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Consider the probability space $(\Omega,\mathcal{F},\mathbb{P})$, where $\mathbb{P}$ is defined as : $\mathbb{P}:\mathcal{F} \rightarrow [0,1]$ satisfies

$$ (D1)\\~\mathbb{P}(\Omega)=1 \\~ \text{if}~ A_n\in \mathcal{F}, A_i \cap A_j=\phi, i \neq j ~\text{then}~ \mathbb{P} \left( \cup A_n \right)=\sum \mathbb{P}(A_n) $$ I found another definition of $\mathbb{P}$ satisfying the axioms :

$$(D2)\\\mathbb{P}(\Omega)=1 \\\mathbb{P} \left(A_1 \cup A_2 \right)=\mathbb{P}(A_1)+\mathbb{P}(A_2), A_1,A_2 \in \mathcal{F}, A_1 \cap A_2 =\phi \\ \text{If}~A_n ~\text{is decreasing}~ \mathbb{P} \left(\cup A_n\right)= \lim \mathbb{P}(A_n)$$ How these two definitions are equivalent? i.e $(D1) \iff (D2)$ ?

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    $\begingroup$ There seems to be a mistake in your D2. Either "decreasing" should be "increasing", or else $\cup$ should be $\cap$. $\endgroup$ Aug 4, 2019 at 18:29
  • $\begingroup$ But with that correction, yes, they are equivalent. Please tell us what you tried in attempting to prove this. $\endgroup$ Aug 4, 2019 at 18:30
  • $\begingroup$ @NateEldredge i was thinking of taking a family of disjoint sets $\{B_i : i \ge 1 \}$ i.e $B_1=A_11, B_2=A_2 \setminus A_1, B_3=A_3 \setminus A_1 \cup A_2 ...$ then taking union over them and use and set equality condition, then im unable to proceed $\endgroup$
    – Siddhartha
    Aug 4, 2019 at 18:42

1 Answer 1

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Hints:

(i) $\underset{n\in \mathbb{N}}{\cup}A_n= \underset{N\rightarrow \infty}{\lim}\Big( \underset{n=1}{\overset{N}{\bigcup}}A_n \Big)$

(ii) $\mathbb{P}\Bigg( \underset{n=1}{\overset{N}{\bigcup}}A_n\Bigg)= \sum \limits_{n=1}^N\mathbb{P}(B_n)$ in both definitions, given your what you wrote as $B_n$ in the comments

(iii) $ \underset{n=1}{\overset{N}{\bigcup}}A_n$ is an increasing event for any sequence of events $\{ A_n\} $

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