I have just started learning about stochastic processes and I am confused with the notion of Brownian motion. The text defines (linear) Brownian motion under measure $\mathbb{P}$ as $B=(B_t; t\geq 0)$ where each $B_t$ is some random variable such that:
$\bullet$ $t\to B_t$ is a continuous function
$\bullet$ $B_t$ is distributed as $N(0,t)$.
$\bullet$ $B_{s+t}-B_{s}$ is distributed as $N(0,t)$
I have some questions regarding this definition. What is the sample space (space of events) on which each random variable $B_t$ is defined? What does it mean that $t\to B_t$ is continuous? Is it that for every $\omega$ in the sample space $t\to B_t(\omega)$ is continuous? What kind of values is $B_t$ taking?
Hopefully I formulated my questions clearly. Any help is appreciated.