When dealing with U process I meet with such a uniform entropy to calculate.
For any $\eta>0$, function class $\mathcal{F}$ containing functions $f=\left(f_{i, j}\right)_{1 \leq i \neq j \leq n}: \chi \times \chi \rightarrow$ $\mathbb{R}$ and a discrete probability measure $Q$ with support $\left\{z_1, \ldots, z_t\right\}$, let $N\left(\eta, \mathcal{F},\|\cdot\|_{2, Q}\right)$ denotes the minimum $m$ such that there exists $f^{(1)}, f^{(2)}, \ldots, f^{(m)} \in \mathcal{F}$ satisfying $$ \sup _{f \in \mathcal{F}} \inf _{1 \leq j \leq m}\left\|f-f^{(j)}\right\|_{2, Q} \leq \eta, $$ where for $f \in \mathcal{F}$, $$ \|f\|_{2, Q}^2:=\frac{\sum_{1 \leq i \neq j \leq t} f_{i, j}^2\left(z_i, z_j\right) Q\left(\left\{z_i\right\}\right) Q\left(\left\{z_j\right\}\right)}{\sum_{1 \leq i \neq j \leq t} Q\left(\left\{z_i\right\}\right) Q\left(\left\{z_j\right\}\right)} . $$ The uniform entropy I need to calculate is $$ J_2\left(\delta, \mathcal{F},\|\cdot\|_2\right):=\sup _Q \int_0^\delta \log N\left(\eta , \mathcal{F},\|\cdot\|_{2, Q}\right) d \eta . $$ The function class I need to deal with is $$ f_{ij}(z_{i}, z_{j}) = 1(z_{i} - z_{j}\leq t^{T}c_{ij}), \quad t \in \mathrm{R}^{d} $$ $c_{ij}\in \mathrm{R}^{d}$ is given parameter.
