Locally octahedral and locally almost square Köthe-Bochner spaces

It has been proved in [J.-D. Hardtke, J. Math. Phys. Anal. Geom. 16, no.2, 119--137 (2020)] that a K\"othe-Bochner space \(E(X)\) is locally octahedral/locally almost square if \(X\) has the respective property and the simple functions are dense in \(E(X)\). Here we show that the result still holds true without the density assumption. The proof makes use of the Kuratowski-Ryll-Nardzewski Theorem on measurable selections.