Complex interpolation of spaces of integrable functions with respect to a vector measure

Let $(\Omega,\Sigma)$ be a measurable space and $m:\Sigma \rightarrow X$ be a vector measure with values in the complex Banach space $X.$ We apply the Calder\'on interpolation methods to the family of spaces of scalar $p-$integrable functions with respect to $m$ with $1\leq p \leq \infty$. Moreover we obtain a result about the relation between the complex interpolation spaces $[X_0, X_1] _{[\theta]}$ and $[X_0, X_1] ^{[\theta]}$ for a Banach couple of interpolation $(X_0, X_1)$ such that $X_1 \subset X_0$ with continuous inclusion.