VC-subgraph classes, envelopes, and the first-unshattered index

Let $(\mathcal{X},\mathcal{A})$ be a measurable space and let $\mathcal{F}$ be a class of measurable real-valued functions on it. The subgraph of $f\in \mathcal{F}$ is

$C_f:=\{(x,t)\in \mathcal{X}\times \mathbb{R}:t<f(x)\}\mathrm{.}$

It is measurable in $\mathcal{A}\otimes \mathcal{B}(\mathbb{R})$: the coordinate lifts are measurable by The product sigma-algebra and its finite iterates, their difference is measurable by Arithmetic and lattice operations preserve measurability whenever they are defined, and its strict negative set is measurable by Threshold characterisations of real-valued and extended-real-valued measurability. The class $\mathcal{F}$ is a VC-subgraph class when the set family ${\mathcal{C}}_{\mathcal{F}}:=\{C_f:f\in \mathcal{F}\}$ has finite VC index as defined in Shattering, VC dimension, and first-unshattered index for set classes. For nonempty $\mathcal{F}$, write $d(\mathcal{F})$ for the greatest size of a finite shattered set (zero if none is shattered) and

$V(\mathcal{F}):=d(\mathcal{F})+1,$

equivalently the least positive integer $V$ for which no $V$-point set is shattered. If no finite such integer exists, set $V=+\mathrm{\infty }$. The empty class is treated separately and assigned index $V=0$ by convention. Thus for every nonempty finite-index class, the VC dimension is exactly $V-1$.

A measurable envelope of $\mathcal{F}$ is a nonnegative measurable function $F$ on $\mathcal{X}$ such that $\mathrm{\mid }f(x)\mathrm{\mid }\le F(x)$ for every $f\in \mathcal{F}$ and $x\in \mathcal{X}$. In an $L^r(Q)$ assertion, changing $F$ on a $Q$-null set does not change its norm or the distances in that assertion.