Assume we have red and a black cube (normal cubes with 6 sides). We roll these two dice. Is it true that the sample space is $$\Omega = \left\{ (r,s) \mid r \in \{ 1, \dots, 6\}, s \in \{ 1, \dots, 6\} \right\}?$$
I'm now interested in the event $A =$'The red cube shows an even number' for example. Can I define the set $A$ like
$$A = \{(r,s) | r \in \{2,4,6\}, s \in \{ 1 \dots 6\} \}?$$
How can I define $A$'s probability density function and probability measure for this probability space?
your event $A$ is defined correctly. Before defining a probability measure you have to define a simga-field. intuitively, the sigma-field contains all sets which you want to be able to assign a probability to. Most often we simply choose the power set of $\Omega$ as the corresponding sigma field $\mathcal F$, i.e. $$ \mathcal F = \mathcal P (\Omega). $$
By doing so we are able to define a probability measure $\Bbb P$ on $\mathcal{F}$ which assigns a probability to every possible event. Since every event has the same probability and there is a total of 36 events we could define: $$ \Bbb P: \mathcal F \rightarrow [0,1]: A \mapsto \frac{|A|}{36} $$ In particular, if we want to compute the probability that the red dice is even we have to count all events where this is the case. It is easy to verify that in 18 ($ = 3*6$) cases the red dice is red. Therefore $$ \Bbb(P)(A) = \frac{18}{36} = \frac 12, $$ in this case.