Theorem: MeasureTheory.lintegral_tsum
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: simp only [ENNReal.tsum_eq_iSup_sum]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ∫⁻ (a : α), ∑' (i : β), f i a ∂μ = ∑' (i : β), ∫⁻ (a : α), f i a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ∫⁻ (a : α), ⨆ s, ∑ i ∈ s, f i a ∂μ = ⨆ s, ∑ i ∈ s, ∫⁻ (a : α), f i a ∂μ

tactic: rw [lintegral_iSup_directed]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ∫⁻ (a : α), ⨆ s, ∑ i ∈ s, f i a ∂μ = ⨆ s, ∑ i ∈ s, ∫⁻ (a : α), f i a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ⨆ b, ∫⁻ (a : α), ∑ i ∈ b, f i a ∂μ = ⨆ s, ∑ i ∈ s, ∫⁻ (a : α), f i a ∂μ

case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ∀ (b : Finset β), AEMeasurable (fun a => ∑ i ∈ b, f i a) μ

case h_directed
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ Directed (fun x x_1 => x ≤ x_1) fun s a => ∑ i ∈ s, f i a

tactic: simp [lintegral_finset_sum' _ fun i _ => hf i]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ⨆ b, ∫⁻ (a : α), ∑ i ∈ b, f i a ∂μ = ⨆ s, ∑ i ∈ s, ∫⁻ (a : α), f i a ∂μ
state_after: no goals

tactic: intro b
state_before: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ ∀ (b : Finset β), AEMeasurable (fun a => ∑ i ∈ b, f i a) μ
state_after: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
b : Finset β
⊢ AEMeasurable (fun a => ∑ i ∈ b, f i a) μ

tactic: exact Finset.aemeasurable_sum _ fun i _ => hf i
state_before: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
b : Finset β
⊢ AEMeasurable (fun a => ∑ i ∈ b, f i a) μ
state_after: no goals

tactic: intro s t
state_before: case h_directed
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
⊢ Directed (fun x x_1 => x ≤ x_1) fun s a => ∑ i ∈ s, f i a
state_after: case h_directed
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ ∃ z,
    (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) z) ∧
      (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) z)

tactic: use s ∪ t
state_before: case h_directed
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ ∃ z,
    (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) z) ∧
      (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) z)
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t)) ∧
    (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))

tactic: constructor
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t)) ∧
    (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))
state_after: case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))

case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))

tactic: exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_left
state_before: case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))
state_after: no goals

tactic: exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_right
state_before: case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x x_1 => x ≤ x_1) ((fun s a => ∑ i ∈ s, f i a) t) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))
state_after: no goals