A

**Borel function**(or

*Baire function*) is defined as a function where

**{ t: f(t) < a} ∈ B(ℝ^{n})**

for every *a *∈ ℝ.

To define this slightly differently, if a function is Borel measurable, then that function is a Borel function. Measurable functions are, generally speaking, a generalization of the idea of continuous functions.

## Borel Function Properties

A single-variable, real-valued Borel function has a domain and range on the real number line; these functions are measurable with respect to Borel sets.

Borel sets are certain subsets of topological space, which form the Borel σ-algebra of the space. Given a topological space

X, there is a σ-algerbra on spaceXgenerated by open subsets of X. Borel sets are elements of this σ-algebra [1].

Therefore, the inverse image (i.e., the set of points that map to a given point) of any Borel set in the range space is a Borel set in the domain space [2]. Every real-valued continuous function is a Borel function. A complex-valued function is a Borel if both its imaginary and real parts are Borel functions [3]. We can generalize these statements to define Borel functions as any real-valued function or complex-valued function that has a Borel set as an inverse image f^{-1} for every open subset *U *of ℝ or ℂ [4].

A *simple *Borel function is a step function; Borel functions are step functions or limits of step functions.

Given a Borel function, every pointwise limit is also a Borel function, so is every:

- Countable infinimum and countable supremum.
- Every countable limit inferior and limit superior.
- Bounded pointwise limit of a continuous function (for example, the indicator function) [4].

Every Borel function is Lebesgue measurable [5], but the converse is not true; In other words, Lebesgue measurable functions are defined in a larger set.

## References

[1] Pfeiffer, P. (2013). Concepts of Probability Theory: Second Revised Edition. Dover Publications.

[2] Wan, K. (2019). Quantum Mechanics. A Fundamental Approach. Jenny Stanford Publishing.

[3] Bhat, B. Modern Probability Theory. New Age International.

[4] Sternberg, S. (2019). A Mathematical Companion to Quantum Mechanics. Dover Publications.

[5] Swartz, C. (1994). Measure, Integration and Function Spaces. World Scientific.