Definition of pmf of two random variables is as follows: If $X,Y$ are discrete random variables, on $(\Omega,\mathcal F,\mathsf P)$, then PMF of vector $(x,y)$ is $$\mathsf P_{(X,Y)}(x,y)~{=\mathsf P(X=x,Y=y)\\=\mathsf P\{\omega \in \Omega : X(\omega)=x, Y(\omega)=y\}}$$
My question is why is it only one sample space $\Omega$? shouldn't it be $\Omega_x$ for X and $Ω_y$ for y? can anyone give me an example? Thanks.
Additionally, $\sum_{x∈ImgX, y∈imgY} P(X=x, Y=y)=1$ and $P_{(X,Y)}>=0$ according to the definition of probability measure; however, why is $\sum_{x∈ImgX} P(X=x)=1$ not necessarily true (or $\sum_{y∈ImgX} P(Y=x)=1$)? and does this mean $P_{X}>=0$ does not have to be true ($P_{(X,Y)}$ can be $<0$)?
Please give an example of this as well if possible.
The random variables are on the same sample space and there is not and should not be two different sample spaces.
The outcome in $\Omega$ that occurs will fully specify the values of all the random variables under consideration. A (real-valued) random variable is a measurable function $X:\Omega\to \mathbb R.$ We think of this as meaning that if an outcome $\omega\in \Omega$ occurs then $X$ takes the value $X(\omega)$ and $Y$ takes the value $Y(\omega).$
We can do things very explicitly in the simple situation where both $X$ and $Y$ are binary random variables, taking the value of either zero or one. Here our sample space must consist of at least four points, corresponding to $(X,Y) = (0,0), (0,1),(1,0),(1,1).$ (Note I said at least. The sample space could easily have far more points than this, even infinitely many. $X$ and $Y$ aren't necessarily the only things $\Omega$ "knows about".)
Let's say we're int the simplest case where $\Omega$ has exactly four points. Then these four correspond to the four possible outcomes for $(X,Y).$ Let's label the elements as $$\Omega = \{\omega_{0,0},\omega_{1,0},\omega_{0,1},\omega_{1,1}\}$$ where the meaning of the subscripts should be clear. Then any event is a subset of $\Omega.$ For instance the event that $X=0$ and $Y=0$ corresponds to the subset $\{\omega_{0,0}\}$ and the event that $X=0$ corresponds to the subset $\{\omega_{0,0}, \omega_{0,1}\}.$ So we can write things like $$ P(X=0,Y=0) = P(\{\omega_{0,0}\}) \\ P(X=0) = P(\{\omega_{0,0}, \omega_{0,1}\}) = P(\{\omega_{0,0}\})+P(\{\omega_{0,1}\})$$ where in the last example we used the additivity of the measure. By playing around a little more, you should be able to see that everything comes out as expected: the space is characterized by four probabilities, corresponding to the four possible values of $(x,y)$ in $P(X=x,Y=y).$
If $X$ and $Y$ lived on different probability spaces, it would be unclear how they were related to one another, i.e. what their dependency is. Perhaps we could think of random variables in different homework problems (that have nothing to do with one another) as being on different sample spaces, but ones that are part of the same situation must be on the same. More generally they must be on the same space if we want to think about their joint distribution.