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Lebesgue Measurable Function

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Lebesgue measurable functions play an important role in Lebesgue integration. Lebesgue measure is a natural extension of the concept of area, length, or volume, depending on dimension. Several different definitions are available. One general definition is [1]:


“Let f be a function defined on a measurable domain E taking values in the extended real number line. We say f is a Lebesgue measurable function if for every real number c the set
{x ∈ E | f(x) > c}
is measurable.”

We can also say that an extended real-valued function f defined on E ∈ M is Lebesgue if it satisfies the following (equivalent) statements for each number c in ℝ [2]:

  • {x ∈ E | f(x) > c} ∈ M.
  • {x ∈ E | f(x) ≥ c} ∈ M.
  • {x ∈ E | f(x) < c} ∈ M.
  • {x ∈ E | f(x) ≤ c} ∈ M.

A Lebesgue measurable function f can also be defined as follows: R → C is a function such that f-1(V) ∈ ℛ for any open set V ⊂ C. “ℛ” here is the σ-algebra of all m-measurable sets [3]. In other words, Lebesgue measurable functions are functions in σ-algebra generated by open sets (as well as null sets).

In general terms, if a function g is labeled as “measurable,” it means that g is a Lebesgue measurable function.

Lebesgue Measurable Function Example

Let’s take a look at f(x) = x2 on the closed interval [−1, 5]. One definition for a Lebesgue measurable function is that it is Lebesgue measurable on I if, for every s ∈ ℝ the set
{x ∈ I | f(x) > s}
is a Lebesgue measurable set [4].


If we let s ∈ ℝ:

  1. If s ≥ 25, then {x ∈ I | f(x) > s} = ∅.
  2. If s < 0, then {x ∈ I | f(x) > s} = [−1, 5].
  3. If 0 ≤ s < 1, then {x ∈ I | f(x) > s} = [−1, −√s) ∪ (√s, 5].
  4. If 1 ≤ s < 25, then {x ∈ I | f(x) > s} = (√s, 5].

All of the above are Lebesgue measurable sets. Therefore, f is a Lebesgue measurable function on the interval [−1, 5].

Lebesgue Measurable Functions vs. Borel & Gauge

All Borel measurable functions are Lebesgue measurable, but the converse is not always true. There are a few subsets which are Lebesgue measurable but not Borel measurable. These subsets are tedious to construct and involve defining a continuous function on the Cantor set (a set of points on a single line segment with a number of unintuitive properties) [5].

A Lebesgue measurable function is Gauge integrable under one of the following conditions [6]:

  1. It has locally small Riemann sums or,
  2. It has functionally small Riemann sums.

*Note: a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.

References

[1] Oliver, J. (2019). Survey of Lebesgue and Hausdorff Measures.
[2] Chapter 3: Lebesgue Measurable Functions. Retrieved August 6, 2021 from:
https://faculty.etsu.edu/gardnerr/5210/notes/3-1.pdf
[3] Math 73/103 Assignment Three.
[4] Nelson, G. (2015). A User-Friendly Introduction to Lebesgue Measure and Integration. Student Mathematical Library, Volume 78. American Mathematical Society.
[5] Nelson, B. The Lebesgue Integral. Retrieved August 6, 2021 from: https://math.berkeley.edu/~brent/files/lebesgue_integral.pdf
[6] Bongiorno, B. (2002). The Henstock-Kurzweil Integral. In Handbook of Measure Theory.

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Stephanie Glen. "Lebesgue Measurable Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/lebesgue-measurable-function/
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