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I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like $\left(\Omega, \mathfrak{F}, \{\mathfrak{F}_n\}, \mathbb{P}\right)$ where $\mathfrak{F}_0\subset\mathfrak{F}_1\subset\cdots\subset\mathfrak{F}$. $n$ is the index indicating time. Let us keep ourselves to discrete time and sample space. This is just to make the notations clear.

If I consider tossing a coin giving head($H$) or tail($T$) nine times (completely random, just to keep it finite), then $\Omega=\{H, T\}^9$. Obviously $\mathfrak{F}=\mathfrak{F}_9=\mathcal{P}(\Omega)$ where $\mathcal{P}$ is the power set. But, how about $\mathfrak{F}_2$? I have the feeling that $\mathfrak{F}_2=\mathcal{P}\left(\{H, T\}^2\right)$, since that is the set of yes/no questions we can answer after two tosses. But can we really say that $\mathcal{P}\left(\{H, T\}\right)^2\subset\mathcal{P}\left(\{H, T\}^3\right)\cdots\subset\mathcal{P}(\Omega)$?

Can you connect the simple repeated toss experiment with the abstract definition of filtered space?

Della
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    You have to begin with the full (infinite) sample space $\Omega={x_1x_2\cdots x_n\cdots: x_i\in{H,T}, \ \forall i}$ – zoli Oct 27 '15 at 07:07
  • Indeed one chooses the probability space once and for all and large enough to "receive" every random variable at play. Or rather, one does not choose it but leaves it unspecified, which is acceptable because the choice is irrelevant for all practical purposes and because general theorems ensure that such "large enough" sample spaces exist. True, this might enter in contradiction with the emphasis on describing one space for each situation, that some curricula lose an awful amount of time on, misleading students about what probability is really about. Which is not choosing $\Omega$! – Did Oct 27 '15 at 08:00
  • User @zoli recommands ${0,1}^\mathbb N$ to build the sequence of heads and tails $(X_n)$, this works, one could also consider $\Omega=[0,1]$, but the key fact is that, if ever one fixes $\Omega$ and one is said afterwards to consider another random experiment in addition, say a random integer $N$ independent of $(X_n)$, one would change $\Omega$, switching to $\Omega={0,1}^\mathbb N\times\mathbb N$ for example? And again when another random variable enters the frame? This simply would not make sense. – Did Oct 27 '15 at 08:04
  • @Did Thanks a lot for clarifying the concept of $\Omega$. I have edited the question. Can you show me the simplified case of how to construct $\mathfrak{F}_2$? – Della Oct 28 '15 at 10:32
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    $$\mathfrak{F}_k=\left{A\times{H,T}^{9-k}\mid A\in\mathcal{P}\left({H, T}^k\right)\right}$$ – Did Oct 28 '15 at 12:03
  • Excellent. You showed the whole thing in a new perspective. It should have been an answer. – Della Oct 29 '15 at 08:00

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