ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measurePreserving_prod_mul_swap
@[to_additive measurePreserving_prod_add_swap
" The map `(x, y) ↦ (y, y + x)` sends the measure `μ × ν` to `ν × μ`. "]
theorem measurePreserving_prod_mul_swap [IsMulLeftInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.2 * z.1)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsMulLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.2 * z.1)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_prod_add_swap
@[to_additive measurePreserving_prod_add_swap
" The map `(x, y) ↦ (y, y + x)` sends the measure `μ × ν` to `ν × μ`. "]
theorem measurePreserving_prod_mul_swap [IsMulLeftInvariant μ] :
    MeasurePreserving (fun z : G × G => (z.2, z.2 * z.1)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.2 + z.1)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurable_measure_mul_right
@[to_additive]
theorem measurable_measure_mul_right (hs : MeasurableSet s) :
    Measurable fun x => μ ((fun y => y * x) ⁻¹' s) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} (hs : MeasurableSet s)
Goal: Measurable fun x => μ ((fun y => y * x) ⁻¹' s)
-- MeasureTheory.measurable_measure_add_right
@[to_additive]
theorem measurable_measure_mul_right (hs : MeasurableSet s) :
    Measurable fun x => μ ((fun y => y * x) ⁻¹' s) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} (hs : MeasurableSet s)
Goal: Measurable fun x => μ ((fun y => y + x) ⁻¹' s)
-- MeasureTheory.measurePreserving_prod_neg_add
@[to_additive measurePreserving_prod_neg_add
"The map `(x, y) ↦ (x, - x + y)` is measure-preserving."]
theorem measurePreserving_prod_inv_mul [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.1⁻¹ * z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [ν.IsAddLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, -z.1 + z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_inv_mul
@[to_additive measurePreserving_prod_neg_add
"The map `(x, y) ↦ (x, - x + y)` is measure-preserving."]
theorem measurePreserving_prod_inv_mul [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.1⁻¹ * z.2)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [ν.IsMulLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.1⁻¹ * z.2)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_inv_mul_swap
@[to_additive measurePreserving_prod_neg_add_swap
"The map `(x, y) ↦ (y, - y + x)` sends `μ × ν` to `ν × μ`."]
theorem measurePreserving_prod_inv_mul_swap :
    MeasurePreserving (fun z : G × G => (z.2, z.2⁻¹ * z.1)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, z.2⁻¹ * z.1)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_prod_neg_add_swap
@[to_additive measurePreserving_prod_neg_add_swap
"The map `(x, y) ↦ (y, - y + x)` sends `μ × ν` to `ν × μ`."]
theorem measurePreserving_prod_inv_mul_swap :
    MeasurePreserving (fun z : G × G => (z.2, z.2⁻¹ * z.1)) (μ.prod ν) (ν.prod μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2, -z.2 + z.1)) (μ.prod ν) (ν.prod μ)
-- MeasureTheory.measurePreserving_mul_prod_inv
@[to_additive measurePreserving_add_prod_neg
"The map `(x, y) ↦ (y + x, - x)` is measure-preserving."]
theorem measurePreserving_mul_prod_inv [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] [ν.IsMulLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_add_prod_neg
@[to_additive measurePreserving_add_prod_neg
"The map `(x, y) ↦ (y + x, - x)` is measure-preserving."]
theorem measurePreserving_mul_prod_inv [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] [ν.IsAddLeftInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.2 + z.1, -z.1)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.quasiMeasurePreserving_inv
@[to_additive]
theorem quasiMeasurePreserving_inv : QuasiMeasurePreserving (Inv.inv : G → G) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Inv.inv μ μ
-- MeasureTheory.quasiMeasurePreserving_neg
@[to_additive]
theorem quasiMeasurePreserving_inv : QuasiMeasurePreserving (Inv.inv : G → G) μ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Neg.neg μ μ
-- MeasureTheory.measure_neg_null
@[to_additive (attr := simp)]
theorem measure_inv_null : μ s⁻¹ = 0 ↔ μ s = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: μ (-s) = 0 ↔ μ s = 0
-- MeasureTheory.measure_inv_null
@[to_additive (attr := simp)]
theorem measure_inv_null : μ s⁻¹ = 0 ↔ μ s = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: μ s⁻¹ = 0 ↔ μ s = 0
-- MeasureTheory.neg_ae
@[to_additive (attr := simp)]
theorem inv_ae : (ae μ)⁻¹ = ae μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: -MeasureTheory.ae μ = MeasureTheory.ae μ
-- MeasureTheory.inv_ae
@[to_additive (attr := simp)]
theorem inv_ae : (ae μ)⁻¹ = ae μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: (MeasureTheory.ae μ)⁻¹ = MeasureTheory.ae μ
-- MeasureTheory.eventuallyConst_inv_set_ae
@[to_additive (attr := simp)]
theorem eventuallyConst_inv_set_ae :
    EventuallyConst (s⁻¹ : Set G) (ae μ) ↔ EventuallyConst s (ae μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: Filter.EventuallyConst s⁻¹ (MeasureTheory.ae μ) ↔ Filter.EventuallyConst s (MeasureTheory.ae μ)
-- MeasureTheory.eventuallyConst_neg_set_ae
@[to_additive (attr := simp)]
theorem eventuallyConst_inv_set_ae :
    EventuallyConst (s⁻¹ : Set G) (ae μ) ↔ EventuallyConst s (ae μ) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: Filter.EventuallyConst (-s) (MeasureTheory.ae μ) ↔ Filter.EventuallyConst s (MeasureTheory.ae μ)
-- MeasureTheory.neg_absolutelyContinuous
@[to_additive]
theorem inv_absolutelyContinuous : μ.inv ≪ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: μ.neg ≪ μ
-- MeasureTheory.inv_absolutelyContinuous
@[to_additive]
theorem inv_absolutelyContinuous : μ.inv ≪ μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: μ.inv ≪ μ
-- MeasureTheory.absolutelyContinuous_neg
@[to_additive]
theorem absolutelyContinuous_inv : μ ≪ μ.inv 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant]
Goal: μ ≪ μ.neg
-- MeasureTheory.absolutelyContinuous_inv
@[to_additive]
theorem absolutelyContinuous_inv : μ ≪ μ.inv 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant]
Goal: μ ≪ μ.inv
-- MeasureTheory.lintegral_lintegral_add_neg
@[to_additive]
theorem lintegral_lintegral_mul_inv [IsMulLeftInvariant ν] (f : G → G → ℝ≥0∞)
    (hf : AEMeasurable (uncurry f) (μ.prod ν)) :
    (∫⁻ x, ∫⁻ y, f (y * x) x⁻¹ ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] [ν.IsAddLeftInvariant] (f : G → G → ℝ≥0∞) (hf : AEMeasurable (Function.uncurry f) (μ.prod ν))
Goal: ∫⁻ (x : G), ∫⁻ (y : G), f (y + x) (-x) ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
-- MeasureTheory.lintegral_lintegral_mul_inv
@[to_additive]
theorem lintegral_lintegral_mul_inv [IsMulLeftInvariant ν] (f : G → G → ℝ≥0∞)
    (hf : AEMeasurable (uncurry f) (μ.prod ν)) :
    (∫⁻ x, ∫⁻ y, f (y * x) x⁻¹ ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] [ν.IsMulLeftInvariant] (f : G → G → ℝ≥0∞) (hf : AEMeasurable (Function.uncurry f) (μ.prod ν))
Goal: ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
-- MeasureTheory.measure_add_right_null
@[to_additive]
theorem measure_mul_right_null (y : G) : μ ((fun x => x * y) ⁻¹' s) = 0 ↔ μ s = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableNeg G] [μ.IsAddLeftInvariant] (y : G)
Goal: μ ((fun x => x + y) ⁻¹' s) = 0 ↔ μ s = 0
-- MeasureTheory.measure_mul_right_null
@[to_additive]
theorem measure_mul_right_null (y : G) : μ ((fun x => x * y) ⁻¹' s) = 0 ↔ μ s = 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableInv G] [μ.IsMulLeftInvariant] (y : G)
Goal: μ ((fun x => x * y) ⁻¹' s) = 0 ↔ μ s = 0
-- MeasureTheory.measure_add_right_ne_zero
@[to_additive]
theorem measure_mul_right_ne_zero (h2s : μ s ≠ 0) (y : G) : μ ((fun x => x * y) ⁻¹' s) ≠ 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableNeg G] [μ.IsAddLeftInvariant] (h2s : μ s ≠ 0) (y : G)
Goal: μ ((fun x => x + y) ⁻¹' s) ≠ 0
-- MeasureTheory.measure_mul_right_ne_zero
@[to_additive]
theorem measure_mul_right_ne_zero (h2s : μ s ≠ 0) (y : G) : μ ((fun x => x * y) ⁻¹' s) ≠ 0 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] {s : Set G} [MeasurableInv G] [μ.IsMulLeftInvariant] (h2s : μ s ≠ 0) (y : G)
Goal: μ ((fun x => x * y) ⁻¹' s) ≠ 0
-- MeasureTheory.absolutelyContinuous_map_mul_right
@[to_additive]
theorem absolutelyContinuous_map_mul_right (g : G) : μ ≪ map (· * g) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] (g : G)
Goal: μ ≪ MeasureTheory.Measure.map (fun x => x * g) μ
-- MeasureTheory.absolutelyContinuous_map_add_right
@[to_additive]
theorem absolutelyContinuous_map_mul_right (g : G) : μ ≪ map (· * g) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] (g : G)
Goal: μ ≪ MeasureTheory.Measure.map (fun x => x + g) μ
-- MeasureTheory.absolutelyContinuous_map_sub_left
@[to_additive]
theorem absolutelyContinuous_map_div_left (g : G) : μ ≪ map (fun h => g / h) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] (g : G)
Goal: μ ≪ MeasureTheory.Measure.map (fun h => g - h) μ
-- MeasureTheory.absolutelyContinuous_map_div_left
@[to_additive]
theorem absolutelyContinuous_map_div_left (g : G) : μ ≪ map (fun h => g / h) μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ : MeasureTheory.Measure G) [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] (g : G)
Goal: μ ≪ MeasureTheory.Measure.map (fun h => g / h) μ
-- MeasureTheory.measure_mul_lintegral_eq
@[to_additive "This is the computation performed in the proof of [Halmos, §60 Th. A]."]
theorem measure_mul_lintegral_eq [IsMulLeftInvariant ν] (sm : MeasurableSet s) (f : G → ℝ≥0∞)
    (hf : Measurable f) : (μ s * ∫⁻ y, f y ∂ν) = ∫⁻ x, ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] {s : Set G} [MeasurableInv G] [μ.IsMulLeftInvariant] [ν.IsMulLeftInvariant] (sm : MeasurableSet s) (f : G → ℝ≥0∞) (hf : Measurable f)
Goal: μ s * ∫⁻ (y : G), f y ∂ν = ∫⁻ (x : G), ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ
-- MeasureTheory.measure_add_lintegral_eq
@[to_additive "This is the computation performed in the proof of [Halmos, §60 Th. A]."]
theorem measure_mul_lintegral_eq [IsMulLeftInvariant ν] (sm : MeasurableSet s) (f : G → ℝ≥0∞)
    (hf : Measurable f) : (μ s * ∫⁻ y, f y ∂ν) = ∫⁻ x, ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] {s : Set G} [MeasurableNeg G] [μ.IsAddLeftInvariant] [ν.IsAddLeftInvariant] (sm : MeasurableSet s) (f : G → ℝ≥0∞) (hf : Measurable f)
Goal: μ s * ∫⁻ (y : G), f y ∂ν = ∫⁻ (x : G), ν ((fun z => z + x) ⁻¹' s) * f (-x) ∂μ
-- MeasureTheory.absolutelyContinuous_of_isAddLeftInvariant
@[to_additive
" Any two nonzero left-invariant measures are absolutely continuous w.r.t. each other. "]
theorem absolutelyContinuous_of_isMulLeftInvariant [IsMulLeftInvariant ν] (hν : ν ≠ 0) : μ ≪ ν 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableNeg G] [μ.IsAddLeftInvariant] [ν.IsAddLeftInvariant] (hν : ν ≠ 0)
Goal: μ ≪ ν
-- MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant
@[to_additive
" Any two nonzero left-invariant measures are absolutely continuous w.r.t. each other. "]
theorem absolutelyContinuous_of_isMulLeftInvariant [IsMulLeftInvariant ν] (hν : ν ≠ 0) : μ ≪ ν 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasurableInv G] [μ.IsMulLeftInvariant] [ν.IsMulLeftInvariant] (hν : ν ≠ 0)
Goal: μ ≪ ν
-- MeasureTheory.ae_measure_preimage_mul_right_lt_top
@[to_additive]
theorem ae_measure_preimage_mul_right_lt_top (hμs : μ' s ≠ ∞) :
    ∀ᵐ x ∂μ', ν' ((· * x) ⁻¹' s) < ∞ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] {s : Set G} [MeasurableInv G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsMulLeftInvariant] [ν'.IsMulLeftInvariant] (hμs : μ' s ≠ ⊤)
Goal: ∀ᵐ (x : G) ∂μ', ν' ((fun x_1 => x_1 * x) ⁻¹' s) < ⊤
-- MeasureTheory.ae_measure_preimage_add_right_lt_top
@[to_additive]
theorem ae_measure_preimage_mul_right_lt_top (hμs : μ' s ≠ ∞) :
    ∀ᵐ x ∂μ', ν' ((· * x) ⁻¹' s) < ∞ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] {s : Set G} [MeasurableNeg G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsAddLeftInvariant] [ν'.IsAddLeftInvariant] (hμs : μ' s ≠ ⊤)
Goal: ∀ᵐ (x : G) ∂μ', ν' ((fun x_1 => x_1 + x) ⁻¹' s) < ⊤
-- MeasureTheory.ae_measure_preimage_add_right_lt_top_of_ne_zero
@[to_additive]
theorem ae_measure_preimage_mul_right_lt_top_of_ne_zero (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    ∀ᵐ x ∂μ', ν' ((fun y => y * x) ⁻¹' s) < ∞ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] {s : Set G} [MeasurableNeg G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsAddLeftInvariant] [ν'.IsAddLeftInvariant] (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: ∀ᵐ (x : G) ∂μ', ν' ((fun y => y + x) ⁻¹' s) < ⊤
-- MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero
@[to_additive]
theorem ae_measure_preimage_mul_right_lt_top_of_ne_zero (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    ∀ᵐ x ∂μ', ν' ((fun y => y * x) ⁻¹' s) < ∞ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] {s : Set G} [MeasurableInv G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsMulLeftInvariant] [ν'.IsMulLeftInvariant] (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: ∀ᵐ (x : G) ∂μ', ν' ((fun y => y * x) ⁻¹' s) < ⊤
-- MeasureTheory.measure_lintegral_sub_measure
@[to_additive
"A technical lemma relating two different measures. This is basically [Halmos, §60 Th. A]. Note that
if `f` is the characteristic function of a measurable set `t` this states that `μ t = c * μ s` for a
constant `c` that does not depend on `μ`.

Note: There is a gap in the last step of the proof in [Halmos]. In the last line, the equality
`g(-x) + ν(s - x) = f(x)` holds if we can prove that `0 < ν(s - x) < ∞`. The first inequality
follows from §59, Th. D, but the second inequality is not justified. We prove this inequality for
almost all `x` in `MeasureTheory.ae_measure_preimage_add_right_lt_top_of_ne_zero`."]
theorem measure_lintegral_div_measure (sm : MeasurableSet s) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞)
    (f : G → ℝ≥0∞) (hf : Measurable f) :
    (μ' s * ∫⁻ y, f y⁻¹ / ν' ((· * y⁻¹) ⁻¹' s) ∂ν') = ∫⁻ x, f x ∂μ' 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] {s : Set G} [MeasurableNeg G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsAddLeftInvariant] [ν'.IsAddLeftInvariant] (sm : MeasurableSet s) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤) (f : G → ℝ≥0∞) (hf : Measurable f)
Goal: μ' s * ∫⁻ (y : G), f (-y) / ν' ((fun x => x + -y) ⁻¹' s) ∂ν' = ∫⁻ (x : G), f x ∂μ'
-- MeasureTheory.measure_lintegral_div_measure
@[to_additive
"A technical lemma relating two different measures. This is basically [Halmos, §60 Th. A]. Note that
if `f` is the characteristic function of a measurable set `t` this states that `μ t = c * μ s` for a
constant `c` that does not depend on `μ`.

Note: There is a gap in the last step of the proof in [Halmos]. In the last line, the equality
`g(-x) + ν(s - x) = f(x)` holds if we can prove that `0 < ν(s - x) < ∞`. The first inequality
follows from §59, Th. D, but the second inequality is not justified. We prove this inequality for
almost all `x` in `MeasureTheory.ae_measure_preimage_add_right_lt_top_of_ne_zero`."]
theorem measure_lintegral_div_measure (sm : MeasurableSet s) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞)
    (f : G → ℝ≥0∞) (hf : Measurable f) :
    (μ' s * ∫⁻ y, f y⁻¹ / ν' ((· * y⁻¹) ⁻¹' s) ∂ν') = ∫⁻ x, f x ∂μ' 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] {s : Set G} [MeasurableInv G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsMulLeftInvariant] [ν'.IsMulLeftInvariant] (sm : MeasurableSet s) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤) (f : G → ℝ≥0∞) (hf : Measurable f)
Goal: μ' s * ∫⁻ (y : G), f y⁻¹ / ν' ((fun x => x * y⁻¹) ⁻¹' s) ∂ν' = ∫⁻ (x : G), f x ∂μ'
-- MeasureTheory.measure_add_measure_eq
@[to_additive]
theorem measure_mul_measure_eq (s t : Set G) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    μ' s * ν' t = ν' s * μ' t 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsAddLeftInvariant] [ν'.IsAddLeftInvariant] (s t : Set G) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: μ' s * ν' t = ν' s * μ' t
-- MeasureTheory.measure_mul_measure_eq
@[to_additive]
theorem measure_mul_measure_eq (s t : Set G) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    μ' s * ν' t = ν' s * μ' t 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] [MeasurableInv G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsMulLeftInvariant] [ν'.IsMulLeftInvariant] (s t : Set G) (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: μ' s * ν' t = ν' s * μ' t
-- MeasureTheory.measure_eq_div_smul
@[to_additive
" Left invariant Borel measures on an additive measurable group are unique (up to a scalar). "]
theorem measure_eq_div_smul (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    μ' = (μ' s / ν' s) • ν' 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] {s : Set G} [MeasurableInv G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsMulLeftInvariant] [ν'.IsMulLeftInvariant] (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: μ' = (μ' s / ν' s) • ν'
-- MeasureTheory.measure_eq_sub_vadd
@[to_additive
" Left invariant Borel measures on an additive measurable group are unique (up to a scalar). "]
theorem measure_eq_div_smul (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ∞) :
    μ' = (μ' s / ν' s) • ν' 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] {s : Set G} [MeasurableNeg G] (μ' ν' : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ'] [MeasureTheory.SigmaFinite ν'] [μ'.IsAddLeftInvariant] [ν'.IsAddLeftInvariant] (h2s : ν' s ≠ 0) (h3s : ν' s ≠ ⊤)
Goal: μ' = (μ' s / ν' s) • ν'
-- MeasureTheory.measurePreserving_prod_mul_right
@[to_additive measurePreserving_prod_add_right]
theorem measurePreserving_prod_mul_right [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.2 * z.1)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [ν.IsMulRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.2 * z.1)) (μ.prod ν) (μ.prod ν)
-- MeasureTheory.measurePreserving_prod_add_right
@[to_additive measurePreserving_prod_add_right]
theorem measurePreserving_prod_mul_right [IsMulRightInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.2 * z.1)) (μ.prod ν) (μ.prod ν) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd₂ G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [ν.IsAddRightInvariant]
Goal: MeasureTheory.MeasurePreserving (fun z => (z.1, z.2 + z.1)) (μ.prod ν) (μ.prod ν)