 Solution

Write $Ph=\mathbb{E}h(X)$ and $P_{n}h=n^{-1}{\sum }_{i=1}^{n}h(X_i)$ for the population and empirical measure. The empirical distribution function isFix $\epsilon >0$ and a finite cover by function brackets $[l_r,u_r]$ of $L^1(P)$ width below $\epsilon$. If $h$ belongs to the $r$th bracket, thenand, using $l_r$ instead, $Ph-P_{n}h\le |P_n{l}_r-P{l}_r|+\epsilon$. ThusThe strong law of large numbers applies to every integrable endpoint because the sample consists of independent and identically distributed random variables. For this finite collection the maximum tends to zero on a common probability-one event. Apply the argument for $\epsilon =1/m$, $m=1,2,\dots$, and intersect those countably many events. On the resulting event the limsup is at most $1/m$ for every $m$, proving the uniform strong law from finite L1 bracketing. This pathwise argument also avoids assuming without justification that an arbitrary uncountable supremum is measurable.

For the Glivenko-Cantelli theorem, take $h_t(x)=1_{\{x\le t\}}$. Here is a finite bracketing construction which also handles atoms of the distribution. Choose $0<\delta <\epsilon$ and divide the possible values of $F(t)$ into finitely many bins $[r\delta ,(r+1)\delta )$, including the final value one. For each nonempty parameter set $T_r=\{t:r\delta \le F(t)<(r+1)\delta \}$, set $l_r={\inf }_{t\in T_r}{h}_t$ and $u_r={\sup }_{t\in T_r}{h}_t$ pointwise. These functions are indicators of rays, possibly with an open endpoint, so they are measurable. Monotonicity of $h_t$ and monotone limits at the extreme thresholds give $P{l}_r={\inf }_{t\in T_r}F(t)$ and $P{u}_r={\sup }_{t\in T_r}F(t)$. Hence $P(u_r-l_r)\le \delta <\epsilon$. No continuity of $F$ is required. Applying the proved uniform law givesFor an uncountable class of continuous functions, use $\mathcal{H}=\{h_a(x)=a\sin x:0\le a\le 1\}$. The functions are distinct, bounded and integrable for every probability law. Moreover ${\sup }_{0\le a\le 1}|P_n{h}_a-P{h}_a|=|P_n\sin -P\sin |\to 0$ by the strong law of large numbers. This supplies the requested example without additional compactness machinery.