Finitely overlapping family of sets

Definition

A family of sets ${E_j}_{j\in J}$ is finitely overlapping with multiplicity at most $N$ if

\sum_{j\in J}\mathbf 1_{E_j}(x)\le N

for every $x$ , or for almost every $x$ when the statement is measure-theoretic. The bound $N$ is required to be independent of the size and location of the family members.

Integral consequence

For nonnegative measurable $g$ , Tonelli's theorem gives

\sum_j\int_{E_j}g\le N\int_{\bigcup_jE_j}g.

Thus local estimates may be summed without accumulating a factor equal to the number of sets.

Typical construction

Euclidean annuli can be covered by equal-width cubes so that suitably enlarged cubes have bounded overlap, with a multiplicity depending only on dimension. This is the form used to assemble smooth weights from bump functions.

References

Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Publisher record.