Bochner Integral - Properties

Properties

Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Particularly useful is Bochner's criterion for integrability, which states that if (X, Σ, μ) is a measure space, then a Bochner-measurable function ƒ : X → B is Bochner integrable if and only if

A function ƒ : X → B  is called Bochner-measurable if it is equal μ-almost everywhere to a function g taking values in a separable subspace B₀ of B, and such that the inverse image g−1(U) of every open set U  in B  belongs to Σ. Equivalently, ƒ is limit μ-almost everywhere of a sequence of simple functions.

If is a continuous linear operator, and is Bochner-integrable, then is Bochner-integrable and integration and may be interchanged:

This also holds for closed operators, given that be itself integrable (which, via the criterium mentioned above is trivially true for bounded ).

A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if ƒ_n : X → B is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function ƒ, and if

for almost every x ∈ X, and g ∈ L1(μ), then

as n → ∞ and

for all E ∈ Σ.

If ƒ is Bochner integrable, then the inequality

holds for all E ∈ Σ. In particular, the set function

defines a countably-additive B-valued vector measure on X which is absolutely continuous with respect to μ.