Theorem: MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst
File: Mathlib/MeasureTheory/Integral/SetToL1.lean

tactic: have h_empty : T ∅ = 0 := h_zero _ MeasurableSet.empty measure_empty
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.716015
G : Type ?u.716018
𝕜 : Type ?u.716021
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
x : E
⊢ setToL1S T (indicatorConst 1 hs (_ : ↑↑μ s ≠ ⊤) x) = ↑(T s) x

tactic: rw [setToL1S_eq_setToSimpleFunc]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.716015
G : Type ?u.716018
𝕜 : Type ?u.716021
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
x : E
h_empty : T ∅ = 0
⊢ setToL1S T (indicatorConst 1 hs (_ : ↑↑μ s ≠ ⊤) x) = ↑(T s) x

tactic: refine' Eq.trans _ (SimpleFunc.setToSimpleFunc_indicator T h_empty hs x)
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.716015
G : Type ?u.716018
𝕜 : Type ?u.716021
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
x : E
h_empty : T ∅ = 0
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : ↑↑μ s ≠ ⊤) x)) = ↑(T s) x

tactic: refine' SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) _
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.716015
G : Type ?u.716018
𝕜 : Type ?u.716021
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
x : E
h_empty : T ∅ = 0
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (indicatorConst 1 hs (_ : ↑↑μ s ≠ ⊤) x)) =
    SimpleFunc.setToSimpleFunc T (SimpleFunc.piecewise s hs (SimpleFunc.const α x) (SimpleFunc.const α 0))

tactic: exact toSimpleFunc_indicatorConst hs hμs.ne x
state_before: α : Type u_1
E : Type u_2
F : Type u_3
F' : Type ?u.716015
G : Type ?u.716018
𝕜 : Type ?u.716021
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
s : Set α
h_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hs : MeasurableSet s
hμs : ↑↑μ s < ⊤
x : E
h_empty : T ∅ = 0
⊢ ↑(toSimpleFunc (indicatorConst 1 hs (_ : ↑↑μ s ≠ ⊤) x)) =ᵐ[μ]
    ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α x) (SimpleFunc.const α 0))