Theorem: MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀
File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

tactic: filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk,
  ae_le_of_forall_setLIntegral_le_of_sigmaFinite hf.measurable_mk hg.measurable_mk h']
  with a haf hag ha
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
h' :
  ∀ (s : Set α),
    MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, AEMeasurable.mk f hf x ∂μ ≤ ∫⁻ (x : α) in s, AEMeasurable.mk g hg x ∂μ
⊢ f ≤ᶠ[ae μ] g
state_after: case h
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
h' :
  ∀ (s : Set α),
    MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, AEMeasurable.mk f hf x ∂μ ≤ ∫⁻ (x : α) in s, AEMeasurable.mk g hg x ∂μ
a : α
haf : f a = AEMeasurable.mk f hf a
hag : g a = AEMeasurable.mk g hg a
ha : AEMeasurable.mk f hf a ≤ AEMeasurable.mk g hg a
⊢ f a ≤ g a

tactic: rwa [haf, hag]
state_before: case h
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
h' :
  ∀ (s : Set α),
    MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, AEMeasurable.mk f hf x ∂μ ≤ ∫⁻ (x : α) in s, AEMeasurable.mk g hg x ∂μ
a : α
haf : f a = AEMeasurable.mk f hf a
hag : g a = AEMeasurable.mk g hg a
ha : AEMeasurable.mk f hf a ≤ AEMeasurable.mk g hg a
⊢ f a ≤ g a
state_after: no goals

tactic: refine fun s hs hμs ↦ (setLIntegral_congr_fun hs ?_).trans_le
  ((h s hs hμs).trans_eq (setLIntegral_congr_fun hs ?_))
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
⊢ ∀ (s : Set α),
    MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, AEMeasurable.mk f hf x ∂μ ≤ ∫⁻ (x : α) in s, AEMeasurable.mk g hg x ∂μ
state_after: case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → AEMeasurable.mk f hf x = f x

case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → g x = AEMeasurable.mk g hg x

tactic: filter_upwards [hf.ae_eq_mk] with a ha using fun _ ↦ ha.symm
state_before: case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → AEMeasurable.mk f hf x = f x
state_after: no goals

tactic: filter_upwards [hg.ae_eq_mk] with a ha using fun _ ↦ ha
state_before: case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → g x = AEMeasurable.mk g hg x
state_after: no goals