The sample space is the set where you pick up the events (it's called in general).
The support is the set where the random variable you're considering is non-zero.
You consider an event and a random variable (which is a mapping). If , then it belongs to the support of X.
From what I understand of a state-space, it is similar to the support, except that we're considering the different possible states of a stochastic process (e.g. a Markov chain)