-- Mathlib/Probability/Kernel/WithDensity.lean
theorem withDensity_kernel_sum [Countable ι] (κ : ι → Kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
    (f : α → β → ℝ≥0∞) :
    withDensity (Kernel.sum κ) f = Kernel.sum fun i => withDensity (κ i) f

Proof state:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
⊢ (Kernel.sum κ).withDensity f = Kernel.sum fun i => (κ i).withDensity f

Tactic:
ext1 a

Result:
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
a : α
⊢ ((Kernel.sum κ).withDensity f) a = (Kernel.sum fun i => (κ i).withDensity f) a

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
⊢ (Kernel.sum κ).withDensity f = Kernel.sum fun i => (κ i).withDensity f

Tactic:
by_cases hf : Measurable (Function.uncurry f)

Result:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
⊢ (Kernel.sum κ).withDensity f = Kernel.sum fun i => (κ i).withDensity f

case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ (Kernel.sum κ).withDensity f = Kernel.sum fun i => (κ i).withDensity f

Proof state:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ 0 = Kernel.sum fun i => 0

Tactic:
exact sum_zero.symm

Result:
no goals

Proof state:
case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
a : α
⊢ ((Kernel.sum κ).withDensity f) a = (Kernel.sum fun i => (κ i).withDensity f) a

Tactic:
simp_rw [sum_apply, Kernel.withDensity_apply _ hf, sum_apply,
  withDensity_sum (fun n => κ n a) (f a)]

Result:
no goals

Proof state:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ (Kernel.sum κ).withDensity f = Kernel.sum fun i => (κ i).withDensity f

Tactic:
simp_rw [withDensity_of_not_measurable _ hf]

Result:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ 0 = Kernel.sum fun i => 0