/-- Auxiliary lemma for `setToFun_congr_measure`: the function sending `f : α →₁[μ] G` to
`f : α →₁[μ'] G` is continuous when `μ' ≤ c' • μ` for `c' ≠ ∞`. -/
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f := by
  by_cases hc'0 : c' = 0
  · have hμ'0 : μ' = 0 := by rw [← Measure.nonpos_iff_eq_zero']; refine' hμ'_le.trans _; simp [hc'0]
    have h_im_zero :
      (fun f : α →₁[μ] G =>
          (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f) =
        0 := by
      ext1 f; ext1; simp_rw [hμ'0]; simp only [ae_zero, EventuallyEq, eventually_bot]
    rw [h_im_zero]
    exact continuous_zero
  rw [Metric.continuous_iff]
  intro f ε hε_pos
  use ε / 2 / c'.toReal
  refine' ⟨div_pos (half_pos hε_pos) (toReal_pos hc'0 hc'), _⟩
  intro g hfg
  rw [Lp.dist_def] at hfg ⊢
  let h_int := fun f' : α →₁[μ] G => (L1.integrable_coeFn f').of_measure_le_smul c' hc' hμ'_le
  have :
    snorm (⇑(Integrable.toL1 g (h_int g)) - ⇑(Integrable.toL1 f (h_int f))) 1 μ' =
      snorm (⇑g - ⇑f) 1 μ' :=
    snorm_congr_ae ((Integrable.coeFn_toL1 _).sub (Integrable.coeFn_toL1 _))
  rw [this]
  have h_snorm_ne_top : snorm (⇑g - ⇑f) 1 μ ≠ ∞ := by
    rw [← snorm_congr_ae (Lp.coeFn_sub _ _)]; exact Lp.snorm_ne_top _
  have h_snorm_ne_top' : snorm (⇑g - ⇑f) 1 μ' ≠ ∞ := by
    refine' ((snorm_mono_measure _ hμ'_le).trans_lt _).ne
    rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]
    refine' ENNReal.mul_lt_top _ h_snorm_ne_top
    simp [hc', hc'0]
  

Proof state:
case h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g✝ : α → E
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1)
hfg : ENNReal.toReal (snorm (↑↑g - ↑↑f) 1 μ) < ε / 2 / ENNReal.toReal c'
h_int : ∀ (f' : ↥(Lp G 1)), Integrable ↑↑f' :=
  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')
this :
  snorm (↑↑(Integrable.toL1 ↑↑g (_ : Integrable ↑↑g)) - ↑↑(Integrable.toL1 ↑↑f (_ : Integrable ↑↑f))) 1 μ' =
    snorm (↑↑g - ↑↑f) 1 μ'
h_snorm_ne_top : snorm (↑↑g - ↑↑f) 1 μ ≠ ⊤
h_snorm_ne_top' : snorm (↑↑g - ↑↑f) 1 μ' ≠ ⊤
⊢ ENNReal.toReal (snorm (↑↑g - ↑↑f) 1 μ') < ε

Next tactic:
calc
    (snorm (⇑g - ⇑f) 1 μ').toReal ≤ (c' * snorm (⇑g - ⇑f) 1 μ).toReal := by
      rw [toReal_le_toReal h_snorm_ne_top' (ENNReal.mul_ne_top hc' h_snorm_ne_top)]
      refine' (snorm_mono_measure (⇑g - ⇑f) hμ'_le).trans _
      rw [snorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]
      simp
    _ = c'.toReal * (snorm (⇑g - ⇑f) 1 μ).toReal := toReal_mul
    _ ≤ c'.toReal * (ε / 2 / c'.toReal) :=
      (mul_le_mul le_rfl hfg.le toReal_nonneg toReal_nonneg)
    _ = ε / 2 := by
      refine' mul_div_cancel' (ε / 2) _; rw [Ne.def, toReal_eq_zero_iff]; simp [hc', hc'0]
    _ < ε := half_lt_self hε_pos