Let $B$ be a standard Brownian motion. Define a Brownian bridge $b$ by $b_t=B_t-tB_1$. Let $\mathbb{W'}$ be the law of this process.
According to Wikipedia,
A Brownian bridge is a continuous-time stochastic process B(t) whose probability distribution is the conditional probability distribution of a Wiener process W(t) (a mathematical model of Brownian motion) given the condition that B(0) = B(1) = 0.
Surely it makes no sense to condition on a probability 0 event? So I'm trying to show that $\mathbb{W'}$ is the weak limit as $\epsilon\to 0$ of Brownian motion conditioned upon the event $\{|B_1|\leq \epsilon\}$. How do we prove this?
Thank you.