Abstract We prove a selection theorem for convex-valued lower semicontinuous mappings F to Frechet spaces under the assumption that every value of F contains all interior (in the convex sense) points of its own closure. This is an extension of E. Michael's theorem 3.1‴ in [7] for mappings to Banach spaces. The desired continuous single-valued selection is constructed as a pointwise barycenter mapping with respect to a suitable family of probability measures concentrated on values of F. As an application, we show that, for any metric space M there is a continuous mapping which to every compact set K ⊂ M assigns a probability measure whose support coincides with K. Earlier this fact was proved for a complete separable metric space M by using a fundamentally different technique based on Milyutin mappings.

中文翻译：

---

非规范空间中的连续内部选择

摘要 我们证明了凸值下半连续映射 F 到 Frechet 空间的选择定理，假设 F 的每个值都包含其自身闭包的所有内部（凸意义上）点。这是 [7] 中 E. Michael 定理 3.1‴ 的扩展，用于映射到 Banach 空间。期望的连续单值选择被构造为相对于集中在 F 值上的合适概率测度系列的逐点重心映射。作为一个应用，我们表明，对于任何度量空间 M，都有一个连续映射，该映射到每个紧致集合 K ⊂ M 分配一个概率测度，其支持度与 K 一致。早些时候，通过使用基于 Milyutin 映射的完全不同的技术，这一事实已被证明用于完全可分离的度量空间 M。