[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
x : R
⊢ MeasurableSet (floor ⁻¹' {⌊x⌋})
[PROOFSTEP]
simpa only [Int.preimage_floor_singleton] using measurableSet_Ico
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
x : R
⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})
[PROOFSTEP]
simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : BorelSpace R
⊢ Measurable Int.fract
[PROOFSTEP]
intro s hs
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : BorelSpace R
s : Set R
hs : MeasurableSet s
⊢ MeasurableSet (Int.fract ⁻¹' s)
[PROOFSTEP]
rw [Int.preimage_fract]
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : BorelSpace R
s : Set R
hs : MeasurableSet s
⊢ MeasurableSet (⋃ (m : ℤ), (fun x => x - ↑m) ⁻¹' (s ∩ Ico 0 1))
[PROOFSTEP]
exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico)
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : BorelSpace R
s : Set R
hs : MeasurableSet s
⊢ MeasurableSet (Int.fract '' s)
[PROOFSTEP]
simp only [Int.image_fract, sub_eq_add_neg, image_add_right']
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedRing R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : BorelSpace R
s : Set R
hs : MeasurableSet s
⊢ MeasurableSet (⋃ (m : ℤ), (fun x => x + ↑m) ⁻¹' s ∩ Ico 0 1)
[PROOFSTEP]
exact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})
[PROOFSTEP]
cases' eq_or_ne ⌊n⌋₊ 0 with h h
[GOAL]
case inl
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
h : ⌊n⌋₊ = 0
⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})
[PROOFSTEP]
simp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]
[GOAL]
case inr
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
h : ⌊n⌋₊ ≠ 0
⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})
[PROOFSTEP]
simp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]
[GOAL]
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
⊢ MeasurableSet (ceil ⁻¹' {⌈n⌉₊})
[PROOFSTEP]
cases' eq_or_ne ⌈n⌉₊ 0 with h h
[GOAL]
case inl
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
h : ⌈n⌉₊ = 0
⊢ MeasurableSet (ceil ⁻¹' {⌈n⌉₊})
[PROOFSTEP]
simp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]
[GOAL]
case inr
α : Type u_1
R : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : LinearOrderedSemiring R
inst✝⁴ : FloorSemiring R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
f : α → R
n : R
h : ⌈n⌉₊ ≠ 0
⊢ MeasurableSet (ceil ⁻¹' {⌈n⌉₊})
[PROOFSTEP]
simp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]
