Bochnerintegrable

A Bochner integrable function is a vector‑valued function that can be integrated in the sense of the Bochner integral, an extension of the Lebesgue integral to Banach‑space‑valued functions. Given a measure space \((X, \Sigma, \mu)\) and a Banach space \(B\), a function \(f:X \to B\) is said to be Bochner integrable if it is measurable in the strong (Bochner) sense and the integral of its norm is finite, i.e.

\[

\int_X \|f(x)\|_B\,d\mu(x) < \infty .

\]

Strong measurability requires that \(f\) be the pointwise limit almost everywhere of simple functions taking values in \(B\). Under this condition, the Bochner integral \(\int_X f\,d\mu\) exists and uniquely satisfies the property that for every continuous linear functional \(\phi\) on \(B\),

\[

\phi\!\left(\int_X f\,d\mu\right)=\int_X \phi(f(x))\,d\mu(x).

\]

Thus the Bochner integral generalises the scalar Lebesgue integral while preserving linearity and continuity.

Key properties include linearity, additivity over disjoint sets, and dominated convergence: if \(|f_n(x)| \le g(x)\) for an integrable scalar function \(g\), then \(\int f_n \to \int f\) when \(f_n \to f\) almost everywhere. The Fubini and Tonelli theorems also extend to Bochner integrals provided the Banach space satisfies appropriate measurability and separability conditions.

Common applications appear in probability theory for random elements in Banach spaces, in the theory of partial differential equations for time‑dependent vector fields, and in functional analysis where operators are expressed as integrals of operator‑valued functions. In many contexts, Bochner integrable functions are called “strongly integrable” to emphasize that measurability is with respect to the Banach‑space norm.