stochastic process

One thing I was confused on initially was the outcome space. Consider a stochastic process with the probability space \((\Omega, \mathcal{F}, P)\). I think of an \(\omega \in \Omega\) as corresponding with an infinite dimensional vector that represents a single realization of a stochastic process. I can think of the random variable \(X_t(\omega)\) as selecting the value at slot \(t\). In general, \(P(X_t = a)\) is \(P\{\omega \in \Omega \mid X_t(\omega) = a\}\).