mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∉ t
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))

pp: case pos
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∉ t
⊢ Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hle : f ≤ᵐ[μ] g
hg : AEMeasurable g μ
ε : ℝ≥0∞
φ : α → ℝ≥0∞
hφm : Measurable φ
hφ_le : φ ≤ f
hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ
⊢ ∫⁻ (x : α), φ x ∂μ + ε * μ {x | f x + ε ≤ g x} ≤ ∫⁻ (x : α), φ x ∂μ + ε * μ {x | φ x + ε ≤ g x}

Tactic:
by
      gcongr
      exact hφ_le _

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → x ∈ S m
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t

pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := ⋯
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := ⋯
fs : ℕ → α →ₛ β := ⋯
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → x ∈ S m
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∈ s
⊢ f x ≤ s.indicator 1 x

pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∉ s
⊢ f x ≤ s.indicator 1 x

Tactic:
·

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∈ s
⊢ f x ≤ s.indicator 1 x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∉ s
⊢ f x ≤ s.indicator 1 x

Tactic:
·

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : ∀ (a : α), f a ≤ 1
h'f : ∀ a ∈ sᶜ, f a = 0
x : α
hx : x ∉ s
⊢ f x ≤ s.indicator 1 x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a ↦ if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a
n : ℕ
a : α
h : a ∈ s
⊢ 0 ≤ 0

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a ↦ if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a
n : ℕ
a : α
h : a ∉ s
⊢ f n a ≤ f (n + 1) a

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a ↦ if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a
n : ℕ
a : α
h : a ∈ s
⊢ 0 ≤ 0
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a ↦ if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a
n : ℕ
a : α
h : a ∉ s
⊢ f n a ≤ f (n + 1) a

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0
g : ℕ → α → ℝ≥0∞ := fun n a ↦ if a ∈ s then 0 else f n a
g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a
n : ℕ
a : α
h : a ∉ s
⊢ f n a ≤ f (n + 1) a
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n : ℕ
hn : x ∈ S n
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m

pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := ⋯
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := ⋯
fs : ℕ → α →ₛ β := ⋯
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
⊢ ∃ n, x ∈ S n

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n : ℕ
hn : x ∈ S n
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case intro.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : IsEmpty β
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

pp: case intro.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
·

Goal state after:
case intro.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : IsEmpty β
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
n : ℕ
⊢ ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ ≤ ⨆ b, ∫⁻ (a : α), f b a ∂μ

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
b : β
⊢ ∫⁻ (a : α), f b a ∂μ ≤ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
n : ℕ
⊢ ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ ≤ ⨆ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
b : β
⊢ ∫⁻ (a : α), f b a ∂μ ≤ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a
b : β
⊢ ∫⁻ (a : α), f b a ∂μ ≤ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hμ : μ ≠ 0
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
h : ∀ᵐ (x : α) ∂μ, f x < g x
⊢ ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ

Tactic:
by
  rw [Ne, ← Measure.measure_univ_eq_zero] at hμ
  refine lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) hμ ?_
  simpa using h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₁ ≤ aeSeq hf p z

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₁ ≤ aeSeq hf p z
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x

Tactic:
aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ aeSeq hf p z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ¬∫⁻ (x : α), f x ∂μ = ∞
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
g : α →ₛ ℝ≥0∞
hg : ⇑g ≤ fun a ↦ s.indicator f a
this : ⇑g ≤ f
t : ℝ≥0∞
H : t = 0
⊢ t * μ (⇑g ⁻¹' {t}) = t * μ (⇑g ⁻¹' {t} ∩ s)

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
g : α →ₛ ℝ≥0∞
hg : ⇑g ≤ fun a ↦ s.indicator f a
this : ⇑g ≤ f
t : ℝ≥0∞
H : ¬t = 0
⊢ t * μ (⇑g ⁻¹' {t}) = t * μ (⇑g ⁻¹' {t} ∩ s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
g : α →ₛ ℝ≥0∞
hg : ⇑g ≤ fun a ↦ s.indicator f a
this : ⇑g ≤ f
t : ℝ≥0∞
H : t = 0
⊢ t * μ (⇑g ⁻¹' {t}) = t * μ (⇑g ⁻¹' {t} ∩ s)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ aeSeq hf p z x

Tactic:
aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ f z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ f z x

Tactic:
aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ f b₁ x ≤ f z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₁ x ≤ aeSeq hf p z x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₂ ≤ aeSeq hf p z

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
⊢ aeSeq hf p b₂ ≤ aeSeq hf p z
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∉ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
b₁ b₂ z : β
hz₁ : f b₁ ≤ f z
hz₂ : f b₂ ≤ f z
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p b₂ x ≤ aeSeq hf p z x