Special Measurable Functions
- If (X, Σ) and (Y, Τ) are Borel spaces, a measurable function f: (X, Σ) → (Y, Τ) is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of some map, it is called a Borel section.
- A Lebesgue measurable function is a measurable function, where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.
- Random variables are by definition measurable functions defined on sample spaces.
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