Theorem: MeasureTheory.sfinite_sum_of_countable
File: Mathlib/MeasureTheory/Measure/Typeclasses.lean

tactic: obtain ⟨f, hf⟩ : ∃ f : ι → ℕ, Function.Injective f := Countable.exists_injective_nat ι
state_before: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
⊢ SFinite (sum m)
state_after: case intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
⊢ SFinite (sum m)

tactic: refine ⟨_, fun n ↦ ?_, (sum_extend_zero hf m).symm⟩
state_before: case intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
⊢ SFinite (sum m)
state_after: case intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
⊢ IsFiniteMeasure (Function.extend f m 0 n)

tactic: rcases em (n ∈ range f) with ⟨i, rfl⟩ | hn
state_before: case intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
⊢ IsFiniteMeasure (Function.extend f m 0 n)
state_after: case intro.inl.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (Function.extend f m 0 (f i))

case intro.inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure (Function.extend f m 0 n)

tactic: rw [hf.extend_apply]
state_before: case intro.inl.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (Function.extend f m 0 (f i))
state_after: case intro.inl.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (m i)

tactic: infer_instance
state_before: case intro.inl.intro
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
i : ι
⊢ IsFiniteMeasure (m i)
state_after: no goals

tactic: rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply]
state_before: case intro.inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure (Function.extend f m 0 n)
state_after: case intro.inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure 0

tactic: infer_instance
state_before: case intro.inr
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝¹ : Countable ι
m : ι → Measure α
inst✝ : ∀ (n : ι), IsFiniteMeasure (m n)
f : ι → ℕ
hf : Injective f
n : ℕ
hn : n ∉ range f
⊢ IsFiniteMeasure 0
state_after: no goals