Theorem: MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
File: Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean

tactic: have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

tactic: rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
  ⟨g0, f_lt_g0, g0_cont, g0_int⟩
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: case intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

tactic: rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
state_before: case intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

tactic: rcases exists_le_lowerSemicontinuous_lintegral_ge μ (s.indicator fun _x => ∞)
    (measurable_const.indicator smeas) this with
  ⟨g1, le_g1, g1_cont, g1_int⟩
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

tactic: refine ⟨fun x => g0 x + g1 x, fun x => ?_, g0_cont.add g1_cont, ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), (fun x => g0 x + g1 x) x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

tactic: by_cases h : x ∈ s
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

tactic: have := le_g1 x
state_before: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
this : s.indicator (fun _x => ⊤) x ≤ g1 x
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

tactic: simp only [h, Set.indicator_of_mem, top_le_iff] at this
state_before: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
this : s.indicator (fun _x => ⊤) x ≤ g1 x
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
this : g1 x = ⊤
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

tactic: simp [this]
state_before: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
this : g1 x = ⊤
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: no goals

tactic: have : f x = fmeas.mk f x := by rw [Set.compl_subset_comm] at hs; exact hs h
state_before: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

tactic: rw [this]
state_before: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(f x) < (fun x => g0 x + g1 x) x
state_after: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x

tactic: exact (f_lt_g0 x).trans_le le_self_add
state_before: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this✝ : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
this : f x = AEMeasurable.mk f fmeas x
⊢ ↑(AEMeasurable.mk f fmeas x) < (fun x => g0 x + g1 x) x
state_after: no goals

tactic: rw [Set.compl_subset_comm] at hs
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ f x = AEMeasurable.mk f fmeas x
state_after: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : sᶜ ⊆ {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ f x = AEMeasurable.mk f fmeas x

tactic: exact hs h
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : sᶜ ⊆ {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ f x = AEMeasurable.mk f fmeas x
state_after: no goals

tactic: refine add_le_add ?_ ?_
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g0 x ∂μ + ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + (0 + ε / 2)
state_after: case refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2

case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g1 x ∂μ ≤ 0 + ε / 2

tactic: convert g0_int using 2
state_before: case refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2
state_after: case h.e'_4.h.e'_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ = ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ

tactic: exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
state_before: case h.e'_4.h.e'_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ = ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ
state_after: no goals

tactic: convert g1_int
state_before: case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g1 x ∂μ ≤ 0 + ε / 2
state_after: case h.e'_4.h.e'_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ 0 = ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ

tactic: simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
  lintegral_indicator, mul_zero, restrict_apply]
state_before: case h.e'_4.h.e'_5
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ 0 = ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ
state_after: no goals

tactic: simp only [add_assoc, ENNReal.add_halves, zero_add]
state_before: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + (0 + ε / 2) = ∫⁻ (x : α), ↑(f x) ∂μ + ε
state_after: no goals