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

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: set s : ℕ → Set α := disjointed (spanningSets μ)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: have : ∀ n, μ (s n) < ∞ := fun n =>
    (measure_mono <| disjointed_subset _ _).trans_lt (measure_spanningSets_lt_top μ n)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: obtain ⟨δ, δpos, δsum⟩ : ∃ δ : ℕ → ℝ≥0, (∀ i, 0 < δ i) ∧ (∑' i, μ (s i) * δ i) < ε :=
    ENNReal.exists_pos_tsum_mul_lt_of_countable ε0 _ fun n => (this n).ne
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: set N : α → ℕ := spanningSetsIndex μ
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: have hN_meas : Measurable N := measurable_spanningSetsIndex μ
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
tactic: have hNs : ∀ n, N ⁻¹' {n} = s n := preimage_spanningSetsIndex_singleton μ
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∫⁻ (x : α), ↑((δ ∘ N) x) ∂μ < ε
tactic: refine ⟨δ ∘ N, fun x => δpos _, measurable_from_nat.comp hN_meas, ?_⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∫⁻ (x : α), ↑((δ ∘ N) x) ∂μ < ε
state_after: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∫⁻ (a : ℕ), ↑(δ a) ∂Measure.map N μ < ε
tactic: erw [lintegral_comp measurable_from_nat.coe_nnreal_ennreal hN_meas]
state_before: case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
⊢ ∫⁻ (a : ℕ), ↑(δ a) ∂Measure.map N μ < ε
state_after: no goals
tactic: simpa [N, hNs, lintegral_countable', measurable_spanningSetsIndex, mul_comm] using δsum