Theorem: ProbabilityTheory.kernel.withDensity_add_left
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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
⊢ withDensity (κ + η) f = withDensity κ f + withDensity η 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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
⊢ withDensity (κ + η) f = withDensity κ f + withDensity η 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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ withDensity (κ + η) f = withDensity κ f + withDensity η f

tactic: ext a s
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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
⊢ withDensity (κ + η) f = withDensity κ f + withDensity η f
state_after: case pos.h.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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
a : α
s : Set β
a✝ : MeasurableSet s
⊢ ((withDensity (κ + η) f) a) s = ((withDensity κ f + withDensity η f) a) s

tactic: simp only [kernel.withDensity_apply _ hf, coeFn_add, Pi.add_apply, withDensity_add_measure,
  Measure.add_apply]
state_before: case pos.h.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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
a : α
s : Set β
a✝ : MeasurableSet s
⊢ ((withDensity (κ + η) f) a) s = ((withDensity κ f + withDensity η f) a) s
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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ withDensity (κ + η) f = withDensity κ f + withDensity η 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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ 0 = 0 + 0

tactic: rw [zero_add]
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∞
κ η : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → ℝ≥0∞
hf : ¬Measurable (Function.uncurry f)
⊢ 0 = 0 + 0
state_after: no goals