The Banach–Steinhaus Uniform Boundedness Principle for Operators in Banach–Kantorovich Spaces over $L^0$

We consider a vector-valued version of the Banach–Steinhaus uniform boundedness principle for universally complete Banach–Kantorovich spaces over the ring of measurable functions. We prove that, if a family of bounded linear operators in a universally complete Banach–Kantorovich space is pointwise bounded, then it is uniformly bounded. We also present applications to weak convergence and weak boundedness in universally complete Banach–Kantorovich spaces.