the [*slicing measures*]{} of $\eta$ along the subspace $W$ (cf. [@Mattila1995 §10.1]). Following Furstenberg [@Furstenberg2008], we give the following.

$\eta$ is said to be dimension conserving with respect to the projection $P_{W^\perp}$, if $$\dim_{\rm H}\eta =\dim_{\rm H} \eta_{W, z}+\dim_{\rm H}\left( \eta\circ (P_{W^\perp})^{-1}\right)$$ for $\eta$-a.e. $z\in \R^d$.

For $i\in \{0, \ldots, s-1\}$, define $\Pi_i:\Sigma'\to G(d, \sum_{j=i+1}^sk_j)$ by $$\label{e-Pi}
\Pi_i(x)= V_x^i.$$ The push-forward measures $m\circ(\Pi_i)^{-1}$, $i=1,\ldots, s-1$, are called the [*Furstenberg measures*]{} or [*Furstenberg-Oseledets measures*]{} associated with $(M_j)_{j\in \Lambda}$ and $m$. An ergodic measure $\nu\in \M_\sigma(\Sigma)$ is said to be [*quasi-Bernoulli*]{} if there exists a positive constant $C$ such that $$C^{-1}\nu([I]_0)\nu([J]_0)\leq \nu([IJ]_0)\leq C\nu([I]_0) \nu([J]_0)$$ for any finite words $I$, $J$ over $\Lambda$, where $$[I]_0:=\{x\in \Sigma:\; x_j=i_j \mbox{ for }0\leq  j\leq n-1\}$$ for $I=i_0\ldots i_{n-1}$. Similarly, we say that $\nu$ is [*sub-multiplicative*]{} if there exists a positive constant $C$ such that $\nu([IJ]_0)\leq C\nu([I]_0) \nu([J]_0)$ for any finite words $I$, $J$ over $\Lambda$.

\[cor-1.0\] Under the assumptions of Theorem \[thm-1.1\], we further assume that $s\geq 2$. Let $i\in \{1,\ldots, s-1\}$. Then the following statements hold.

-   For $m$-a.e. $x\in \Sigma'$, $m_x^{\xi_0}\circ \pi^{-1}$ is dimension conserving with respect to $P_{{(V_x^i)}^\perp}$ and moreover, the projected measure $(m_x^{\xi_0}\circ \pi^{-1})\circ P_{{(V_x^i)}^\perp}^{-1}$ is exact dimensional, and so are the 