Lp estimate for sums with bounded overlap

Statement

Let $(f_j)$ be a countable family of measurable scalar or finite-dimensional vector-valued functions. Suppose that at almost every point at most $N\ge1$ terms are nonzero. For $1\le p<\infty$ ,

\left\|\sum_jf_j\right\|_p^p \le N^{p-1}\sum_j\|f_j\|_p^p.

For $p=\infty$ , the corresponding estimate is

\left\|\sum_jf_j\right\|_\infty\le N\sup_j\|f_j\|_\infty.

The sum is pointwise finite almost everywhere; it can be assigned any value on the exceptional null set.

Proof

At each ordinary point, the triangle inequality and finite-sum Hölder inequality give $|\sum_j f_j|^p\le N^{p-1}\sum_j|f_j|^p$ . Integrate and use Tonelli's theorem. The supremum estimate follows directly from the triangle inequality.

This estimate does not assert differentiability of the sum. That requires local finiteness or appropriate derivative convergence.