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$ ?