## What does $\lim_{\varepsilon \to 0}\mathbb P\{|\int_0^{t_0}b(\varepsilon ,t,\omega )|>\delta \}=0$ uniformly in $t_0\geq 0$ mean?

What does : for all $$\delta >0$$, $$\lim_{\varepsilon \to 0}\mathbb P\left\{\left|\int_0^{t_0}b(\varepsilon ,t,\omega )\right|>\delta \right\}=0$$ uniformly in $$t_0\geq 0$$ mean ? Is it that :

1) For all $$\delta >0$$ $$\lim_{\varepsilon \to 0}\sup_{t_0\geq 0}\mathbb P\left\{\left|\int_0^{t_0}b(\varepsilon ,t,\omega )\right|>\delta \right\}=0$$ uniformly in $$t_0\geq 0$$

2) or for all $$\delta >0$$

$$\lim_{\varepsilon \to 0}\mathbb P\left\{\sup_{t_0\geq 0}\left|\int_0^{t_0}b(\varepsilon ,t,\omega )\right|>\delta \right\}=0$$ uniformly in $$t_0\geq 0$$ ?