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Can someone give me examples to the following problem: Exist 2 different probability space on the same sample space? a probability space is a triple (Ω, σ-algebra , P) P - probability function, Ω - sample space

Thank you very much :) I try to understand.

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2 Answers 2

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Let $\Omega = \{0,1\}$ and $\Sigma = \{\{0\}, \{1\}, \{0,1\}, \{\} \}$. Here are two different probability spaces on the same sample space:

For the first probability space $(\Omega, \Sigma, P_1)$ assign $P_1(\{0\}) = 3/4, P_1(\{1\}) = 1/4, P_1(\{\}) = 0, P_1(\{0,1\})=1. $

To get another space, $(\Omega, \Sigma, P_2)$ assign $P_2(\{0\}) = P_2(\{1\}) = 1/2, P_2(\{\}) = 0, P_2(\{0,1\})=1.$

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  • $\begingroup$ if I know correct the Ω={0,1}, Σ={{0},{1},{0,1},{}}. is the sample space not probability space. The probability space must fit the 3 condition: I know that, the probability function fit 3 condition: P(a)≥0 P(Ω)=1 P(⋃j∈JAj)=∑j∈JP(Aj) $\endgroup$ Commented Oct 15, 2012 at 21:17
  • $\begingroup$ $\Omega$ is the sample space. $\Sigma$ is the $\sigma$-algebra. Probability space is the triplet $(\Omega, \Sigma, P)$. In the example above, I defined two different P's that satisfy the properties you mentioned, therefore I got two different probability spaces. $\endgroup$
    – Atul Ingle
    Commented Oct 15, 2012 at 21:42
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Given we have a $\Omega$ we can construct various $\sigma$-algebras on $\Omega$. Let's say we construct two different $\sigma$-algebras: $\mathcal{F}_1, \mathcal{F}_2$ on the same $\Omega$. For each $\sigma$-algebra we then define a probability measure $P_1, P_2$ (one for each $\sigma$-algebra). Then, we have two different probability spaces on the same $\Omega$, namely: $(\Omega, \mathcal{F}_1,P_1)$ and $(\Omega, \mathcal{F}_2,P_2)$.

If we were to use the same probability measure $P$ for both $\sigma$-algebras. Then, we have the two different probability spaces: $(\Omega, \mathcal{F}_1,P)$ and $(\Omega, \mathcal{F}_2,P)$.


Background:

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Yes, two different $\sigma$-algebras can have the same probability measure. Remember: a $\sigma$-algebra on $\Omega$ must (among other things) include $\Omega$ and $\varnothing$, and then it may or may not include more sets all of which must be subsets of $\Omega$. A probability measure gives probability to each set in a $\sigma$-algebra so for any $\sigma$-algebra defined on $\Omega$ the measure 1 will be assigned to $\Omega$ and the measure 0 to $\varnothing$, then if the $\sigma$-algebra contains more sets those will be assigned probabilities some where in between 0 and 1.

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A probability space (a.k.a. probability triple/probability model) is represented as: $(\Omega,\mathcal{F},P)$, where

$\Omega$ is a sample space (a set of objects),

$\mathcal{F}$ is a $\sigma$-algebra on $\Omega$ (a set of subsets of $\Omega$),

$P$ is a probability measure on $\mathcal{F}$ (a function assigning probabilities to each subset of $\mathcal{F}$).

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