Theorem: ProbabilityTheory.StieltjesFunction.measurable_measure
File: Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean

tactic: refine Measure.measurable_measure.mpr fun s hs ↦ ?_
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
⊢ Measurable fun a => (f a).measure
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
⊢ Measurable fun b => (f b).measure s

tactic: have : ∀ a, IsProbabilityMeasure (f a).measure :=
  fun a ↦ (f a).isProbabilityMeasure (hf_bot a) (hf_top a)
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
⊢ Measurable fun b => (f b).measure s
state_after: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ Measurable fun b => (f b).measure s

tactic: refine MeasurableSpace.induction_on_inter
  (C := fun s ↦ Measurable fun b ↦ StieltjesFunction.measure (f b) s)
  (borel_eq_generateFrom_Iic ℝ) isPiSystem_Iic ?_ ?_ ?_ ?_ hs
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ Measurable fun b => (f b).measure s
state_after: case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ (fun s => Measurable fun b => (f b).measure s) ∅

case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ t ∈ range Iic, (fun s => Measurable fun b => (f b).measure s) t

case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ (t : Set ℝ),
    MeasurableSet t →
      (fun s => Measurable fun b => (f b).measure s) t → (fun s => Measurable fun b => (f b).measure s) tᶜ

case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ (f_1 : ℕ → Set ℝ),
    Pairwise (Disjoint on f_1) →
      (∀ (i : ℕ), MeasurableSet (f_1 i)) →
        (∀ (i : ℕ), (fun s => Measurable fun b => (f b).measure s) (f_1 i)) →
          (fun s => Measurable fun b => (f b).measure s) (⋃ i, f_1 i)

tactic: simp only [measure_empty, measurable_const]
state_before: case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ (fun s => Measurable fun b => (f b).measure s) ∅
state_after: no goals

tactic: rintro S ⟨u, rfl⟩
state_before: case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ t ∈ range Iic, (fun s => Measurable fun b => (f b).measure s) t
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
u : ℝ
⊢ Measurable fun b => (f b).measure (Iic u)

tactic: simp_rw [StieltjesFunction.measure_Iic (f _) (hf_bot _), sub_zero]
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
u : ℝ
⊢ Measurable fun b => (f b).measure (Iic u)
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
u : ℝ
⊢ Measurable fun b => ENNReal.ofReal (↑(f b) u)

tactic: exact (hf _).ennreal_ofReal
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
u : ℝ
⊢ Measurable fun b => ENNReal.ofReal (↑(f b) u)
state_after: no goals

tactic: intro t ht ht_cd_meas
state_before: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ (t : Set ℝ),
    MeasurableSet t →
      (fun s => Measurable fun b => (f b).measure s) t → (fun s => Measurable fun b => (f b).measure s) tᶜ
state_after: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
⊢ Measurable fun b => (f b).measure tᶜ

tactic: have : (fun a ↦ (f a).measure tᶜ) = (fun a ↦ (f a).measure univ) - fun a ↦ (f a).measure t := by
  ext1 a
  rw [measure_compl ht, Pi.sub_apply]
  exact measure_ne_top _ _
state_before: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
⊢ Measurable fun b => (f b).measure tᶜ
state_after: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this✝ : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
this : (fun a => (f a).measure tᶜ) = (fun a => (f a).measure univ) - fun a => (f a).measure t
⊢ Measurable fun b => (f b).measure tᶜ

tactic: simp_rw [this, measure_univ]
state_before: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this✝ : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
this : (fun a => (f a).measure tᶜ) = (fun a => (f a).measure univ) - fun a => (f a).measure t
⊢ Measurable fun b => (f b).measure tᶜ
state_after: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this✝ : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
this : (fun a => (f a).measure tᶜ) = (fun a => (f a).measure univ) - fun a => (f a).measure t
⊢ Measurable ((fun a => 1) - fun a => (f a).measure t)

tactic: exact Measurable.sub measurable_const ht_cd_meas
state_before: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this✝ : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
this : (fun a => (f a).measure tᶜ) = (fun a => (f a).measure univ) - fun a => (f a).measure t
⊢ Measurable ((fun a => 1) - fun a => (f a).measure t)
state_after: no goals

tactic: ext1 a
state_before: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
⊢ (fun a => (f a).measure tᶜ) = (fun a => (f a).measure univ) - fun a => (f a).measure t
state_after: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
a : α
⊢ (f a).measure tᶜ = ((fun a => (f a).measure univ) - fun a => (f a).measure t) a

tactic: rw [measure_compl ht, Pi.sub_apply]
state_before: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
a : α
⊢ (f a).measure tᶜ = ((fun a => (f a).measure univ) - fun a => (f a).measure t) a
state_after: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
a : α
⊢ (f a).measure t ≠ ⊤

tactic: exact measure_ne_top _ _
state_before: case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
t : Set ℝ
ht : MeasurableSet t
ht_cd_meas : Measurable fun b => (f b).measure t
a : α
⊢ (f a).measure t ≠ ⊤
state_after: no goals

tactic: intro f hf_disj hf_meas hf_cd_meas
state_before: case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ (f_1 : ℕ → Set ℝ),
    Pairwise (Disjoint on f_1) →
      (∀ (i : ℕ), MeasurableSet (f_1 i)) →
        (∀ (i : ℕ), (fun s => Measurable fun b => (f b).measure s) (f_1 i)) →
          (fun s => Measurable fun b => (f b).measure s) (⋃ i, f_1 i)
state_after: case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝¹ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f✝ : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f✝ a) q
hf_bot : ∀ (a : α), Tendsto (↑(f✝ a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f✝ a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f✝ a).measure
f : ℕ → Set ℝ
hf_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_cd_meas : ∀ (i : ℕ), (fun s => Measurable fun b => (f✝ b).measure s) (f i)
⊢ Measurable fun b => (f✝ b).measure (⋃ i, f i)

tactic: simp_rw [measure_iUnion hf_disj hf_meas]
state_before: case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝¹ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f✝ : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f✝ a) q
hf_bot : ∀ (a : α), Tendsto (↑(f✝ a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f✝ a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f✝ a).measure
f : ℕ → Set ℝ
hf_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_cd_meas : ∀ (i : ℕ), (fun s => Measurable fun b => (f✝ b).measure s) (f i)
⊢ Measurable fun b => (f✝ b).measure (⋃ i, f i)
state_after: case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝¹ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f✝ : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f✝ a) q
hf_bot : ∀ (a : α), Tendsto (↑(f✝ a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f✝ a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f✝ a).measure
f : ℕ → Set ℝ
hf_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_cd_meas : ∀ (i : ℕ), (fun s => Measurable fun b => (f✝ b).measure s) (f i)
⊢ Measurable fun b => ∑' (i : ℕ), (f✝ b).measure (f i)

tactic: exact Measurable.ennreal_tsum hf_cd_meas
state_before: case refine_4
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α (β × ℝ))
ν✝ : ↥(kernel α β)
x✝ : MeasurableSpace β
f✝¹ : α × β → StieltjesFunction
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f✝ : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f✝ a) q
hf_bot : ∀ (a : α), Tendsto (↑(f✝ a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f✝ a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f✝ a).measure
f : ℕ → Set ℝ
hf_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_cd_meas : ∀ (i : ℕ), (fun s => Measurable fun b => (f✝ b).measure s) (f i)
⊢ Measurable fun b => ∑' (i : ℕ), (f✝ b).measure (f i)
state_after: no goals