11. Integration and Measure Theory

11.1. Elementary Integration

We first focus on integration of functions on finite intervals in ℝ. We can integrate elementary functions.

open MeasureTheory intervalIntegral

open Interval
-- this introduces the notation `[[a, b]]` for the segment from `min a b` to `max a b`

example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 :=
  integral_id

example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) :=
  integral_one_div h

The fundamental theorem of calculus relates integration and differentiation. Below we give simplified statements of the two parts of this theorem. The first part says that integration provides an inverse to differentiation and the second one specifies how to compute integrals of derivatives. (These two parts are very closely related, but their optimal versions, which are not shown here, are not equivalent.)

example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u ↦ ∫ x : ℝ in a..u, f x) b = f b :=
  (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _)
        hf.continuousAt).hasDerivAt.deriv

example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x)
    (h' : IntervalIntegrable f' volume a b) : (∫ y in a..b, f' y) = f b - f a :=
  integral_eq_sub_of_hasDerivAt h h'

Convolution is also defined in Mathlib and its basic properties are proved.

open Convolution

example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x ↦ ∫ t, f t * g (x - t) :=
  rfl

11.2. Measure Theory

The general context for integration in Mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space, not necessarily finite dimensional.

The first component in the development of measure theory is the notion of a \(\sigma\)-algebra of sets, which are called the measurable sets. The type class MeasurableSpace serves to equip a type with such a structure. The sets empty and univ are measurable, the complement of a measurable set is measurable, and a countable union or intersection of measurable sets is measurable. Note that these axioms are redundant; if you #print MeasurableSpace, you will see the ones that Mathlib uses. As the examples below show, countability assumptions can be expressed using the Encodable type class.

variable {α : Type*} [MeasurableSpace α]

example : MeasurableSet (∅ : Set α) :=
  MeasurableSet.empty

example : MeasurableSet (univ : Set α) :=
  MeasurableSet.univ

example {s : Set α} (hs : MeasurableSet s) : MeasurableSet (sᶜ) :=
  hs.compl

example : Encodable ℕ := by infer_instance

example (n : ℕ) : Encodable (Fin n) := by infer_instance

variable {ι : Type*} [Encodable ι]

example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) :=
  MeasurableSet.iUnion h

example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋂ b, f b) :=
  MeasurableSet.iInter h

Once a type is measurable, we can measure it. On paper, a measure on a set (or type) equipped with a \(\sigma\)-algebra is a function from the measurable sets to the extended non-negative reals that is additive on countable disjoint unions. In Mathlib, we don’t want to carry around measurability assumptions every time we write an application of the measure to a set. So we extend the measure to any set s as the infimum of measures of measurable sets containing s. Of course, many lemmas still require measurability assumptions, but not all.

open MeasureTheory
variable {μ : Measure α}

example (s : Set α) : μ s = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), μ t :=
  measure_eq_iInf s

example (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) :=
  measure_iUnion_le s

example {f : ℕ → Set α} (hmeas : ∀ i, MeasurableSet (f i)) (hdis : Pairwise (Disjoint on f)) :
    μ (⋃ i, f i) = ∑' i, μ (f i) :=
  μ.m_iUnion hmeas hdis

Once a type has a measure associated with it, we say that a property P holds almost everywhere if the set of elements where the property fails has measure 0. The collection of properties that hold almost everywhere form a filter, but Mathlib introduces special notation for saying that a property holds almost everywhere.

example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in μ.ae, P x :=
  Iff.rfl

11.3. Integration

Now that we have measurable spaces and measures we can consider integrals. As explained above, Mathlib uses a very general notion of integration that allows any Banach space as the target. As usual, we don’t want our notation to carry around assumptions, so we define integration in such a way that an integral is equal to zero if the function in question is not integrable. Most lemmas having to do with integrals have integrability assumptions.

section
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E}

example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ a, f a + g a ∂μ = ∫ a, f a ∂μ + ∫ a, g a ∂μ :=
  integral_add hf hg

As an example of the complex interactions between our various conventions, let us see how to integrate constant functions. Recall that a measure μ takes values in ℝ≥0∞, the type of extended non-negative reals. There is a function ENNReal.toReal : ℝ≥0∞ → ℝ which sends ⊤, the point at infinity, to zero. For any s : Set α, if μ s = ⊤, then nonzero constant functions are not integrable on s. In that case, their integrals are equal to zero by definition, as is (μ s).toReal. So in all cases we have the following lemma.

example {s : Set α} (c : E) : ∫ x in s, c ∂μ = (μ s).toReal • c :=
  set_integral_const c

We now quickly explain how to access the most important theorems in integration theory, starting with the dominated convergence theorem. There are several versions in Mathlib, and here we only show the most basic one.

open Filter

example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ)
    (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
    (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) :=
  tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim

Then we have Fubini’s theorem for integrals on product type.

example {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type*}
    [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E)
    (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂ μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ :=
  integral_prod f hf

There is a very general version of convolution that applies to any continuous bilinear form.

open Convolution

variable {𝕜 : Type*} {G : Type*} {E : Type*} {E' : Type*} {F : Type*} [NormedAddCommGroup E]
  [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
  [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F]
  [Sub G]

example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) :
    f ⋆[L, μ] g = fun x ↦ ∫ t, L (f t) (g (x - t)) ∂μ :=
  rfl

Finally, Mathlib has a very general version of the change-of-variables formula. In the statement below, BorelSpace E means the \(\sigma\)-algebra on E is generated by the open sets of E, and IsAddHaarMeasure μ means that the measure μ is left-invariant, gives finite mass to compact sets, and give positive mass to open sets.

example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type*}
    [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E}
    {f' : E → E →L[ℝ] E} (hs : MeasurableSet s)
    (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) :
    ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ :=
  integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g