$Y$. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for any $\mathbb R$-divisor $Q$ on $Y$, we have $$\kappa _\sigma(X, D+f^*Q)\geq \kappa _\sigma(F, D|_F)
+\kappa (Y, Q)$$ and $$\kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F)
+\kappa _\sigma(Y, Q)$$ where $F$ is a sufficiently general fiber of $f:X\to Y$.

We note that $\kappa_\sigma(X, D)$ and $\kappa (X, D)$ denote Nakayama’s numerical dimension and the Iitaka dimension of $D$, respectively. Theorem \[y-thm1.11\] already played a crucial role in the theory of minimal models.

We explain the organization of this paper. In Section \[x-sec2\], we collect some basic definitions. In Section \[x-sec3\], we prepare some useful and important lemmas. They will play a crucial role in this paper. In Section \[x-sec4\], we quickly explain some basic properties of Viehweg’s weakly positive sheaves and big sheaves. In Section \[x-sec5\], we introduce mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves. In Sections \[x-sec6\] and \[x-sec7\], we prove some basic properties of mixed-$\omega$-sheaves based on the theory of mixed Hodge structures. In Section \[x-sec8\], we treat a very special but interesting case. Section \[x-sec9\] is the main part of this paper. We establish a fundamental theorem of the theory of mixed-$\omega$-sheaves. Section \[x-sec10\] is 