Limitations of deducing measures of limsup sets from measures of finite intersections

Early results by Borel and Cantelli and Chung (Rendiconti del Circolo Matematico di Palermo, 1909) and Erdős (Transactions of the AMS, 1952) have provided bounds for the measure of a limsup set in terms of measures of its constituent sets and their intersections. Recent work by Beresnevich and Velani (Journal of Mathematical Analysis and its Applications, 2023) states that for sequences of balls the measure of the corresponding limsup set being positive is equivalent to a condition on the relationship between the measures of these balls and their pairwise intersections. In this paper, we show that the condition that the sets are balls is strictly necessary in the result of Beresnevich and Velani. Moreover, let $d\in \mathbb{N}$ and let $[0,1{]}^d$ be equipped with Lebesgue measure $\mu$. Fix $m\in \mathbb{N}$. When we drop the condition that the sets are balls, we can find two sequences of sets $(A_i{)}_{i\in \mathbb{N}}$ and $(B_i{)}_{i\in \mathbb{N}}$ in $[0,1{]}^d$ such that $\mu (A_i)=\mu (B_i)$ for all $i\in \mathbb{N}$ and for any sequence $(i_1,i_2,\mathrm{.}\mathrm{.}\mathrm{.},i_l)$ where for all $l\le m$ we have $\mu (A_{i_1}\cap A_{i_2}\cap \mathrm{.}\mathrm{.}\mathrm{.}\cap A_{i_l})=\mu (B_{i_1}\cap B_{i_2}\cap \mathrm{.}\mathrm{.}\mathrm{.}\cap B_{i_l})$ but $\mu ({\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}{A}_i)=1$ and $\mu ({\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}{B}_i)=0$.