 Solution

Construct a bipartite graph with left vertices $A_1,\dots ,A_k$, right vertices $B_1,\dots ,B_k$, and an edge $A_i{B}_j$ exactly when $A_i\cap B_j\ne \varnothing$. For $S\subseteq [k]$, every point of ${\bigcup }_{i\in S}{A}_i$ lies in some $B_j$ adjacent to one of those $A_i$. ThusUsing finite additivity of Lebesgue measure and the equal-volume hypotheses,Hence $|N(S)|\ge |S|$ for every $S$. The Hall marriage theorem supplies a perfect matching, which has the form $A_i{B}_{\sigma (i)}$ for a permutation $\sigma$ of $[k]$. Every matched edge means precisely that $A_i\cap B_{\sigma (i)}\ne \varnothing$.