Let \({\mathcal {R}}=\{R_j\}_{j=1}^\infty \) be a sequence of \(d\times d\) expanding integer matrices and \({\mathcal {A}}=\{A_j\}\) be a sequence of finite digit sets in \({\mathbb {Z}}^d\) . In this paper, we give a construction of the frame measures for a class of Moran measures defined by infinite convolution of discrete measures \(\begin{aligned}\mu _{{\mathcal {A}}}:=\delta _{R_1^{-1}A_1}*\delta _{(R_2R_1)^{-1}A_2}\cdots *\delta _{(R_jR_{j-1}\cdots R_1)^{-1}A_j}\cdots ,\end{aligned}\) where the convergence is in the weak sense. Generally speaking, the Hausdorff dimension of the support of \(\mu _{{\mathcal {A}}}\) is not the upper bound of the Beurling dimensions of its frame measures. We give a condition that yields an upper bound of the Beurling dimensions of some frame measures of \(\mu _{{\mathcal {A}}}\) on \({\mathbb {R}}^d\) , and this condition guarantees that the Hausdorff dimension of the support of \(\mu _{{\mathcal {A}}}\) on \({\mathbb {R}}\) is the upper bound of Beurling dimensions of certain frame measures of it. Some examples are given to illuminate our theory.
