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
\[
\phi\!\left(\int_X f\,d\mu\right)=\int_X \phi(f(x))\,d\mu(x).
\]
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
Common applications appear in probability theory for random elements in Banach spaces, in the theory of