In what follows $B$ is real-valued Brownian motion. The book I am following (Schilling and Partzsch, Brownian motion, An introduction to Stochastic Processes) states the following without proof. Let $t\mapsto h(t)$ be a deterministic function. Then, it follows from Blumenthal's $0-1$ law that
$$P\{B_t < h(t) \text{ for all sufficiently small } t\} \in \{0,1\} \label{eqn0}\tag{0}$$
Clearly, what is meant here is
$$\{\omega: B_t(\omega) < h(t) \text{ for all sufficiently small } t\} \in \mathcal{F}_{0}^+ \label{eqn1}\tag{1}$$ where $\mathcal{F}_{0}^+ = \bigcap_{t>0}\mathcal{F}_t$ and $\mathcal{F}_t = \sigma(B_s, s\leq t)$. I am trying to understand what \eqref{eqn1} means and to prove it.
First, the quantifier "for all sufficiently small $t$". I am not entirely sure how to interpret this but I guess the only thing that determines what sufficiently small means is the function $h$. Moving on, I write \eqref{eqn1} as
$$\{\omega: B_t(\omega) < h(t) \quad \forall t \in [0,\varepsilon]\} \in \mathcal{F}_{0}^+ \label{eqn2}\tag{2}$$
Furthermore, $$\{\omega: B_t(\omega) < h(t) \quad \forall t \in [0,\varepsilon]\} = \bigcap_{t\leq\varepsilon}\{\omega: B_t(\omega) < h(t)\}$$ I don't see why the event on the right should belong to $\mathcal{F}_{0}^+$, which makes me think that my understanding of this claim is flawed. I would appreciate it if someone could explain how to interpret \eqref{eqn0} and how to prove it.
Saying that $B_t < h(t)$ for all sufficiently small $t$ means that there exists a positive $\varepsilon$ such that if $t\in [0,\varepsilon]$, then $B_t < h(t)$. Let $C'_t:=\left\{B_t < h(t)\right\}$, $C_\varepsilon:=\bigcap_{0\leqslant t\leqslant \varepsilon}C'_t$. Then we have to prove that $$\bigcup_{\varepsilon\gt 0}C_\varepsilon\in \mathcal F_0^+.$$ Since $C_\varepsilon\subset C_{\varepsilon'}$ if $\varepsilon\geqslant \varepsilon'$ the following equality holds for all $\varepsilon_0\gt 0$: $$ \bigcup_{\varepsilon\gt 0}C_\varepsilon=\bigcup_{0\lt \varepsilon\lt \varepsilon_0}C_\varepsilon $$ and the latter set belongs to $\mathcal F_{\varepsilon_0}$. Arbitrariness of $\varepsilon_0$ allows to conclude.