# 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.

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 $P_2(\{0\}) = P_2(\{1\}) = 1/2, P_2(\{\}) = 0, P_2(\{0,1\})=1.$

• 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) Commented Oct 15, 2012 at 21:17
• $\Omega$ 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. Commented Oct 15, 2012 at 21:42

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:

1)

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.

2)

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 $$\mathcal{F}$$).