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Measurable Functions#

Definition: Measurable Functions

Let \((X, \Sigma_X)\) and \((Y, \Sigma_Y)\) be measurable spaces and let \(f: X \to Y\) be a function.

We say that \(f\) is measurable if the inverse image \(f^{-1}(E)\) of each \(E \in \Sigma_Y\) is in \(\Sigma_X\):

\[ E \in \Sigma_Y \implies f^{-1}(E) \in \Sigma_X \]