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

tactic: have h_sum_a : ∀ a, Summable fun n => f n a := fun a => Pi.summable.mpr fun b => ENNReal.summable
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
state_after: 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)

tactic: have h_sum : Summable fun n => f n := Pi.summable.mpr h_sum_a
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
state_after: 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)

tactic: ext a s hs
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
state_after: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ((withDensity κ (∑' (n : ι), f n)) a) s = ((kernel.sum fun n => withDensity κ (f n)) a) s

tactic: rw [sum_apply' _ a hs, kernel.withDensity_apply' κ _ a s]
state_before: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ((withDensity κ (∑' (n : ι), f n)) a) s = ((kernel.sum fun n => withDensity κ (f n)) a) s
state_after: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s

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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))

tactic: swap
state_before: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s

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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
state_after: 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))

case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s

tactic: have : ∫⁻ b in s, (∑' n, f n) a b ∂κ a = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a := by
  congr with b
  rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
state_before: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
state_after: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s

tactic: rw [this, lintegral_tsum fun n => (Measurable.of_uncurry_left (hf n)).aemeasurable]
state_before: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
state_after: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∑' (i : ι), ∫⁻ (a_1 : β) in s, f i a a_1 ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s

tactic: congr with n
state_before: case 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∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∑' (i : ι), ∫⁻ (a_1 : β) in s, f i a a_1 ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
state_after: case h.h.e_f.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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
n : ι
⊢ ∫⁻ (a_1 : β) in s, f n a a_1 ∂κ a = ((withDensity κ (f n)) a) s

tactic: rw [kernel.withDensity_apply' _ (hf n) a s]
state_before: case h.h.e_f.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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
n : ι
⊢ ∫⁻ (a_1 : β) in s, f n a a_1 ∂κ a = ((withDensity κ (f n)) a) s
state_after: no goals

tactic: have : Function.uncurry (∑' n, f n) = ∑' n, Function.uncurry (f n) := by
  ext1 p
  simp only [Function.uncurry_def]
  rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]
  exact Pi.summable.mpr fun p => ENNReal.summable
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
state_after: 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))

tactic: rw [this]
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
state_after: 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (∑' (n : ι), Function.uncurry (f n))

tactic: exact Measurable.ennreal_tsum' hf
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (∑' (n : ι), Function.uncurry (f n))
state_after: no goals

tactic: ext1 p
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
state_after: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Function.uncurry (∑' (n : ι), f n) p = tsum (fun n => Function.uncurry (f n)) p

tactic: simp only [Function.uncurry_def]
state_before: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Function.uncurry (∑' (n : ι), f n) p = tsum (fun n => Function.uncurry (f n)) p
state_after: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p

tactic: rw [tsum_apply h_sum, tsum_apply (h_sum_a _), tsum_apply]
state_before: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p
state_after: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Summable fun n p => f n p.1 p.2

tactic: exact Pi.summable.mpr fun p => ENNReal.summable
state_before: case 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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Summable fun n p => f n p.1 p.2
state_after: no goals

tactic: congr with b
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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
state_after: case e_f.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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
b : β
⊢ tsum (fun n => f n) a b = ∑' (n : ι), (fun b => f n a b) b

tactic: rw [tsum_apply h_sum, tsum_apply (h_sum_a a)]
state_before: case e_f.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✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
b : β
⊢ tsum (fun n => f n) a b = ∑' (n : ι), (fun b => f n a b) b
state_after: no goals