of radius $r$ centered at $x$. If $$\overline{\dim}_{\rm loc}(\eta, x)=\underline{\dim}_{\rm loc} (\eta, x),$$ the common value is denoted as $\dim_{\rm loc}(\eta,x)$ and is called the [*local dimension*]{} of $\eta$ at $x$. We say that $\eta$ is [*exact dimensional*]{} if there exists a constant $C$ such that the local dimension $\dim_{\rm loc}(\eta, x)$ exists and equals $C$ for $\eta$-a.e. $x\in X$. It is well-known that if $\eta$ is an exact dimensional measure in ${\Bbb R}^d$, the Hausdorff and packing dimensions of $\eta$ coincide and are equal to the involved constant $C$, and so are some other notions of dimension (e.g.  entropy dimension); see [@Young1982; @Falconer1997]. Recall that the Hausdorff and packing dimensions of $\eta$ are defined by $$\begin{aligned}
{\dim}_{\rm H}\eta &=\inf \{\dim_{\rm H}F:\; \eta(F)>0\mbox{ and $F$ is a Borel set}\},\\
{\dim}_{\rm P}\eta &=\inf \{\dim_{\rm P}F:\; \eta(\R^d\setminus F)=0\mbox{ and $F$ is a Borel set}\},\end{aligned}$$ where $\dim_{\rm H}F, \dim_{\rm P}F$ stand for the Hausdorff and packing dimensions of $F$, respectively (cf. [@Falconer2003]).

A folklore open problem in fractal geometry asks whether every ergodic invariant measure for an affine IFS is exact dimensional. As the main result of this paper, we give the following confirmative answer.

\[thm-1.0\] Let ${\mathcal S}=\{M_jx+a_j\}_{j\in \Lambda}$ be an affine 