  Let $\{(p_n, \mathcal{D}_n, L_n)\}$ be a sequence of Hadamard triples on
$\mathbb{R}$. Suppose that the associated Cantor-Moran measure $$
\mu_{\{p_n,\mathcal{D}_n\}}=\delta_{p_1^{-1}\mathcal{D}_1}\ast\delta_{(p_2p_1)^{-1}\mathcal{D}_2}\ast\cdots,
$$ where $\sup_n\{|p_n^{-1}d|:d\in \mathcal{D}_n\}<\infty$ and
$\sup\#\mathcal{D}_n<\infty$. It has been observed that the spectrality of
$\mu_{\{p_n,D_n\}}$ is determined by equi-positivity. A significant problem is
what kind of Moran measures can satisfy this property. In this paper, we
introduce the conception of \textit{Double Points Condition Set}
(\textit{DPCS}) to characterize the equi-positivity equivalently. As
applications of our characterization, we show that all singularly continuous
Cantor-Moran measures are spectral. For the absolutely continuous case, we
study Fuglede's Conjecture on Cantor-Moran set. We show that the
equi-positivity of $\mu_{\{p_n,D_n\}}$ implies the tiling of its support, and
the reverse direction holds under certain conditions.
