MEASURABLE FUNCTION Meaning and
Definition
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A measurable function is a concept in mathematics, specifically in measure theory, that describes a function between two measurable spaces. A measurable space is a set equipped with a sigma-algebra, which is a collection of subsets of the set that satisfies certain properties. A function is said to be measurable if the pre-image of any measurable set is also measurable.
To provide more clarity, let's consider two measurable spaces: (X, Σ) and (Y, τ), where Σ and τ are sigma-algebras on X and Y, respectively. A function f: X → Y is considered measurable if for any measurable set A in Y, its pre-image under f, denoted as f^(-1)(A) = {x ∈ X : f(x) ∈ A}, is also measurable in X.
In simpler terms, a measurable function is one that preserves the structure of measurable sets. It ensures that if we start with a measurable set in the codomain, we can always find a corresponding measurable set in the domain. This property is crucial in measure theory, as it allows us to define integrals, measures, and other fundamental concepts.
Measurable functions are widely used in various branches of mathematics, such as probability theory, real analysis, and functional analysis. They play a fundamental role in describing and analyzing the behavior of functions in the context of measures and integration, providing a basis for more advanced mathematical concepts and applications.
Etymology of MEASURABLE FUNCTION
The word "measurable" comes from the Latin word "mensurabilis", which means "able to measure". It is derived from the Latin word "mensura", meaning "measure". The word "function" comes from the Latin word "functio", meaning "performance, execution". In mathematics, the term "measurable function" refers to a function that possesses certain measurable properties.
