Documentation

Mathlib.MeasureTheory.Constructions.Pi

Product measures #

In this file we define and prove properties about finite products of measures (and at some point, countable products of measures).

Main definition #

To apply Fubini's theorem or Tonelli's theorem along some subset, we recommend using the marginal construction MeasureTheory.lmarginal and (todo) MeasureTheory.marginal. This allows you to apply the theorems without any bookkeeping with measurable equivalences.

Implementation Notes #

We define MeasureTheory.OuterMeasure.pi, the product of finitely many outer measures, as the maximal outer measure n with the property that n (pi univ s) ≤ ∏ i, m i (s i), where pi univ s is the product of the sets {s i | i : ι}.

We then show that this induces a product of measures, called MeasureTheory.Measure.pi. For a collection of σ-finite measures μ and a collection of measurable sets s we show that Measure.pi μ (pi univ s) = ∏ i, m i (s i). To do this, we follow the following steps:

Tags #

finitary product measure

We start with some measurability properties

theorem IsPiSystem.pi {ι : Type u_1} {α : ιType u_3} {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsPiSystem (C i)) :
IsPiSystem (Set.pi Set.univ '' Set.pi Set.univ C)

Boxes formed by π-systems form a π-system.

theorem isPiSystem_pi {ι : Type u_1} {α : ιType u_3} [(i : ι) → MeasurableSpace (α i)] :
IsPiSystem (Set.pi Set.univ '' Set.pi Set.univ fun (i : ι) => {s : Set (α i) | MeasurableSet s})

Boxes form a π-system.

theorem IsCountablySpanning.pi {ι : Type u_1} {α : ιType u_3} [Finite ι] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsCountablySpanning (C i)) :
IsCountablySpanning (Set.pi Set.univ '' Set.pi Set.univ C)

Boxes of countably spanning sets are countably spanning.

theorem generateFrom_pi_eq {ι : Type u_1} {α : ιType u_3} [Finite ι] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsCountablySpanning (C i)) :
MeasurableSpace.pi = MeasurableSpace.generateFrom (Set.pi Set.univ '' Set.pi Set.univ C)

The product of generated σ-algebras is the one generated by boxes, if both generating sets are countably spanning.

theorem generateFrom_eq_pi {ι : Type u_1} {α : ιType u_3} [Finite ι] [h : (i : ι) → MeasurableSpace (α i)] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), MeasurableSpace.generateFrom (C i) = h i) (h2C : ∀ (i : ι), IsCountablySpanning (C i)) :
MeasurableSpace.generateFrom (Set.pi Set.univ '' Set.pi Set.univ C) = MeasurableSpace.pi

If C and D generate the σ-algebras on α resp. β, then rectangles formed by C and D generate the σ-algebra on α × β.

theorem generateFrom_pi {ι : Type u_1} {α : ιType u_3} [Finite ι] [(i : ι) → MeasurableSpace (α i)] :
MeasurableSpace.generateFrom (Set.pi Set.univ '' Set.pi Set.univ fun (i : ι) => {s : Set (α i) | MeasurableSet s}) = MeasurableSpace.pi

The product σ-algebra is generated from boxes, i.e. s ×ˢ t for sets s : set α and t : set β.

def MeasureTheory.piPremeasure {ι : Type u_1} {α : ιType u_3} [Fintype ι] (m : (i : ι) → MeasureTheory.OuterMeasure (α i)) (s : Set ((i : ι) → α i)) :

An upper bound for the measure in a finite product space. It is defined to by taking the image of the set under all projections, and taking the product of the measures of these images. For measurable boxes it is equal to the correct measure.

Equations
Instances For
    theorem MeasureTheory.piPremeasure_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : (i : ι) → Set (α i)} (hs : Set.Nonempty (Set.pi Set.univ s)) :
    MeasureTheory.piPremeasure m (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (m i) (s i)
    theorem MeasureTheory.piPremeasure_pi' {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : (i : ι) → Set (α i)} :
    MeasureTheory.piPremeasure m (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (m i) (s i)
    theorem MeasureTheory.piPremeasure_pi_mono {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : Set ((i : ι) → α i)} {t : Set ((i : ι) → α i)} (h : s t) :
    theorem MeasureTheory.piPremeasure_pi_eval {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : Set ((i : ι) → α i)} :
    def MeasureTheory.OuterMeasure.pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] (m : (i : ι) → MeasureTheory.OuterMeasure (α i)) :
    MeasureTheory.OuterMeasure ((i : ι) → α i)

    OuterMeasure.pi m is the finite product of the outer measures {m i | i : ι}. It is defined to be the maximal outer measure n with the property that n (pi univ s) ≤ ∏ i, m i (s i), where pi univ s is the product of the sets {s i | i : ι}.

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      theorem MeasureTheory.OuterMeasure.pi_pi_le {ι : Type u_1} {α : ιType u_3} [Fintype ι] (m : (i : ι) → MeasureTheory.OuterMeasure (α i)) (s : (i : ι) → Set (α i)) :
      (MeasureTheory.OuterMeasure.pi m) (Set.pi Set.univ s) Finset.prod Finset.univ fun (i : ι) => (m i) (s i)
      theorem MeasureTheory.OuterMeasure.le_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {n : MeasureTheory.OuterMeasure ((i : ι) → α i)} :
      n MeasureTheory.OuterMeasure.pi m ∀ (s : (i : ι) → Set (α i)), Set.Nonempty (Set.pi Set.univ s)n (Set.pi Set.univ s) Finset.prod Finset.univ fun (i : ι) => (m i) (s i)
      def MeasureTheory.Measure.tprod {δ : Type u_4} {π : δType u_5} [(x : δ) → MeasurableSpace (π x)] (l : List δ) (μ : (i : δ) → MeasureTheory.Measure (π i)) :

      A product of measures in tprod α l.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[simp]
        theorem MeasureTheory.Measure.tprod_nil {δ : Type u_4} {π : δType u_5} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i)) :
        @[simp]
        theorem MeasureTheory.Measure.tprod_cons {δ : Type u_4} {π : δType u_5} [(x : δ) → MeasurableSpace (π x)] (i : δ) (l : List δ) (μ : (i : δ) → MeasureTheory.Measure (π i)) :
        theorem MeasureTheory.Measure.tprod_tprod {δ : Type u_4} {π : δType u_5} [(x : δ) → MeasurableSpace (π x)] (l : List δ) (μ : (i : δ) → MeasureTheory.Measure (π i)) [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] (s : (i : δ) → Set (π i)) :
        (MeasureTheory.Measure.tprod l μ) (Set.tprod l s) = List.prod (List.map (fun (i : δ) => (μ i) (s i)) l)
        def MeasureTheory.Measure.pi' {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [Encodable ι] :
        MeasureTheory.Measure ((i : ι) → α i)

        The product measure on an encodable finite type, defined by mapping Measure.tprod along the equivalence MeasurableEquiv.piMeasurableEquivTProd. The definition MeasureTheory.Measure.pi should be used instead of this one.

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          theorem MeasureTheory.Measure.pi'_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [Encodable ι] [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i)) :
          (MeasureTheory.Measure.pi' μ) (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (μ i) (s i)
          theorem MeasureTheory.Measure.pi_caratheodory {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) :
          MeasurableSpace.pi MeasureTheory.OuterMeasure.caratheodory (MeasureTheory.OuterMeasure.pi fun (i : ι) => (μ i))
          @[irreducible]
          def MeasureTheory.Measure.pi {ι : Type u_4} {α : ιType u_5} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) :
          MeasureTheory.Measure ((i : ι) → α i)

          Measure.pi μ is the finite product of the measures {μ i | i : ι}. It is defined to be measure corresponding to MeasureTheory.OuterMeasure.pi.

          Equations
          Instances For
            theorem MeasureTheory.Measure.pi_def {ι : Type u_4} {α : ιType u_5} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) :
            instance MeasureTheory.MeasureSpace.pi {ι : Type u_1} [Fintype ι] {α : ιType u_4} [(i : ι) → MeasureTheory.MeasureSpace (α i)] :
            MeasureTheory.MeasureSpace ((i : ι) → α i)
            Equations
            theorem MeasureTheory.Measure.pi_pi_aux {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i)) (hs : ∀ (i : ι), MeasurableSet (s i)) :
            (MeasureTheory.Measure.pi μ) (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (μ i) (s i)
            def MeasureTheory.Measure.FiniteSpanningSetsIn.pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} {C : (i : ι) → Set (Set (α i))} (hμ : (i : ι) → MeasureTheory.Measure.FiniteSpanningSetsIn (μ i) (C i)) :

            Measure.pi μ has finite spanning sets in rectangles of finite spanning sets.

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              theorem MeasureTheory.Measure.pi_eq_generateFrom {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), MeasurableSpace.generateFrom (C i) = inst✝ i) (h2C : ∀ (i : ι), IsPiSystem (C i)) (h3C : (i : ι) → MeasureTheory.Measure.FiniteSpanningSetsIn (μ i) (C i)) {μν : MeasureTheory.Measure ((i : ι) → α i)} (h₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i C i)μν (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (μ i) (s i)) :

              A measure on a finite product space equals the product measure if they are equal on rectangles with as sides sets that generate the corresponding σ-algebras.

              theorem MeasureTheory.Measure.pi_eq {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {μ' : MeasureTheory.Measure ((i : ι) → α i)} (h : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), MeasurableSet (s i))μ' (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (μ i) (s i)) :

              A measure on a finite product space equals the product measure if they are equal on rectangles.

              theorem MeasureTheory.Measure.pi'_eq_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [Encodable ι] :
              @[simp]
              theorem MeasureTheory.Measure.pi_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i)) :
              (MeasureTheory.Measure.pi μ) (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => (μ i) (s i)
              theorem MeasureTheory.Measure.pi_univ {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] :
              (MeasureTheory.Measure.pi μ) Set.univ = Finset.prod Finset.univ fun (i : ι) => (μ i) Set.univ
              theorem MeasureTheory.Measure.pi_ball {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : } (hr : 0 < r) :
              (MeasureTheory.Measure.pi μ) (Metric.ball x r) = Finset.prod Finset.univ fun (i : ι) => (μ i) (Metric.ball (x i) r)
              theorem MeasureTheory.Measure.pi_closedBall {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : } (hr : 0 r) :
              (MeasureTheory.Measure.pi μ) (Metric.closedBall x r) = Finset.prod Finset.univ fun (i : ι) => (μ i) (Metric.closedBall (x i) r)
              instance MeasureTheory.Measure.pi.sigmaFinite {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] :
              Equations
              instance MeasureTheory.Measure.instSigmaFiniteForAllToMeasurableSpacePiVolume {ι : Type u_1} [Fintype ι] {α : ιType u_4} [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] :
              MeasureTheory.SigmaFinite MeasureTheory.volume
              Equations
              theorem MeasureTheory.Measure.pi_of_empty {α : Type u_4} [Fintype α] [IsEmpty α] {β : αType u_5} {m : (a : α) → MeasurableSpace (β a)} (μ : (a : α) → MeasureTheory.Measure (β a)) (x : optParam ((a : α) → β a) fun (a : α) => isEmptyElim a) :
              theorem MeasureTheory.Measure.volume_pi_eq_dirac {ι : Type u_4} [Fintype ι] [IsEmpty ι] {α : ιType u_5} [(i : ι) → MeasureTheory.MeasureSpace (α i)] (x : optParam ((a : ι) → α a) fun (a : ι) => isEmptyElim a) :
              MeasureTheory.volume = MeasureTheory.Measure.dirac x
              @[simp]
              theorem MeasureTheory.Measure.pi_empty_univ {α : Type u_4} [Fintype α] [IsEmpty α] {β : αType u_5} {m : (α : α) → MeasurableSpace (β α)} (μ : (a : α) → MeasureTheory.Measure (β a)) :
              (MeasureTheory.Measure.pi μ) Set.univ = 1
              theorem MeasureTheory.Measure.pi_eval_preimage_null {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {i : ι} {s : Set (α i)} (hs : (μ i) s = 0) :
              theorem MeasureTheory.Measure.pi_hyperplane {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)] (x : α i) :
              (MeasureTheory.Measure.pi μ) {f : (i : ι) → α i | f i = x} = 0
              theorem MeasureTheory.Measure.ae_eval_ne {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)] (x : α i) :
              ∀ᵐ (y : (i : ι) → α i) ∂MeasureTheory.Measure.pi μ, y i x
              theorem MeasureTheory.Measure.tendsto_eval_ae_ae {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {i : ι} :
              theorem MeasureTheory.Measure.ae_pi_le_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] :
              theorem MeasureTheory.Measure.ae_eq_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {β : ιType u_4} {f : (i : ι) → α iβ i} {f' : (i : ι) → α iβ i} (h : ∀ (i : ι), f i =ᶠ[MeasureTheory.Measure.ae (μ i)] f' i) :
              (fun (x : (i : ι) → α i) (i : ι) => f i (x i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] fun (x : (i : ι) → α i) (i : ι) => f' i (x i)
              theorem MeasureTheory.Measure.ae_le_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {β : ιType u_4} [(i : ι) → Preorder (β i)] {f : (i : ι) → α iβ i} {f' : (i : ι) → α iβ i} (h : ∀ (i : ι), f i ≤ᶠ[MeasureTheory.Measure.ae (μ i)] f' i) :
              (fun (x : (i : ι) → α i) (i : ι) => f i (x i)) ≤ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] fun (x : (i : ι) → α i) (i : ι) => f' i (x i)
              theorem MeasureTheory.Measure.ae_le_set_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {I : Set ι} {s : (i : ι) → Set (α i)} {t : (i : ι) → Set (α i)} (h : iI, s i ≤ᶠ[MeasureTheory.Measure.ae (μ i)] t i) :
              theorem MeasureTheory.Measure.ae_eq_set_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {I : Set ι} {s : (i : ι) → Set (α i)} {t : (i : ι) → Set (α i)} (h : iI, s i =ᶠ[MeasureTheory.Measure.ae (μ i)] t i) :
              theorem MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Iio (f i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Iic (f i)
              theorem MeasureTheory.Measure.pi_Ioi_ae_eq_pi_Ici {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Ioi (f i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Ici (f i)
              theorem MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i} :
              theorem MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i} :
              theorem MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Ioo (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Icc (f i) (g i)
              theorem MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Ioc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Ioo (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Ioc (f i) (g i)
              theorem MeasureTheory.Measure.univ_pi_Ioo_ae_eq_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi Set.univ fun (i : ι) => Set.Ioo (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.Icc f g
              theorem MeasureTheory.Measure.pi_Ioc_ae_eq_pi_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Ioc (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Icc (f i) (g i)
              theorem MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi Set.univ fun (i : ι) => Set.Ioc (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.Icc f g
              theorem MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi s fun (i : ι) => Set.Ico (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.pi s fun (i : ι) => Set.Icc (f i) (g i)
              theorem MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i} {g : (i : ι) → α i} :
              (Set.pi Set.univ fun (i : ι) => Set.Ico (f i) (g i)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.pi μ)] Set.Icc f g
              theorem MeasureTheory.Measure.pi_noAtoms {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)] :

              If one of the measures μ i has no atoms, them Measure.pi µ has no atoms. The instance below assumes that all μ i have no atoms.

              instance MeasureTheory.Measure.pi_noAtoms' {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [h : Nonempty ι] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] :
              Equations
              instance MeasureTheory.Measure.instNoAtomsForAllToMeasurableSpacePiVolume {ι : Type u_1} [Fintype ι] {α : ιType u_4} [Nonempty ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.NoAtoms MeasureTheory.volume] :
              MeasureTheory.NoAtoms MeasureTheory.volume
              Equations
              instance MeasureTheory.Measure.instIsLocallyFiniteMeasureForAllToMeasurableSpacePiTopologicalSpaceVolume {ι : Type u_1} [Fintype ι] {X : ιType u_4} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → MeasureTheory.MeasureSpace (X i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.IsLocallyFiniteMeasure MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsAddLeftInvariantForAllToMeasurableSpacePiInstAddToAddToAddZeroClassToAddMonoidToSubNegAddMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → AddGroup (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableAdd (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsAddLeftInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsMulLeftInvariantForAllToMeasurableSpacePiInstMulToMulToMulOneClassToMonoidToDivInvMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → Group (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableMul (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsMulLeftInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsAddRightInvariantForAllToMeasurableSpacePiInstAddToAddToAddZeroClassToAddMonoidToSubNegAddMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → AddGroup (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableAdd (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsAddRightInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsMulRightInvariantForAllToMeasurableSpacePiInstMulToMulToMulOneClassToMonoidToDivInvMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → Group (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableMul (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsMulRightInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsNegInvariantForAllToMeasurableSpacePiInstNegToNegToNegZeroClassToSubNegZeroAddMonoidToDivisionAddMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → AddGroup (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableNeg (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsNegInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsInvInvariantForAllToMeasurableSpacePiInstInvToInvToInvOneClassToDivInvOneMonoidToDivisionMonoidVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → Group (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableInv (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsInvInvariant MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsOpenPosMeasureForAllTopologicalSpaceToMeasurableSpacePiVolume {ι : Type u_1} [Fintype ι] {X : ιType u_4} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → MeasureTheory.MeasureSpace (X i)] [∀ (i : ι), MeasureTheory.Measure.IsOpenPosMeasure MeasureTheory.volume] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsFiniteMeasureOnCompactsForAllToMeasurableSpacePiTopologicalSpaceVolume {ι : Type u_1} [Fintype ι] {X : ιType u_4} [(i : ι) → MeasureTheory.MeasureSpace (X i)] [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.IsFiniteMeasureOnCompacts MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.pi.isAddHaarMeasure {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → AddGroup (α i)] [(i : ι) → TopologicalSpace (α i)] [∀ (i : ι), MeasureTheory.Measure.IsAddHaarMeasure (μ i)] [∀ (i : ι), MeasurableAdd (α i)] :
              Equations
              instance MeasureTheory.Measure.pi.isHaarMeasure {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → Group (α i)] [(i : ι) → TopologicalSpace (α i)] [∀ (i : ι), MeasureTheory.Measure.IsHaarMeasure (μ i)] [∀ (i : ι), MeasurableMul (α i)] :
              Equations
              instance MeasureTheory.Measure.instIsAddHaarMeasureForAllAddGroupTopologicalSpaceToMeasurableSpacePiVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → AddGroup (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableAdd (G i)] [(i : ι) → TopologicalSpace (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsAddHaarMeasure MeasureTheory.volume] :
              Equations
              instance MeasureTheory.Measure.instIsHaarMeasureForAllGroupTopologicalSpaceToMeasurableSpacePiVolume {ι : Type u_1} [Fintype ι] {G : ιType u_4} [(i : ι) → Group (G i)] [(i : ι) → MeasureTheory.MeasureSpace (G i)] [∀ (i : ι), MeasurableMul (G i)] [(i : ι) → TopologicalSpace (G i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [∀ (i : ι), MeasureTheory.Measure.IsHaarMeasure MeasureTheory.volume] :
              Equations
              theorem MeasureTheory.volume_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] :
              MeasureTheory.volume = MeasureTheory.Measure.pi fun (x : ι) => MeasureTheory.volume
              theorem MeasureTheory.volume_pi_pi {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] (s : (i : ι) → Set (α i)) :
              MeasureTheory.volume (Set.pi Set.univ s) = Finset.prod Finset.univ fun (i : ι) => MeasureTheory.volume (s i)
              theorem MeasureTheory.volume_pi_ball {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : } (hr : 0 < r) :
              MeasureTheory.volume (Metric.ball x r) = Finset.prod Finset.univ fun (i : ι) => MeasureTheory.volume (Metric.ball (x i) r)
              theorem MeasureTheory.volume_pi_closedBall {ι : Type u_1} {α : ιType u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : } (hr : 0 r) :
              MeasureTheory.volume (Metric.closedBall x r) = Finset.prod Finset.univ fun (i : ι) => MeasureTheory.volume (Metric.closedBall (x i) r)

              We intentionally restrict this only to the nondependent function space, since type-class inference cannot find an instance for ι → ℝ when this is stated for dependent function spaces.

              Equations

              We intentionally restrict this only to the nondependent function space, since type-class inference cannot find an instance for ι → ℝ when this is stated for dependent function spaces.

              Equations

              We intentionally restrict this only to the nondependent function space, since type-class inference cannot find an instance for ι → ℝ when this is stated for dependent function spaces.

              Equations

              We intentionally restrict this only to the nondependent function space, since type-class inference cannot find an instance for ι → ℝ when this is stated for dependent function spaces.

              Equations

              Measure preserving equivalences #

              In this section we prove that some measurable equivalences (e.g., between Fin 1 → α and α or between Fin 2 → α and α × α) preserve measure or volume. These lemmas can be used to prove that measures of corresponding sets (images or preimages) have equal measures and functions f ∘ e and f have equal integrals, see lemmas in the MeasureTheory.measurePreserving prefix.

              theorem MeasureTheory.measurePreserving_piEquivPiSubtypeProd {ι : Type u_1} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (p : ιProp) [DecidablePred p] :
              theorem MeasureTheory.measurePreserving_piCongrLeft {ι : Type u_1} {ι' : Type u_2} {α : ιType u_3} [Fintype ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [Fintype ι'] (f : ι' ι) :
              theorem MeasureTheory.volume_measurePreserving_piCongrLeft {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (α : ιType u_4) (f : ι' ι) [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] :
              theorem MeasureTheory.measurePreserving_sumPiEquivProdPi_symm {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] {π : ι ι'Type u_4} {m : (i : ι ι') → MeasurableSpace (π i)} (μ : (i : ι ι') → MeasureTheory.Measure (π i)) [∀ (i : ι ι'), MeasureTheory.SigmaFinite (μ i)] :
              theorem MeasureTheory.volume_measurePreserving_sumPiEquivProdPi_symm {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (π : ι ι'Type u_4) [(i : ι ι') → MeasureTheory.MeasureSpace (π i)] [∀ (i : ι ι'), MeasureTheory.SigmaFinite MeasureTheory.volume] :
              theorem MeasureTheory.measurePreserving_sumPiEquivProdPi {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] {π : ι ι'Type u_4} {_m : (i : ι ι') → MeasurableSpace (π i)} (μ : (i : ι ι') → MeasureTheory.Measure (π i)) [∀ (i : ι ι'), MeasureTheory.SigmaFinite (μ i)] :
              theorem MeasureTheory.volume_measurePreserving_sumPiEquivProdPi {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [Fintype ι'] (π : ι ι'Type u_4) [(i : ι ι') → MeasureTheory.MeasureSpace (π i)] [∀ (i : ι ι'), MeasureTheory.SigmaFinite MeasureTheory.volume] :
              theorem MeasureTheory.measurePreserving_piFinSuccAbove {n : } {α : Fin (n + 1)Type u} {m : (i : Fin (n + 1)) → MeasurableSpace (α i)} (μ : (i : Fin (n + 1)) → MeasureTheory.Measure (α i)) [∀ (i : Fin (n + 1)), MeasureTheory.SigmaFinite (μ i)] (i : Fin (n + 1)) :
              theorem MeasureTheory.volume_preserving_piFinSuccAbove {n : } (α : Fin (n + 1)Type u) [(i : Fin (n + 1)) → MeasureTheory.MeasureSpace (α i)] [∀ (i : Fin (n + 1)), MeasureTheory.SigmaFinite MeasureTheory.volume] (i : Fin (n + 1)) :
              theorem MeasureTheory.measurePreserving_piUnique {ι : Type u_1} [Fintype ι] {π : ιType u_4} [Unique ι] {m : (i : ι) → MeasurableSpace (π i)} (μ : (i : ι) → MeasureTheory.Measure (π i)) :
              theorem MeasureTheory.measurePreserving_pi_empty {ι : Type u} {α : ιType v} [Fintype ι] [IsEmpty ι] {m : (i : ι) → MeasurableSpace (α i)} (μ : (i : ι) → MeasureTheory.Measure (α i)) :
              theorem MeasureTheory.measurePreserving_piFinsetUnion {ι : Type u_1} {α : ιType u_3} {m : (i : ι) → MeasurableSpace (α i)} [DecidableEq ι] {s : Finset ι} {t : Finset ι} (h : Disjoint s t) (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] :
              theorem MeasureTheory.volume_preserving_piFinsetUnion {ι : Type u_1} (α : ιType u_4) [DecidableEq ι] {s : Finset ι} {t : Finset ι} (h : Disjoint s t) [(i : ι) → MeasureTheory.MeasureSpace (α i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] :