Types of Functions > Measurable function

In measure theory, a **measurable function **is analogous to a continuous function in topology and a random variable in probability theory; they can be integrated with respect to *measures *in a similar way to how continuous functions can be integrated with respect to *x*. The building blocks of a topological space are *open sets*, while the building blocks of a measurable space are *measurable sets*.

The exact definition of a measurable function depends on

**the type of measure**. However, a measurable function on a closed interval is defined and finite almost everywhere; it can be represented a sequence of step functions which converge almost everywhere to the measurable function as

*n*→ ∞ [1]. Every step function is measurable, as are continuous functions and the Dirichlet function.

## Formal Definition of Measurable Function

A general definition for a measurable function is as follows [2]:

“Let (ω,

F) and (S,A) be measurable spaces. Letf: ω →Sbe a function that satisfies

f^{−1}(A) ∈Ffor eachA∈A.

ThenfisF/A-measurable. If the σ-field’s are understood from context,thenfis measurable.”

Measure theory is an advanced mathematical topic beyond the scope of this article. However, the good news is that any measurable function can be approximated by simple functions [3]. The *C*-property tells us that a measurable function on a closed interval can be turned into a continuous function on the same interval by changing its values on a set of arbitrarily small measure [4].

Several classes of these functions exist. The most common ones are random variables, Borel functions, and the Lebesgue Measurable Function.

Note that the measurability of a function depends only on σ-algebras, that is the collection of subsets of measurable set in a measurable space. Measures do not actually have to be defined [5].

## References

[1] Polyanin, A. et al. (2008). Handbook of Integral Equations, Second Edition. CRC Press.

[2] Rinaldo, A. (2020). Lecture 05 – Measurable Functions. Retrieved November 11, 2021 from: http://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Scribed_Lectures/Scribed_Lecture05_Sep16(W).pdf

[3] Swartz, C. (1994). Measure, Integration and Function Spaces.

[4] Hazelwinkel, M. (2006). Encyclopaedia of Mathematics, Volume 6. Springer Netherlands.

[5] Measurable functions. Retrieved August 23, 2023 from: https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf