# Different probability space in the same sample space (probability)

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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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 $P_2(\{0\}) = P_2(\{1\}) = 1/2, P_2(\{\}) = 0, P_2(\{0,1\})=1.$
$\Omega$ 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. – Atul Ingle Oct 15 '12 at 21:42