A basic measure theory question on Stochastic Process

Question

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process, so it is measurable map. Let $S^T$ be the collection of all functions from $T$ into $S$. I don't understand why $Φ_X : Ω → S^T$, defined by $Φ_X(\omega)(t) = X_t(\omega)$ is a measurable map. Is Fubini's theorem of help here.