The L^p,∞ quasi-norm of functions on a measure space can be characterized in terms of their pairing with normalized characteristic functions. We generalize this result to the case of the outer L^p,∞ quasi-norms for appropriate sizes. This characterization provides an explicit form for certain implicitly defined quasi-norms appearing in an article of Di Plinio and Fragkos.

The author gratefully acknowledges financial support by the CRC 1060 "The Mathematics of Emergent Effects" at the University of Bonn, funded through the Deutsche Forschungsgemeinschaft. He is also supported by the Basque Government through the BERC 2022-2025 program and by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S / MICIN / AEI / 10.13039/501100011033.