Cut norm ($\Vert H{\Vert }_{\mathrm{cut}}={\max }_{A,B}|\mathbb{E}H{1}_A{1}_B|$)

For a real-valued function $H$ on a nonempty finite Cartesian product $X\times Y$, its normalized cut norm is the maximum of $|{\mathbb{E}}_{x,y}H(x,y)1_A(x)1_B(y)|$ over $A\subseteq X,B\subseteq Y$. It measures all rectangular subset discrepancies. The convention using arbitrary test functions bounded by one is equivalent within a factor of four, by decomposing their positive and negative parts into indicator functions.