  Let (X j , d j , $\mu$ j), j = 0, 1,. .. , m be metric measure spaces. Given
0 < p $\kappa$ $\le$ $\infty$ for $\kappa$ = 1,. .. , m and an analytic family
of multilinear operators T z : L p 1 (X 1) x $\bullet$ $\bullet$ $\bullet$ L p
m (X m) $\rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove
a theorem in the spirit of Stein's complex interpolation for analytic families.
Analyticity and our admissibility condition are defined in the weak (integral)
sense and relax the pointwise definitions given in [9]. Continuous functions
with compact support are natural dense subspaces of Lebesgue spaces over metric
measure spaces and we assume the operators T z are initially defined on them.
Our main lemma concerns the approximation of continuous functions with compact
support by similar functions that depend analytically in an auxiliary parameter
z. An application of the main theorem concerning bilinear estimates for
Schr{\"o}dinger operators on L p is included.
