setLIntegral_nnnorm_condExpL2_indicator_le (hm : m ≤ m0) (hs : Measura
         ((condExpL2_indicator_ae_eq_smul 𝕜 hm hs hμs x).mono fun a ha _ => by rw [ha])
     _ = (∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ) * ‖x‖₊ := by
       simp_rw [nnnorm_smul, ENNReal.coe_mul]
-      rw [lintegral_mul_const, lpMeas_coe]
+      rw [lintegral_mul_const]
       exact (Lp.stronglyMeasurable _).enorm
     _ ≤ μ (s ∩ t) * ‖x‖₊ :=
       mul_le_mul_right' (lintegral_nnnorm_condExpL2_indicator_le_real hs hμs ht hμt) _
@@ -401,7 +397,7 @@ theorem integrable_condExpL2_indicator (hm : m ≤ m0) [SigmaFinite (μ.trim hm)
     Integrable (ε := E') (condExpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x)) μ := by
   refine integrable_of_forall_fin_meas_le' hm (μ s * ‖x‖₊)
     (ENNReal.mul_lt_top hμs.lt_top ENNReal.coe_lt_top) ?_ ?_
-  · rw [lpMeas_coe]; exact Lp.aestronglyMeasurable _
+  · exact Lp.aestronglyMeasurable _
   · refine fun t ht hμt =>
       (setLIntegral_nnnorm_condExpL2_indicator_le hm hs hμs x ht hμt).trans ?_
     gcongr
@@ -474,7 +470,7 @@ theorem setLIntegral_nnnorm_condExpIndSMul_le (hm : m ≤ m0) (hs : MeasurableSe
         ((condExpIndSMul_ae_eq_smul hm hs hμs x).mono fun a ha _ => by rw [ha])
     _ = (∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ) * ‖x‖₊ := by
       simp_rw [nnnorm_smul, ENNReal.coe_mul]
-      rw [lintegral_mul_const, lpMeas_coe]
+      rw [lintegral_mul_const]
       exact (Lp.stronglyMeasurable _).enorm
     _ ≤ μ (s ∩ t) * ‖x‖₊ :=
       mul_le_mul_right' (lintegral_nnnorm_condExpL2_indicator_le_real hs hμs ht hμt) _