Bourgain discretization using Lebesgue-Bochner spaces

We study the Lebesgue-Bochner discretization property of Banach spaces $Y$, which ensures that the Bourgain’s discretization modulus for $Y$ has a good lower estimate. We prove that there exist spaces that do not have the Lebesgue-Bochner discretization property, and we give a class of examples of spaces that enjoy this property.