Measurable function

In mathematics, particularly in measure theory, a measurable function is a structure-preserving function between measurable spaces. For example, the notion of integrability can be defined for a real-valued measurable function on a measurable space. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the definition that a function between topological spaces is continuous if the preimage of each open set is open. A measurable function is said to be bimeasurable if it is bijective and its inverse is also measurable.

For example, in probability theory, a measurable function on a probability space is known as a random variable. The σ-algebra of a probability space 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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