Measurable function: Events in the target sigma algebra correspond to events in the origin sigma algebra

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Let \(\; (\Omega, \mathcal{F}, P) \;\) be a probability space ("origin"), let \(\; (E, \mathcal{E}) \;\) be a measurable space ("target") and let \(\; X: \Omega \to E \;\) be a function.

We say that \(\; X \;\) is \(\; {\color{green} \textrm{measurable}} \;\) if, for every subset \(\; B \in \mathcal{E} \;\), its preimage \(\; X^{-1}(B) = \{ \omega \in \Omega:\; X(\omega) \in B \} \;\) is in \(\; \mathcal{F} \;\).

This correspondence between events in the target sigma algebra \(\; \mathcal{E} \;\) and events in the origin sigma algebra \(\; \mathcal{F} \;\) allows us to measure target events by applying the origin measure \(\; P \;\).