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Let $\Omega$ be the space of continuous functions $\omega: [0,T]\to \mathbb{R}^{d}$, $\mathcal{F}=\mathcal{B}(C[0,T))$ and $\mathbb{P}$ be the Wiener measure. Therefore the coordinate processes $W$ defined by $W_t(\omega)=\omega_t$ is a Brownian motion. Let $(\mathcal{F}_t)_{0\leq t<\infty}$ by the natural filtration generated by $W$. Is the following conclusion true?

For $0<t<s$ and a non-null set $A\in \mathcal{F}_s$, we can always find a non-null set $B$ satisfying that $B\subset A$ and $B\in\mathcal{F}_t$?

Jackie
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  • Intuitively the situation should be reversed. Measurable sets in a filtration are the sets that you can know whether you are in them by observing only up to a certain time. Thus you can make finer distinctions if you have more time to wait, so you would expect that if $A \in \mathbb{F}_t$ then there is a strict subset of it in $\mathcal{F}_s$. This is not to say that what you said doesn't happen, but if it does I think it will have to be somewhat trivial. – Ian Oct 24 '15 at 03:53
  • Thanks for the comment and it makes sense to me. But I am still wondering if what I asked is true. Thanks! – Jackie Oct 24 '15 at 13:25
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    Does anyone of the persons who voted to close this question have an idea how to answer the question....? – saz Oct 24 '15 at 19:05

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