Bochner integral #

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The Bochner integral extends the definition of the Lebesgue integral to functions that map from a measure space into a Banach space (complete normed vector space). It is constructed here by extending the integral on simple functions.

Main definitions #

The Bochner integral is defined through the extension process described in the file set_to_L1, which follows these steps:

1. Define the integral of the indicator of a set. This is weighted_smul μ s x = (μ s).to_real * x. weighted_smul μ is shown to be linear in the value x and dominated_fin_meas_additive (defined in the file set_to_L1) with respect to the set s.

2. Define the integral on simple functions of the type simple_func α E (notation : α →ₛ E) where E is a real normed space. (See simple_func.integral for details.)

3. Transfer this definition to define the integral on L1.simple_func α E (notation : α →₁ₛ[μ] E), see L1.simple_func.integral. Show that this integral is a continuous linear map from α →₁ₛ[μ] E to E.

4. Define the Bochner integral on L1 functions by extending the integral on integrable simple functions α →₁ₛ[μ] E using continuous_linear_map.extend and the fact that the embedding of α →₁ₛ[μ] E into α →₁[μ] E is dense.

5. Define the Bochner integral on functions as the Bochner integral of its equivalence class in L1 space, if it is in L1, and 0 otherwise.

The result of that construction is ∫ a, f a ∂μ, which is definitionally equal to set_to_fun (dominated_fin_meas_additive_weighted_smul μ) f. Some basic properties of the integral (like linearity) are particular cases of the properties of set_to_fun (which are described in the file set_to_L1).

Main statements #

1. Basic properties of the Bochner integral on functions of type α → E, where α is a measure space and E is a real normed space.

• integral_zero : ∫ 0 ∂μ = 0
• integral_add : ∫ x, f x + g x ∂μ = ∫ x, f ∂μ + ∫ x, g x ∂μ
• integral_neg : ∫ x, - f x ∂μ = - ∫ x, f x ∂μ
• integral_sub : ∫ x, f x - g x ∂μ = ∫ x, f x ∂μ - ∫ x, g x ∂μ
• integral_smul : ∫ x, r • f x ∂μ = r • ∫ x, f x ∂μ
• integral_congr_ae : f =ᵐ[μ] g → ∫ x, f x ∂μ = ∫ x, g x ∂μ
• norm_integral_le_integral_norm : ‖∫ x, f x ∂μ‖ ≤ ∫ x, ‖f x‖ ∂μ

2. Basic properties of the Bochner integral on functions of type α → ℝ, where α is a measure space.

• integral_nonneg_of_ae : 0 ≤ᵐ[μ] f → 0 ≤ ∫ x, f x ∂μ
• integral_nonpos_of_ae : f ≤ᵐ[μ] 0 → ∫ x, f x ∂μ ≤ 0
• integral_mono_ae : f ≤ᵐ[μ] g → ∫ x, f x ∂μ ≤ ∫ x, g x ∂μ
• integral_nonneg : 0 ≤ f → 0 ≤ ∫ x, f x ∂μ
• integral_nonpos : f ≤ 0 → ∫ x, f x ∂μ ≤ 0
• integral_mono : f ≤ᵐ[μ] g → ∫ x, f x ∂μ ≤ ∫ x, g x ∂μ

3. Propositions connecting the Bochner integral with the integral on ℝ≥0∞-valued functions, which is called lintegral and has the notation ∫⁻.

• integral_eq_lintegral_max_sub_lintegral_min : ∫ x, f x ∂μ = ∫⁻ x, f⁺ x ∂μ - ∫⁻ x, f⁻ x ∂μ, where f⁺ is the positive part of f and f⁻ is the negative part of f.
• integral_eq_lintegral_of_nonneg_ae : 0 ≤ᵐ[μ] f → ∫ x, f x ∂μ = ∫⁻ x, f x ∂μ

4. tendsto_integral_of_dominated_convergence : the Lebesgue dominated convergence theorem

5. (In the file set_integral) integration commutes with continuous linear maps.

• continuous_linear_map.integral_comp_comm
• linear_isometry.integral_comp_comm

Notes #

Some tips on how to prove a proposition if the API for the Bochner integral is not enough so that you need to unfold the definition of the Bochner integral and go back to simple functions.

One method is to use the theorem integrable.induction in the file simple_func_dense_lp (or one of the related results, like Lp.induction for functions in Lp), which allows you to prove something for an arbitrary integrable function.

Another method is using the following steps. See integral_eq_lintegral_max_sub_lintegral_min for a complicated example, which proves that ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻, with the first integral sign being the Bochner integral of a real-valued function f : α → ℝ, and second and third integral sign being the integral on ℝ≥0∞-valued functions (called lintegral). The proof of integral_eq_lintegral_max_sub_lintegral_min is scattered in sections with the name pos_part.

Here are the usual steps of proving that a property p, say ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻, holds for all functions :

1. First go to the L¹ space.

   For example, if you see ennreal.to_real (∫⁻ a, ennreal.of_real $ ‖f a‖), that is the norm of f in L¹ space. Rewrite using L1.norm_of_fun_eq_lintegral_norm.

2. Show that the set {f ∈ L¹ | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻} is closed in L¹ using is_closed_eq.

3. Show that the property holds for all simple functions s in L¹ space.

   Typically, you need to convert various notions to their simple_func counterpart, using lemmas like L1.integral_coe_eq_integral.

4. Since simple functions are dense in L¹,

univ = closure {s simple}
     = closure {s simple | ∫ s = ∫⁻ s⁺ - ∫⁻ s⁻} : the property holds for all simple functions
     ⊆ closure {f | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻}
     = {f | ∫ f = ∫⁻ f⁺ - ∫⁻ f⁻} : closure of a closed set is itself

Use is_closed_property or dense_range.induction_on for this argument.

Notations #

• α →ₛ E : simple functions (defined in measure_theory/integration)
• α →₁[μ] E : functions in L1 space, i.e., equivalence classes of integrable functions (defined in measure_theory/lp_space)
• α →₁ₛ[μ] E : simple functions in L1 space, i.e., equivalence classes of integrable simple functions (defined in measure_theory/simple_func_dense)
• ∫ a, f a ∂μ : integral of f with respect to a measure μ
• ∫ a, f a : integral of f with respect to volume, the default measure on the ambient type

We also define notations for integral on a set, which are described in the file measure_theory/set_integral.

Note : ₛ is typed using \_s. Sometimes it shows as a box if the font is missing.

Tags #

Bochner integral, simple function, function space, Lebesgue dominated convergence theorem

The Bochner integral of simple functions #

Define the Bochner integral of simple functions of the type α →ₛ β where β is a normed group, and prove basic property of this integral.

The Bochner integral of L1 #

Define the Bochner integral on α →₁ₛ[μ] E by extension from the simple functions α →₁ₛ[μ] E, and prove basic properties of this integral.

The Bochner integral on functions #

Define the Bochner integral on functions generally to be the L1 Bochner integral, for integrable functions, and 0 otherwise; prove its basic properties.

In the notation for integrals, an expression like ∫ x, g ‖x‖ ∂μ will not be parsed correctly, and needs parentheses. We do not set the binding power of r to 0, because then ∫ x, f x = 0 will be parsed incorrectly.