theorem extracted_formal_statement_7.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} [inst_1 : MeasurableSpace 𝒳] [inst_2 : Nonempty 𝒳] [inst_3 : Nonempty ι] [inst_4 : Countable ι]
  [inst_5 : IsProbabilityMeasure μ] (f : ι → 𝒳 → ℝ) (hf : ∀ (i : ι), Measurable (f i)) (X : Ω → 𝒳) (hX : Measurable X)
  {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) {t : ℝ} (ht : 0 ≤ t) (ht' : t * b ^ 2 ≤ 1 / 2) {ε : ℝ}
  (hε : 0 ≤ ε) (hn_1 : ¬n = 0) (hn : 0 < n)
  (h_original_goal :
    ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ≤
      (Measure.pi fun x => μ) fun ω =>
        ε ≤
          uniformDeviation n f μ X (X ∘ ω) -
            ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ) :
  (fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ⊆ fun ω =>
    ε ≤
      uniformDeviation n f μ X (X ∘ ω) -
        ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ := sorry

State before:
n : ℕ
Ω : Type u
inst✝⁵ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
inst✝⁴ : MeasurableSpace 𝒳
inst✝³ : Nonempty 𝒳
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
f : ι → 𝒳 → ℝ
hf : ∀ (i : ι), Measurable (f i)
X : Ω → 𝒳
hX : Measurable X
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
t : ℝ
ht : 0 ≤ t
ht' : t * b ^ 2 ≤ 1 / 2
ε : ℝ
hε : 0 ≤ ε
hn✝ : ¬n = 0
hn : 0 < n
⊢ ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ≤
    (Measure.pi fun x => μ) fun ω =>
      ε ≤ uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ

Tactic: apply measure_mono

State after:
[case h
 n : ℕ
 Ω : Type u
 inst✝⁵ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 inst✝⁴ : MeasurableSpace 𝒳
 inst✝³ : Nonempty 𝒳
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 f : ι → 𝒳 → ℝ
 hf : ∀ (i : ι), Measurable (f i)
 X : Ω → 𝒳
 hX : Measurable X
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 t : ℝ
 ht : 0 ≤ t
 ht' : t * b ^ 2 ≤ 1 / 2
 ε : ℝ
 hε : 0 ≤ ε
 hn✝ : ¬n = 0
 hn : 0 < n
 ⊢ (fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ⊆ fun ω =>
     ε ≤ uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ]

Relevant lemmas:
MeasureTheory.measure_mono : ∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ℝ≥0∞]
  [inst_6 : OuterMeasureClass F α] {μ : F} {s t : Set α}, s ⊆ t → μ s ≤ μ t