For a homogeneous additive process \((X(t))_{t\in T}\) with \(E| X| <\infty\), we consider Wiener’s stochastic integral \(\int_{T}x(t)dX(t)\) in the sense of \(L_ 1\)-convergence. Denote by D(X) the Banach space consisting of integrands \(\{\) x(t)\(\}\) equipped with norm \(E| \int_{T}x(t)dX(t)|\). Define the spectrum of Banach space B by \(Sp(B)=\{1\leq q\leq \infty\); B contains \(\ell^ n_ q\)’s uniformly} (after L. Schwartz). We show that, if \(r<\min (Sp(D)X)\), we have \((E| \int_{T}x(t)dX(t)|^ r)^{1/r}\leq CE| \int_{T}x(t)dX(t)|\) for any \(x\in D(X)\) with a constant C independent of x. In other words, the \(L_ 1\)-stochastic integrals for D(X) can be lifted up arbitrarily close to \(L_{\min Sp(D(X))}\). Adopting the \(L_ p\)-convergence \((0<p<1)\) instead of \(L_ 1\) in the definition of stochastic integrals, we reach similar results though it is not noted in this paper. We also show that, if \(T={\mathbb{N}}\) (discrete time), any D(X) is obtained as the Banach space of differentiable shifts of an appropriate stationary product measure on \({\mathbb{R}}^ N\).