Let $[c,d]$ be a subset or $\mathbb R$. On $\mathscr B([c,d])$ we define the probability $P$ such as
$$P(I)=\frac{b-a}{d-c} $$ if $I$ is a Borelian with lower and upper bound $a$ and $b$
I call this probability the uniform probability (but maybe it's not appropriate..).
Could you provide other examples of probabilities on a continuous sample space, different from the uniform probability?
Especially I notice that if $\Omega=\mathbb R$, the uniform probability cannot work on $\mathscr B(\mathbb R)$. What is the probability corresponding?
PS: I am having a headaching hard time to see the link between the sample space and the event space in the case of a continuous sample space. I've read some stuff on the probability distributions, but as far as I understand it is related to the event space of a R.V.
Also, I have the feeling that most of the time the R.V. is simply $X: \omega \rightarrow \omega$, is it correct ?
Lastly, is it also the case in the continuous case that the pre-image of the Random Variable partitions the sample space? (as it was the case in discrete spaces)