Borel function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces. A measurable function is said to be bimeasurable if it is bijective and its inverse is also measurable.

In probability theory, the σ-algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.

Let ${\displaystyle (X,\Sigma )}$ and ${\displaystyle (Y,\mathrm {T} )}$ be measurable spaces, meaning that ${\displaystyle X}$ and ${\displaystyle Y}$ are sets equipped with respective ${\displaystyle \sigma }$-algebras ${\displaystyle \Sigma }$ and ${\displaystyle \mathrm {T} }$. A function ${\displaystyle f:X\to Y}$ is said to be measurable if the preimage of ${\displaystyle E}$ under ${\displaystyle f}$ is in ${\displaystyle \Sigma }$ for every ${\displaystyle E\in \mathrm {T} }$; i.e.

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