Theorem: MeasureTheory.Measure.map_smul
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: rcases eq_or_ne c 0 with (rfl | hc)
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
⊢ map f (c • μ) = c • map f μ

tactic: by_cases hf : AEMeasurable f μ
state_before: case inr
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
⊢ map f (c • μ) = c • map f μ

tactic: simp
state_before: case inl
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
μ : Measure α
f : α → β
⊢ map f (0 • μ) = 0 • map f μ

tactic: have hfc : AEMeasurable f (c • μ) :=
  ⟨hf.mk f, hf.measurable_mk, (ae_smul_measure_iff hc).2 hf.ae_eq_mk⟩
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : AEMeasurable f
⊢ map f (c • μ) = c • map f μ

tactic: simp only [← mapₗ_mk_apply_of_aemeasurable hf, ← mapₗ_mk_apply_of_aemeasurable hfc,
  LinearMap.map_smulₛₗ, RingHom.id_apply]
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : AEMeasurable f
hfc : AEMeasurable f
⊢ map f (c • μ) = c • map f μ

tactic: congr 1
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : AEMeasurable f
hfc : AEMeasurable f
⊢ c • ↑(mapₗ (AEMeasurable.mk f hfc)) μ = c • ↑(mapₗ (AEMeasurable.mk f hf)) μ

tactic: apply mapₗ_congr hfc.measurable_mk hf.measurable_mk
state_before: case pos.e_a
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : AEMeasurable f
hfc : AEMeasurable f
⊢ ↑(mapₗ (AEMeasurable.mk f hfc)) μ = ↑(mapₗ (AEMeasurable.mk f hf)) μ

tactic: exact EventuallyEq.trans ((ae_smul_measure_iff hc).1 hfc.ae_eq_mk.symm) hf.ae_eq_mk
state_before: case pos.e_a
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : AEMeasurable f
hfc : AEMeasurable f
⊢ AEMeasurable.mk f hfc =ᶠ[ae μ] AEMeasurable.mk f hf

tactic: have hfc : ¬AEMeasurable f (c • μ) := by
  intro hfc
  exact hf ⟨hfc.mk f, hfc.measurable_mk, (ae_smul_measure_iff hc).1 hfc.ae_eq_mk⟩
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : ¬AEMeasurable f
⊢ map f (c • μ) = c • map f μ

tactic: simp [map_of_not_aemeasurable hf, map_of_not_aemeasurable hfc]
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : ¬AEMeasurable f
hfc : ¬AEMeasurable f
⊢ map f (c • μ) = c • map f μ

tactic: intro hfc
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : ¬AEMeasurable f
⊢ ¬AEMeasurable f

tactic: exact hf ⟨hfc.mk f, hfc.measurable_mk, (ae_smul_measure_iff hc).1 hfc.ae_eq_mk⟩
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.187403
δ : Type ?u.187406
ι : Type ?u.187409
R : Type ?u.187412
R' : Type ?u.187415
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
μ : Measure α
f : α → β
hc : c ≠ 0
hf : ¬AEMeasurable f
hfc : AEMeasurable f
⊢ False