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