theorem lintegral_nnnorm_condexpL2_le (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) (f : Lp ℝ 2 μ) :
    ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ := by
  let h_meas := lpMeas.aeStronglyMeasurable' (condexpL2 ℝ ℝ hm f)
  let g := h_meas.choose
  have hg_meas : StronglyMeasurable[m] g := h_meas.choose_spec.1
  have hg_eq : g =ᵐ[μ] condexpL2 ℝ ℝ hm f := h_meas.choose_spec.2.symm
  have hg_eq_restrict : g =ᵐ[μ.restrict s] condexpL2 ℝ ℝ hm f := ae_restrict_of_ae hg_eq
  have hg_nnnorm_eq : (fun x => (‖g x‖₊ : ℝ≥0∞)) =ᵐ[μ.restrict s] fun x =>
      (‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ : ℝ≥0∞) := by
    refine' hg_eq_restrict.mono fun x hx => _
    dsimp only
    simp_rw [hx]
  rw [lintegral_congr_ae hg_nnnorm_eq.symm]
  refine'
    lintegral_nnnorm_le_of_forall_fin_meas_integral_eq hm (Lp.stronglyMeasurable f) _ _ _ _ hs hμs
  · exact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs
  · exact hg_meas
  · rw [IntegrableOn, integrable_congr hg_eq_restrict]
    exact integrableOn_condexpL2_of_measure_ne_top hm hμs f
  · intro t ht hμt
    rw [← integral_condexpL2_eq_of_fin_meas_real f ht hμt.ne]
    

Proof state:
case refine'_4
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
h_meas : AEStronglyMeasurable' m (↑↑↑((condexpL2 ℝ ℝ hm) f)) μ := lpMeas.aeStronglyMeasurable' ((condexpL2 ℝ ℝ hm) f)
g : α → ℝ := Exists.choose h_meas
hg_meas : StronglyMeasurable g
hg_eq : g =ᵐ[μ] ↑↑↑((condexpL2 ℝ ℝ hm) f)
hg_eq_restrict : g =ᵐ[Measure.restrict μ s] ↑↑↑((condexpL2 ℝ ℝ hm) f)
hg_nnnorm_eq : (fun x => ↑‖g x‖₊) =ᵐ[Measure.restrict μ s] fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t < ⊤
⊢ ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, ↑↑↑((condexpL2 ℝ ℝ ?m.655885) f) x ∂μ
α : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹³ : IsROrC 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : InnerProductSpace 𝕜 E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
inst✝⁷ : CompleteSpace E'
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
f : ↥(Lp ℝ 2)
h_meas : AEStronglyMeasurable' m (↑↑↑((condexpL2 ℝ ℝ hm) f)) μ := lpMeas.aeStronglyMeasurable' ((condexpL2 ℝ ℝ hm) f)
g : α → ℝ := Exists.choose h_meas
hg_meas : StronglyMeasurable g
hg_eq : g =ᵐ[μ] ↑↑↑((condexpL2 ℝ ℝ hm) f)
hg_eq_restrict : g =ᵐ[Measure.restrict μ s] ↑↑↑((condexpL2 ℝ ℝ hm) f)
hg_nnnorm_eq : (fun x => ↑‖g x‖₊) =ᵐ[Measure.restrict μ s] fun x => ↑‖↑↑↑((condexpL2 ℝ ℝ hm) f) x‖₊
t : Set α
ht : MeasurableSet t
hμt : ↑↑μ t < ⊤
⊢ m ≤ m0

Next tactic:
exact set_integral_congr_ae (hm t ht) (hg_eq.mono fun x hx _ => hx)