case h
G : Type ?u.6308
M : Type u_2
α : Type u_1
s✝ : Set α
m : MeasurableSpace α
inst✝³ : MeasurableSpace M
inst✝² : SMul M α
inst✝¹ : MeasurableSMul M α
c : M
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
s : Set α
hs : MeasurableSet s
⊢ ↑↑(map (fun x => c • x) μ) s = ↑↑μ s
G : Type ?u.6308
M : Type u_2
α : Type u_1
s : Set α
m : MeasurableSpace α
inst✝³ : MeasurableSpace M
inst✝² : SMul M α
inst✝¹ : MeasurableSMul M α
c : M
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
⊢ map (fun x => c • x) μ = μ
ext1 s hs
case h
G : Type ?u.6308
M : Type u_2
α : Type u_1
s✝ : Set α
m : MeasurableSpace α
inst✝³ : MeasurableSpace M
inst✝² : SMul M α
inst✝¹ : MeasurableSMul M α
c : M
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
s : Set α
hs : MeasurableSet s
⊢ ↑↑μ ((fun x x_1 => x • x_1) c ⁻¹' s) = ↑↑μ s
case h
G : Type ?u.6308
M : Type u_2
α : Type u_1
s✝ : Set α
m : MeasurableSpace α
inst✝³ : MeasurableSpace M
inst✝² : SMul M α
inst✝¹ : MeasurableSMul M α
c : M
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
s : Set α
hs : MeasurableSet s
⊢ ↑↑(map (fun x => c • x) μ) s = ↑↑μ s
rw [map_apply (measurable_const_smul c) hs]
no goals
case h
G : Type ?u.6308
M : Type u_2
α : Type u_1
s✝ : Set α
m : MeasurableSpace α
inst✝³ : MeasurableSpace M
inst✝² : SMul M α
inst✝¹ : MeasurableSMul M α
c : M
μ : Measure α
inst✝ : SMulInvariantMeasure M α μ
s : Set α
hs : MeasurableSet s
⊢ ↑↑μ ((fun x x_1 => x • x_1) c ⁻¹' s) = ↑↑μ s
exact SMulInvariantMeasure.measure_preimage_smul c hs