where $h_i$ are defined as in .

Next we give similar dimension formulas for certain conditional measures associated with $m$. For $i=0,\ldots, s$, let $\{m_x^{\xi_i}\}$ be the system of conditional measures of $m$ associated with the partition $\xi_i$ (cf. Section \[Sect-con\]). For a linear subspace $W$ of $\R^d$, let $W^\perp$ denote the orthogonal complement of $W$ in $\R^d$, and let $P_W:\; \R^d\to W$ denote the orthogonal projection from $\R^d$ to $W$.

\[thm-1.2\] Under the assumptions of Theorem \[thm-1.1\], for any $0\leq i<j\leq s$ and $m$-a.e. $x\in \Sigma'$, the measures $m_x^{\xi_i}\circ \pi^{-1}$, $m_x^{\xi_i}\circ \pi^{-1}\circ (P_{{(V_x^j)}^\perp})^{-1}$ are exact dimensional with $$\begin{aligned}
&\dim_{\rm H}(m_x^{\xi_i}\circ \pi^{-1})=\sum_{\ell=i}^{s-1}\frac{h_{\ell+1}-h_\ell}{\lambda_{\ell+1}}, \label{e-con-dim}\\
&\dim_{\rm H}\left(m_x^{\xi_i}\circ \pi^{-1}\circ (P_{{(V_x^j)}^\perp})^{-1}\right)=\sum_{\ell=i}^{j-1}\frac{h_{\ell+1}-h_{\ell}}{\lambda_{\ell+1}}, \label{e-con-proj-dim}\end{aligned}$$ Moreover, for $m$-a.e. $x\in \Sigma'$ and any $1\leq j\leq s$, $$\label{e-proj-dim}
\dim_{\rm loc}\left(m\circ \pi^{-1}\circ (P_{{(V_x^j)}^\perp})^{-1}, P_{{(V_x^j)}^\perp}( \pi x)\right)=\sum_{\ell=0}^{j-1}\frac{h_{\ell+1}-h_{\ell}}{\lambda_{\ell+1}}.$$

From the above theorem, we can deduce certain dimension conservation property for the measures $m^{\xi_0}_x\circ \pi^{-1}$ and $\mu$. To state the result, let $G(d,k)$ denote the Grassmannian manifold of $k$-dimensional linear subspaces of $\R^d$. For a Borel probability measure $\eta$ on $\R^d$ and $W\in G(d,k)$, let $\{\eta_{W,z}=\eta^{\zeta_W}_z\}_{z\in \R^d}$ denote the the system of conditional measures of $\eta$ associated with the measurable partition $\zeta_W$ given by $$\zeta_W=\{W+a:\; a\in W^\perp\}.$$ These conditional measures are also called 