slicing measures $(m_x^{\xi_0}\circ \pi^{-1})_{V_x^i, y}$ for $(m_x^{\xi_0}\circ \pi^{-1})$-a.e. $y$.

-   Assume that $m$ is quasi-Bernoulli. Then for $m\circ (\Pi_i)^{-1}$-a.e. $W$, $\mu$ is dimension conserving with respect to $P_{W^\perp}$, and moreover, the associated projected measure and almost all slicing measures are exact dimensional.

-   Assume that $m$ is sub-multiplicative. Then for $m\circ (\Pi_i)^{-1}$-a.e. $W$, there exists a subset $A_W$ of $\R^d$ with $\mu(A_W)>0$ such that for every $z\in A_W$, $$\begin{split}
      \dim_{\rm loc} (\mu_{W,z}, z)&=\sum_{\ell=i}^{s-1}\frac{h_{\ell+1}-h_\ell}{\lambda_{\ell+1}},\\
      \dim_{\rm loc} (\mu\circ P_{W^\perp}^{-1}, P_{W^\perp}(z))&=\sum_{\ell=0}^{i-1}\frac{h_{\ell+1}-h_\ell}{\lambda_{\ell+1}},
      \end{split}$$ and so, $\dim_{\rm H}\mu=\dim_{\rm loc} (\mu_{W,z}, z)+\dim_{\rm loc} (\mu\circ P_{W^\perp}^{-1}, P_{W^\perp}(z)).$ When $m$ is quasi-Bernoulli then one can take the set $A_W$ such that $\mu(A_W)=1$.

We remark that part (ii) of Theorem \[cor-1.0\] was previously proved in [@BaranyKaenmaki2015] under the assumptions that ${\mathcal S}$ is contracting, invertible and its linear parts satisfy certain domination condition. According to part (iii) of the theorem, when $m$ is sub-multiplicative, $\mu$ partially satisfies dimension conservation. It is unknown whether part (ii) always holds when $m$ is only assumed to be ergodic. However, as is illustrated in the following theorem, this is true in the special case that the linear parts of the IFS commute.

\[thm-1.7’\] Let ${\mathcal S}=\{M_jx+a_j\}_{j\in \Lambda}$ be an affine IFS on $\R^d$, 