/-- The map `u ↦ f • u` is an isometry between the `L^1` spaces for `μ.withDensity f` and `μ`. -/
noncomputable def withDensitySMulLI {f : α → ℝ≥0} (f_meas : Measurable f) :
    Lp E 1 (μ.withDensity fun x => f x) →ₗᵢ[ℝ] Lp E 1 μ where
  toFun u := (memℒ1_smul_of_L1_withDensity f_meas u).toLp _
  map_add' := by
    intro u v
    ext1
    filter_upwards [(memℒ1_smul_of_L1_withDensity f_meas u).coeFn_toLp,
      (memℒ1_smul_of_L1_withDensity f_meas v).coeFn_toLp,
      (memℒ1_smul_of_L1_withDensity f_meas (u + v)).coeFn_toLp,
      Lp.coeFn_add ((memℒ1_smul_of_L1_withDensity f_meas u).toLp _)
        ((memℒ1_smul_of_L1_withDensity f_meas v).toLp _),
      (ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 (Lp.coeFn_add u v)]
    intro x hu hv huv h' h''
    rw [huv, h', Pi.add_apply, hu, hv]
    rcases eq_or_ne (f x) 0 with (hx | hx)
    · simp only [hx, zero_smul, add_zero]
    · rw [h'' _, Pi.add_apply, smul_add]
      simpa only [Ne.def, ENNReal.coe_eq_zero] using hx
  map_smul' := by
    intro r u
    ext1
    filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 (Lp.coeFn_smul r u),
      (memℒ1_smul_of_L1_withDensity f_meas (r • u)).coeFn_toLp,
      Lp.coeFn_smul r ((memℒ1_smul_of_L1_withDensity f_meas u).toLp _),
      (memℒ1_smul_of_L1_withDensity f_meas u).coeFn_toLp]
    intro x h h' h'' h'''
    rw [RingHom.id_apply, h', h'', Pi.smul_apply, h''']
    rcases eq_or_ne (f x) 0 with (hx | hx)
    · simp only [hx, zero_smul, smul_zero]
    · rw [h _, smul_comm, Pi.smul_apply]
      simpa only [Ne.def, ENNReal.coe_eq_zero] using hx
  norm_map' := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : MeasurableSpace δ
inst✝³ : NormedAddCommGroup β
inst✝² : NormedAddCommGroup γ
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
⊢ ∀ (x : ↥(Lp E 1)),
    ‖{
            toAddHom :=
              { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
                map_add' :=
                  (_ :
                    ∀ (u v : ↥(Lp E 1)),
                      (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) (u + v) =
                        (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                          (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) v) },
            map_smul' :=
              (_ :
                ∀ (r : ℝ) (u : ↥(Lp E 1)),
                  AddHom.toFun
                      { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
                        map_add' :=
                          (_ :
                            ∀ (u v : ↥(Lp E 1)),
                              (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1))
                                  (u + v) =
                                (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                                  (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) v) }
                      (r • u) =
                    (RingHom.id ℝ) r •
                      AddHom.toFun
                        { toFun := fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1),
                          map_add' :=
                            (_ :
                              ∀ (u v : ↥(Lp E 1)),
                                (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1))
                                    (u + v) =
                                  (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1)) u +
                                    (fun u => Memℒp.toLp (fun x => f x • ↑↑u x) (_ : Memℒp (fun x => f x • ↑↑u x) 1))
                                      v) }
                        u) }
          x‖ =
      ‖x‖

Next tactic:
intro u