no goals
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
a : α
ha : ∀ (n : ℕ), g n a = f n a
⊢ (fun a => ⨆ (n : ℕ), f n a) a = (fun a => ⨆ (n : ℕ), g n a) a
simp only [ha]
no goals
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
⊢ g n a ≤ g (n + 1) a
simp [if_pos h]
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
⊢ g n a ≤ g (n + 1) a
simp only [if_neg h]
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s
⊢ f n a ≤ f (n + 1) a
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
have := hs.1
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
⊢ f n a ≤ f (n + 1) a
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s
⊢ f n a ≤ f (n + 1) a
rw [subset_def] at this
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
this : ¬a ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a}
⊢ f n a ≤ f (n + 1) a
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
⊢ f n a ≤ f (n + 1) a
have := mt (this a) h
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
this : ∀ (i : ℕ), f i a ≤ f (Nat.succ i) a
⊢ f n a ≤ f (n + 1) a
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
this : ¬a ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a}
⊢ f n a ≤ f (n + 1) a
simp only [Classical.not_not, mem_setOf_eq] at this
no goals
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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
this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s
this : ∀ (i : ℕ), f i a ≤ f (Nat.succ i) a
⊢ f n a ≤ f (n + 1) a
exact this n
no goals
α : Type u_1
β : Type ?u.976729
γ : Type ?u.976732
δ : Type ?u.976735
m : MeasurableSpace α
μ ν : 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 (Nat.succ n) a
s : Set α
hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) 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), g n a ∂μ) = ⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ
simp only [lintegral_congr_ae (g_eq_f.mono fun _a ha => ha _)]