Measurable Function - Formal Definition

Formal Definition

Let (X, Σ) and (Y, Τ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras Σ and Τ. A function f: X → Y is said to be measurable if f−1(E) ∈ Σ for every E ∈ Τ. The notion of measurability depends on the sigma algebras Σ and Τ. To emphasize this dependency, if f: X → Y is a measurable function, we will write

f: (X, Σ) → (Y, Τ).