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Given some probability space $(\Omega, \mathscr F, \mathbb P)$, I guess every event $B \in \mathscr F$ w/ positive probability induces some kind of 'probability subspace' $(B, \mathscr G^B, \mathbb Q^B)$

  • $\mathscr G^B := \{ C \subseteq B | C = D \cap B \ \text{for some} \ D \in \mathscr F \}$
  • $\mathbb Q^B(C) := P(D|B) := \frac{P(D \cap B)}{P(B)}$

Assuming $\mathscr G^B$ is indeed a $\sigma$-algebra on $B$ and $\mathbb Q^B$ is a probability measure on $(B,\mathscr G^B)$ and assuming well-defined-ness in choosing any $D$ works.

I guess it's like how groups with subsets that are closed under the group operation can be subgroups (if they also have identity and inverse) and then manifolds with subsets that are so and so (eg open in the topology) are submanifolds, etc. In this case, the condition for a subset to become a sub-object is simply to be an event with positive probability.

Question 1: Where can I read about this? I couldn't find these in some elementary (Larsen & Marx) or advanced probability texts (Rosenthal, David Williams), but I read something like this in Rosenthal in proving the Kolmogorov Zero-One Law. (Btw, when David Williams proves Kolmogorov Zero-One Law in Probability w/ Martingales there's technically no division needed.)

Question 2: Btw, it's well-defined because identical events $D \cap B = D' \cap B$, a fortiori have equal probability?

Question 3: Or more generally what about measure subspaces?

BCLC
  • 14,197
  • I am getting the impression that you are reading too many textbooks. It should be a fairly easy exercise to show that ${\mathscr G}^B$ is a $\sigma$-algebra and $\mathbb Q^B$ a well-defined probability measure on it. It models the conditional world in which we are certain that the event $B$ occurs. Rosenthal had a different $\sigma$-algebra and needed the measure on it to agree with $\mathbb P,.$ Don't get confused. Do math. Don't read math. – Kurt G. Feb 18 '25 at 09:40

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