Bracketing numbers for products of functions from two spaces

Question

I am reading materials related to empirical processes. The statement below is about measuring the complexity of function classes using bracketing numbers. I wonder if the inequality therein for bracketing numbers holds without any restrictions on signs of functions in the two spaces? It seems to me that it holds for positive (negative) functions. But can this result be valid for more general cases without such sign restrictions?

Let $\mathcal{F}$ and $\mathcal{G}$ be classes of measurable functions bounded by 1. Then for any probability measure $Q$ and any $1\leq r\leq\infty$, $$ N_{[]}(2\epsilon,\mathcal{F}\cdot\mathcal{G},L_{r}(Q))\leq N_{[]}(\epsilon,\mathcal{F},L_{r}(Q))N_{[]}(\epsilon,\mathcal{G},L_{r}(Q)). $$

Answer 1

I think this inequality holds without any restriction. For any $f\in \mathcal{F}$ and $g\in\mathcal{G}$, we can find $\epsilon-$bracket $[l_1, u_1]$ and $[l_2,u_2]$ contain $f$ and $g$ respectively.

No matter the sign of $l$ and $u$, $fg$ can always be controlled by the combination of two elements in $l_1l_2$, $u_1u_2$, $l_1u_2$ and $l_2u_1$. And the largest distance is \begin{align*} ||u_1u_2 - l_1l_2||_{Q,r} & = ||u_1u_2 - l_1u_2 + l_1u_2 - l_1l_2||_{Q,r}\\ & \le ||u_1u_2 - l_1u_2||_{Q,r} + ||l_1u_2 - l_1l_2||_{Q,r}\\ &\le 2\epsilon \end{align*}

Answer 2

This proof is inspired by Andrews(1994).

The key step of the proof is to construct the $2\varepsilon$-bracket of the space of the product of the functions. Assume the minimum $\varepsilon$-bracket of $\mathcal{F}$ and $\mathcal{G}$ are $[l_i^{(1)},u_i^{(1)}]$ and $[l_j^{(2)},u_j^{(2)}]$'s, respectively. Then we can show that $$ \left[\frac{l_i^{(1)}+u_i^{(1)}}2\frac{l_j^{(2)}+u_j^{(2)}}2\pm \left(\frac{u_i^{(1)} - l_i^{(1)}}2 + \frac{u_j^{(2)} - l_j^{(2)}}2\right)\right] $$ is a $2\varepsilon$-bracket of $\mathcal{F}\cdot\mathcal{G}$.

Thus, the inequality holds.