Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $T$ a stopping time. Let $(\mathcal F_t)$ a filtration and $$\mathcal F_T=\{A\in \mathcal F\mid A\cap\{T\leq t\}\in \mathcal F_t, \forall t\geq 0\}.$$ Let $(X_t)$ a stochastic process. In my course it's written that $$\boldsymbol 1_{T<\infty}X_T$$ is $\mathcal F_T$ measurable. In what is this a good thing ? First, I have difficulties to really understand what is $\mathcal F_T$, but also, I don't understand in what the fact that $\boldsymbol 1_{T<\infty}X_T$ is $\mathcal F_T-$measurable can be helpful.
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what does "in what is this a good thing?" mean? – mathworker21 Oct 01 '18 at 07:11
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maybe he meant "in what way is this a good thing"? – user10354138 Oct 01 '18 at 07:32
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yes, in what way is this a good way ? What does it change to have this function $\mathcal F_T$ measurable ? – bob Oct 01 '18 at 10:23
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I mean, in general, you want functions to be measurable so that you can play around with them – mathworker21 Oct 01 '18 at 17:51