$L_{p}(\mu ,X)$ $(1<p<\infty )$ has the Radon-Nikodým property if $X$ does by martingales

Abstract:

Using the fact that ${L_p}[0,\;1]\;(1 < p < \infty )$ has an unconditional basis, Sundaresan has shown that ${L_p}(\mu ,\;X)$ has the Radon-Nikodým property if $1 < p < \infty$ and $X$ has the Radon-Nikodým property. In this note, Sundaresan’s theorem is proved by direct martingale methods. Then it is shown how to adapt this argument to the context of Orlicz spaces in which Sundaresan’s argument is not applicable.