Let Σ be a σ-algebra of subsets of a set Ω and $B(\Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let $\tau(B(\Sigma),ca(\Sigma))$ denote the natural Mackey topology on $B(\Sigma)$. It is shown that a linear operator T from $B(\Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space $(B(\Sigma),\tau(B(\Sigma),ca(\Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator $T:B({\cal B} o)\rightarrow B({\cal B} o)$ generated by a function $f\in L^1({\cal B} o, C(\Omega))$, where Ω is a compact Hausdorff space.

设 Σ 为集合 Ω 子集的σ代数， $B(\Sigma)$为 Ω 上所有有界 Σ 可测标量函数的巴拿赫空间。令$\tau(B(\Sigma),ca(\Sigma))$表示$B(\Sigma)$上的自然 Mackey 拓扑。结果表明，从$B(\Sigma)$到 Banach 空间E的线性算子T是博赫纳可表示的当且仅当T是局部凸空间$(B(\Sigma),\tau(B (\Sigma),ca(\Sigma)))$和 Banach 空间E。我们推导出由函数$ f\in L^1({\ cal B} o, C(\Omega))$，其中 Ω 是紧致豪斯多夫空间。