exactness behaving as follows $$\label{limita}
\displaystyle \lim_{n \to \infty} \frac{m(n)}{2 n}=a \in [0,\, 1].$$ They characterized all possible weak\*-limit points of the sequence of counting measures associated with distribution of nodes corresponding to a convergent scheme.

A sequence $\left\{\nu_n\right\}_{n\in\mathbb{N}}$ of measures is said to converge weakly to the measure $\nu$ provided that there exists a compact set $K$ containing the support of $\nu$ and of each $\nu_n$, and that $$\lim_{n\in\mathbb{N}}\int f d\nu_n=\int f d\nu$$ for each continuous function $f$ on $K$. In such a case, we write $\nu_n\overset{*}{\to}\nu$. We say that $\nu$ is a weak\*-limit of the sequence $\left\{\eta_n\right\}_{n\in\mathbb{N}}$ if some subsequence of $\left\{\nu_n\right\}_{n\in \Lambda\subset \mathbb{N}}$ is weakly convergent to $\nu$.

Set two schemes of numbers as in (\[nodessequence\]) associated to an interpolatory quadrature rule $\left\{I_{n}\right\}_{n\in \Gamma}$ where the degree of exactness satisfies (\[limita\]) for certain $a\in [0,\, 1]$. We also consider its corresponding sequence $\left\{\eta_n\right\}_{n\in \mathbb{N}}$ of probability counting measures $$\label{zerocountingmeasures}
\eta_n:=\frac{1}{n} \sum_{j=1}^n \delta_{x_{j,n}}, \quad n \in \mathbb{N}.$$ According to [@BLS1], if the rule $\{I_n\}_{n \in \Lambda}$ is convergent then every weak\*-limit $\nu$ of the the sequence $\left\{\eta_n\right\}_{n\in\mathbb{N}}$ satisfies that $$\label{sigmainequality}
 \nu \geq  a\, \lambda_0.$$ Also from [@BLS1] we have that this necessary condition is not sufficient. Theorem \[main\] 