-- Mathlib/MeasureTheory/Measure/MutuallySingular.lean
lemma absolutelyContinuous_of_add_of_mutuallySingular {ν₁ ν₂ : Measure α}
    (h : μ ≪ ν₁ + ν₂) (h_ms : μ ⟂ₘ ν₂) : μ ≪ ν₁

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ μ (s ∩ tᶜ) = 0

Tactic:
refine h ?_

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ (ν₁ + ν₂) (s ∩ tᶜ) = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
⊢ μ s = 0

Tactic:
have htμ : μ t = 0 := h_ms.measure_nullSet

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
⊢ μ s = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
⊢ μ s = 0

Tactic:
have ht : MeasurableSet t := h_ms.measurableSet_nullSet

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
⊢ μ s = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ μ (s ∩ t ∪ s ∩ tᶜ) = μ (s ∩ tᶜ)

Tactic:
rw [measure_union, measure_inter_null_of_null_right _ htμ, zero_add]

Result:
case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ Disjoint (s ∩ t) (s ∩ tᶜ)

case h
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ MeasurableSet (s ∩ tᶜ)

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ μ s = μ (s ∩ tᶜ)

Tactic:
conv_lhs => rw [← inter_union_compl s t]

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ μ (s ∩ t ∪ s ∩ tᶜ) = μ (s ∩ tᶜ)

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
⊢ μ s = 0

Tactic:
have htν₂ : ν₂ tᶜ = 0 := h_ms.measure_compl_nullSet

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ μ s = 0

Proof state:
case h
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ MeasurableSet (s ∩ tᶜ)

Tactic:
exact hs.inter ht.compl

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ μ s = 0

Tactic:
rw [this]

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ μ (s ∩ tᶜ) = 0

Proof state:
case hd
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
⊢ Disjoint (s ∩ t) (s ∩ tᶜ)

Tactic:
exact (disjoint_compl_right.inter_right' _ ).inter_left' _

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
⊢ μ s = 0

Tactic:
let t := h_ms.nullSet

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
⊢ μ s = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
⊢ μ ≪ ν₁

Tactic:
refine AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
⊢ μ s = 0

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ ν₁ (s ∩ tᶜ) = 0 ∧ ν₂ (s ∩ tᶜ) = 0

Tactic:
exact ⟨measure_inter_null_of_null_left _ hs_zero, measure_inter_null_of_null_right _ htν₂⟩

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ (ν₁ + ν₂) (s ∩ tᶜ) = 0

Tactic:
simp only [Measure.coe_add, Pi.add_apply, add_eq_zero]

Result:
α : Type u_1
m0 : MeasurableSpace α
μ ν₁ ν₂ : Measure α
h : μ ≪ ν₁ + ν₂
h_ms : μ ⟂ₘ ν₂
s : Set α
hs : MeasurableSet s
hs_zero : ν₁ s = 0
t : Set α := h_ms.nullSet
ht : MeasurableSet t
htμ : μ t = 0
htν₂ : ν₂ tᶜ = 0
this : μ s = μ (s ∩ tᶜ)
⊢ ν₁ (s ∩ tᶜ) = 0 ∧ ν₂ (s ∩ tᶜ) = 0