PettisIntegral

The Pettis integral is a notion of integrating vector-valued functions taking values in a Banach space X, named after Giovanni Pettis. For a measure space (Ω, Σ, μ) and a function f: Ω → X, f is Pettis integrable if it is weakly measurable and for every continuous linear functional x on X, the scalar function ω ↦ x(f(ω)) is μ-integrable, and there exists a vector I ∈ X such that x(I) = ∫_Ω x(f(ω)) dμ for all x ∈ X. The vector I is called the Pettis integral of f, denoted ∫_Ω f dμ.

Key properties include linearity in f and the relation to the Bochner integral. If f is Bochner integrable, then f is Pettis integrable and the Pettis integral equals the Bochner integral. The converse is not guaranteed in general; there exist Pettis integrable functions that are not Bochner integrable, so Pettis integration is weaker than Bochner integration. In practice, weak measurability and scalar integrability are used to verify Pettis integrability, and in separable Banach spaces weak measurability implies strong measurability, which clarifies the relationship between the two notions in that setting.

Generalizations and context: The Pettis integral extends to broader classes of locally convex spaces by working with dual pairings with continuous linear functionals. It is a standard tool in the study of vector-valued random variables and in functional analysis, where the Bochner integral may be too restrictive for certain spaces or functions.