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If I have the following expression:

$E[M_{t\wedge n}|\mathcal{F}_s]$

Where the set of random variables $M_{t\wedge n}$ is bounded in $L^2$, i.e.

$\sup_{t} E[M^2_{t\wedge n}]<\infty$

Hence they are uniformaly integrable. In fact $(M_t)$ is a local martingale. Now my question, why can I take the limit for $n\to \infty$ insight the expectation in the first expression to obtain

$E[M_t|\mathcal{F}_s]$ ?

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    I edited the question! I think I forgot something important2012-06-05

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