Here is an example for the construction of a probability space in Shreve's stochastic calculus, page 4, what is the meaning for $2^{(2^0)}$? it seems like a method to calculate number of subsets, but I still cannot fully understand.Thanks for help!
2 Answers
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The probability space corresponding to the experiment of flipping $x$ coins has $2^x$ possible outcomes. So these outcomes generate a sigma-algebra of size $2^{2^x}$.
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I am not exactly sure, but because $\Omega_\infty$ is a set of possible outcomes, the 2 sets must refer to $\emptyset$ and $\Omega$. Then, based on this, I would guess that $\textbf{2}^{2^0}$ refers to the number of sets, $2^{\textbf{2}^0}$ (the exponential two) refers to the outcomes, which are heads or tails.
Not too sure about the zero that is the exponential's exponential though.
