What does it mean to be in the same probability space? Sometimes I see the expression "two random variables are defined on the same probability space $(\Omega,F,P)$", and I'm curious what it means to be in the same probability space. Does it mean the same as being in the same sample space $(\Omega)$?
For example, let's say I have an object flying along the x-axis, and I pin one point in the x-coordinate, than the flying object can be anywhere in the y-axis and in the z-axis. So two probability distributions are defined as [$g(y) dy$] and [$h(z) dz$].
And
$\int_{-\infty}^\infty g(y)\,dy = 1$
$\int_{-\infty}^\infty h(z)\,dz = 1$
Are the random variables $Y$ and $Z$ defined on the same probability space $(\Omega,F,P)$ ? (It seems like they are in a different sample space..)
 A: When it says $X, Y, Z$ are "on the same probability space" it means they are all part of the same probability experiment.  So indeed there is just one sample space $\Omega$ and the random variables are all functions from that sample space to the real numbers:
\begin{align}
X:\Omega\rightarrow \mathbb{R}\\
Y:\Omega\rightarrow\mathbb{R}\\
Z:\Omega\rightarrow\mathbb{R}
\end{align}
So the particular outcome $\omega \in \Omega$ in the experiment determines the values $X(\omega)$, $Y(\omega)$ and $Z(\omega)$. It means that it makes sense to ask about the probability that $X+e^Y\leq Z$, and to define new random variables such as $W=X+Y+Z$ (otherwise these things would not make sense). Note that if $X, Y, Z$ are on the same probability space, they are not necessarily distributed the same, and they are not necessarily independent.
Further, it only makes sense to say that $X$ and $Y$ are independent if they are on the same probability space. Otherwise, the probability $P[\{X\leq 2.3\} \cap \{Y\leq 1.2\}]$ would not make sense because $\{X\leq 2.3\} \cap \{Y\leq 1.2\}$ would not be a valid event.
For a given problem, it is usually assumed that all random variables are on the same probability space (unless otherwise stated). For example, suppose we flip 10 coins, with all outcomes equally likely, and define $X$ as the number of HEADS, $Y$ as the number of TAILS, and
$$ Z = \left\{\begin{array}{c}
1 & \mbox{ if $X=Y$} \\
0 & \mbox{ else} 
\end{array}\right.$$
So $X, Y, Z$ are all on the same probability space, but that is so obvious that it is usually not stated. Usually you would only mention that random variables are on the same probability space if you want to be formal, or if you want to derive some general result without giving the details of the probability experiment.
