-- Mathlib/MeasureTheory/Function/LpSpace/Basic.lean
theorem tsum_enorm_sub_ae_lt_top {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i) :
    ∀ᵐ x ∂μ, ∑' i, ‖f (i + 1) x - f i x‖ₑ < ∞

Proof state:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ⊤
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ⊤
⊢ ∀ᵐ (x : α) ∂μ, ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ⊤

Tactic:
refine rpow_ae_lt_top.mono fun x hx => ?_

Result:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ⊤
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ⊤
x : α
hx : (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ⊤
⊢ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ⊤

Proof state:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ⊤
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ⊤
x : α
hx : ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ⊤ ^ p⁻¹
⊢ 0 < p⁻¹

Tactic:
simp [hp_pos]

Result:
no goals

Proof state:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ⊤
⊢ ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ⊤

Tactic:
refine ae_lt_top' (AEMeasurable.pow_const ?_ _) h_integral.ne

Result:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ⊤
⊢ AEMeasurable (fun x => ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) μ

Proof state:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_tsum_lt_top : (∑' (i : ℕ), B i) ^ p < ⊤
⊢ 0 < p⁻¹

Tactic:
simp [hp_pos]

Result:
no goals