Measurable Function

Definition

A measurable function is a function between measurable spaces that preserves the measurable structure. It is the morphism in the category of measurable spaces.

Let $(X,{\Sigma }_X)$ and $(Y,{\Sigma }_Y)$ be measurable spaces. A function $f:X\to Y$ is measurable if for every measurable set $B\in {\Sigma }_Y$, the preimage is measurable:

$f^{-1}(B)\in {\Sigma }_X$

Properties

• The composition of measurable functions is measurable
• The identity function on a measurable space is measurable
• Continuous functions between topological spaces (with their Borel σ-algebras) are measurable
• Measurable functions are closed under pointwise limits

Measurable Functions on $\mathbb{R}$

For real-valued functions $f:X\to \mathbb{R}$, it suffices to check that $f^{-1}((-\infty ,a))$ is measurable for all $a\in \mathbb{R}$, since these sets generate the Borel σ-algebra on $\mathbb{R}$.

Examples

• Any continuous function between topological spaces is Borel measurable
• Indicator functions $1_A$ are measurable if and only if $A$ is a measurable set
• Limits of sequences of measurable functions are measurable
• Sums and products of measurable real-valued functions are measurable

Relationship to Integration

Measurable functions are precisely the functions that can be integrated with respect to a measure. The Lebesgue integral is defined first for simple measurable functions, then extended to non-negative measurable functions, and finally to integrable functions.

Related Concepts

• Measurable Space: The domain and codomain of measurable functions
• Measure: Measurable functions are integrated with respect to measures
• Sigma Algebra: The structure preserved by measurable functions
• Borel Set: Measurable sets in topological spaces
• Continuous Map: Always measurable with respect to Borel σ-algebras
• Meas (Category): The category with measurable functions as morphisms