Theorem: MeasureTheory.integral_add_measure
File: Mathlib/MeasureTheory/Integral/Bochner.lean

tactic: have hfi := hμ.add_measure hν
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
⊢ (∫ (x : α), f x ∂μ + ν) = (∫ (x : α), f x ∂μ) + ∫ (x : α), f x ∂ν

tactic: simp_rw [integral_eq_setToFun]
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
⊢ (∫ (x : α), f x ∂μ + ν) = (∫ (x : α), f x ∂μ) + ∫ (x : α), f x ∂ν

tactic: have hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ : Set α → E →L[ℝ] E) 1 :=
  DominatedFinMeasAdditive.add_measure_right μ ν (dominatedFinMeasAdditive_weightedSMul μ)
    zero_le_one
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>
      f a) =
    (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +
      setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a

tactic: have hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν : Set α → E →L[ℝ] E) 1 :=
  DominatedFinMeasAdditive.add_measure_left μ ν (dominatedFinMeasAdditive_weightedSMul ν)
    zero_le_one
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1
⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>
      f a) =
    (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +
      setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a

tactic: rw [← setToFun_congr_measure_of_add_right hμ_dfma (dominatedFinMeasAdditive_weightedSMul μ) f hfi,
  ← setToFun_congr_measure_of_add_left hν_dfma (dominatedFinMeasAdditive_weightedSMul ν) f hfi]
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1
hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1
⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>
      f a) =
    (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +
      setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a

tactic: refine' setToFun_add_left' _ _ _ (fun s _ hμνs => _) f
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1
hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1
⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>
      f a) =
    setToFun (μ + ν) (weightedSMul μ) hμ_dfma f + setToFun (μ + ν) (weightedSMul ν) hν_dfma f

tactic: rw [Measure.coe_add, Pi.add_apply, add_lt_top] at hμνs
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1
hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1
s : Set α
x✝ : MeasurableSet s
hμνs : ↑↑(μ + ν) s < ⊤
⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s

tactic: rw [weightedSMul, weightedSMul, weightedSMul, ← add_smul, Measure.coe_add, Pi.add_apply,
  toReal_add hμνs.1.ne hμνs.2.ne]
state_before: α : Type u_2
E : Type u_1
F : Type ?u.1290399
𝕜 : Type ?u.1290402
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type ?u.1293093
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → E
hμ : Integrable f
hν : Integrable f
hfi : Integrable f
hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1
hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1
s : Set α
x✝ : MeasurableSet s
hμνs : ↑↑μ s < ⊤ ∧ ↑↑ν s < ⊤
⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s