Conjecture on null boundaries of limiting colored regions

Let $t$ be a time, let $\Xi_{\le t}$ be the set of particles arriving by time $t$, and let $A(t,\xi)$ denote the measurable region associated with $\xi\in\Xi_{\le t}$. The regions are defined only up to Lebesgue-null sets. Null-boundary conjecture. For each $t$ one can identify the regions $\{A(t,\xi):\ \xi\in\Xi_{\le t}\}$ so that the topological boundary of each region has Lebesgue measure zero. This would provide a natural topological representative for each limiting colored region and make questions about membership of points in regions well posed.