Font Size: 

We present the construction of a continual family of positive series, the set of incomplete sums of which is Cantorval. Each series of this family has a property$$\sum\limits_{n=1}^{\infty}a_{n}=1,~~~\overline{\lim_{n\rightarrow\infty}}\frac{a_n}{\sum_{k=1}^{\infty}a_{n+k}}=+\infty.$$Moreover for any $\varepsilon>0$  there exists a series in this family, the Lebesgue measure of an incomplete sums set of which is greater than $ 1- \varepsilon $.