  Let $\{ A_k\}_{k=1}^\infty$ be a sequence of finite subsets of $\mathbb{R}^d$
satisfying that $\# A_k \ge 2$ for all integers $k \ge 1$. In this paper, we
first give a sufficient and necessary condition for the existence of the
infinite convolution $$\nu =\delta_{A_1}*\delta_{A_2} * \cdots
*\delta_{A_n}*\cdots, $$ where all sets $A_k \subseteq \mathbb{R}_+^d$ and
$\delta_A = \frac{1}{\# A} \sum_{a \in A} \delta_a$. Then we study the
spectrality of a class of infinite convolutions generated by Hadamard triples
in $\mathbb{R}$ and construct a class of singular spectral measures without
compact support. Finally we show that such measures are abundant, and the
dimension of their supports has the intermediate-value property.
