Theorem: ProbabilityTheory.kernel.withDensity_kernel_sum
File: Mathlib/Probability/Kernel/WithDensity.lean

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

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

tactic: ext1 a
state_before: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝ : Countable ι
κ : ι → ↥(kernel α β)
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
⊢ withDensity (kernel.sum κ) f = kernel.sum fun i => withDensity (κ i) f
state_after: case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
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
⊢ (withDensity (kernel.sum κ) f) a = (kernel.sum fun i => withDensity (κ i) f) a

tactic: simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,
  withDensity_sum (fun n => κ n a) (f a)]
state_before: case pos.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
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
⊢ (withDensity (kernel.sum κ) f) a = (kernel.sum fun i => withDensity (κ i) f) a
state_after: no goals

tactic: simp_rw [withDensity_of_not_measurable _ hf]
state_before: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝ : Countable ι
κ : ι → ↥(kernel α β)
hκ : ∀ (i : ι), IsSFiniteKernel (κ i)
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ withDensity (kernel.sum κ) f = kernel.sum fun i => withDensity (κ i) f
state_after: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
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
state_before: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
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
state_after: no goals