I am studying stochastic calculus on my own, and currently stuck to the following issue.

Say my probability space is $(\Omega, \mathcal F, \mathbb P)$. Now when my $\Omega$ has sequences of finite coin tosses, I can understand that I can draw the brownian motion based upon $Head/Tail$. For example, if $\omega=HH$, then my brownian path would be two upmoves (say). Now say $\Omega$ consists of infnite toss sequences. Now the only way I can get $W(t)$ is by sampling from normal distribution having variance $t$. Because for $t=1$, and for $t=2$, both consist of infinite sequences, and only way out is to sample from corresponding normal distribution with variance $1$ and $2$ respectively!

Now my question is, say we want to place a restraint on brownian motion, $W(t)$, to be less than , say $[0, 0.5]$. Now it corresponds to certain set $A \in \mathcal F$. Now this set $A$ has many elements in it, and each element corresponds to different Brownian motion. Now if I have certain variable $X(t, W)$, and we want to estimate this variable at time $t$, then what I think is logical is to average the variable $X$ over all browinan paths containing in set $A$.

Am I right in my thinking above?

Also a side question, if $\Omega$ is a Borel set on $\mathcal R$, then how do I map a borel set to brownian path. In case of infinite toss sequences, I can understand I have the toss, and I can simulate the path. But with Borel set, how can I draw the Brownian path? For example if $\omega=[1, 1.2]$, then how this closed set maps to a Brownian path?

Any clue would be greatly appreciated.


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