The existence of $p$-convex tensor products of $L_p(X)$-spaces for the case of an arbitrary measure

We obtain a far-reaching generalization (in several directions) of the theorem of A. Lambert on the existence of the projective tensor product of operator sequence spaces. This result is obtained in the context of spaces, generalizing $p$-multinormed spaces of Dales et al. which are based on an arbitrary, perhaps non-discrete measure.