Measurable Function: Overview

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.

measurable function
All step functions are measurable functions.


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. Let f : ω → S be a function that satisfies
f−1(A) ∈ F for each AA.
Then f is F/A-measurable. If the σ-field’s are understood from context,then f is 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


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