# sigma algebra stochastic process

what is the sigma field created by X(t,w) where t belongs to [0,1] and X(t,w) =1 if t=w and zero otherwise.

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$X_{t}$ is a stochastic process. Maybe you mean a natural filtration?
The sigma algebra generated by $X_{t}$ is $\sigma(X_{t})=\{\emptyset,[0;1],t,[0;t)\cup (t;1]\}$.
The natural filtration $\mathcal{F}_{t}$ is the minimal sigma algebra which contains all $\sigma(X_{u})$ for $u\in [0;t]$.
So $\mathcal{F}_{t}$ consists of all countable subsets of $[0;t]$ and their complements.
@none: it does seem that the answer should be $\sigma(X_t) = \lbrace {\rm null}, {\rm omega}, \lbrace t \rbrace, [0,t) \cup (t,1] \rbrace$ (the only difference is in the curly brackets around $t$). –  Shai Covo Nov 16 '10 at 15:45