Duality for outer $L^{p,\mathrm{\infty }}$ spaces

The $L^{p,\mathrm{\infty }}$ 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 $L^{p,\mathrm{\infty }}$ 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.