Suppose we have the probability space $(\Omega,\mathcal{A},P)$. Which of the following are right?
- $P$ is the probability measure defined on the events $\mathcal{A}$ as follows: $P:\mathcal{A}\rightarrow[0,1]$
- $P$ is the probability measure defined on the outcome space $\Omega$ as follows: $P:\Omega\rightarrow[0,1]$
- $X$ is a function $X:\mathcal{A}\rightarrow(E,\mathcal{E})$, where $(E,\mathcal{E})$ is a measurable space.
- $X$ is a function $X:\Omega\rightarrow(E,\mathcal{E})$, where $(E,\mathcal{E})$ is a measurable space.
Basically, I am unsure whether probability measures and random variables are defined on the state space $\Omega$, or the $\sigma-algebra$ $\mathcal{A}$, or both?