Mathematics.

measurable maps

Measurable Functions

Measure Theory65 minDifficulty7 out of 10

You should know: sigma algebras, measure spaces

Overview

A measurable function is a map between measurable spaces that respects their σ-algebraic structure: the pre-image of every measurable set is measurable. This is the natural notion of 'structure-preserving map' in measure theory, analogous to continuous functions in topology. Measurable functions are the integrands of the Lebesgue integral and the random variables of probability theory.

Intuition

Just as a continuous function pulls back open sets to open sets, a measurable function pulls back measurable sets to measurable sets. The condition ensures that asking 'for which x is f(x) ∈ B?' always has a measurable answer, making it possible to compute probabilities or integrals involving f.

Formal Definition

Definition

Let (X, ℱ) and (Y, 𝒢) be measurable spaces. A function f : X → Y is measurable (or (ℱ, 𝒢)-measurable) if the pre-image of every 𝒢-measurable set is ℱ-measurable.

f:X→Y is measurable  ⟺  ∀B∈G,  f−1(B)∈Ff : X \to Y \text{ is measurable} \iff \forall B \in \mathcal{G},\; f^{-1}(B) \in \mathcal{F}
Definition of measurability
f−1(B):={x∈X:f(x)∈B}f^{-1}(B) := \{x \in X : f(x) \in B\}
Pre-image notation
f:(X,F)→(R,B(R)) is measurable  ⟺  ∀a∈R,  {x:f(x)>a}∈Ff : (X,\mathcal{F}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R})) \text{ is measurable} \iff \forall a \in \mathbb{R},\; \{x : f(x) > a\} \in \mathcal{F}
Practical criterion for real-valued functions

Notation

NotationMeaning
f−1(B)f^{-1}(B)Pre-image of B under f
f:(X,F)→(Y,G)f:(X,\mathcal{F})\to(Y,\mathcal{G})Measurable function between measurable spaces
σ(f)\sigma(f)σ-algebra generated by f — the smallest making f measurable
f+,f−f^+, f^-Positive and negative parts of f: f⁺ = max(f,0), f⁻ = max(−f,0)

Properties

Compositions of measurable functions are measurable

f:(X,F)→(Y,G),  g:(Y,G)→(Z,H) measurable  ⟹  g∘f measurablef:(X,\mathcal{F})\to(Y,\mathcal{G}),\; g:(Y,\mathcal{G})\to(Z,\mathcal{H}) \text{ measurable} \implies g\circ f \text{ measurable}

Pointwise limits of measurable functions are measurable

fn measurable for all n,  f=lim⁡n→∞fn (pointwise)  ⟹  f measurablef_n \text{ measurable for all } n,\; f = \lim_{n\to\infty}f_n \text{ (pointwise)} \implies f \text{ measurable}

Sup/inf of countable families

sup⁡nfn,  inf⁡nfn,  lim sup⁡nfn,  lim inf⁡nfn are all measurable when each fn is\sup_n f_n,\; \inf_n f_n,\; \limsup_n f_n,\; \liminf_n f_n \text{ are all measurable when each } f_n \text{ is}

Every continuous function is Borel measurable

f:R→R continuous  ⟹  f is (B(R),B(R))−measurablef : \mathbb{R} \to \mathbb{R} \text{ continuous} \implies f \text{ is } (\mathcal{B}(\mathbb{R}),\mathcal{B}(\mathbb{R}))-\text{measurable}

Worked Examples

  1. The indicator function 1_A : X → {0,1} takes value 1 on A and 0 on Aᶜ.

    1A(x)={1x∈A0x∉A\mathbf{1}_A(x) = \begin{cases}1 & x \in A \\ 0 & x \notin A\end{cases}

  2. The pre-image of the set {1} is A, and of {0} is Aᶜ. These are the only non-trivial pre-images.

    1A−1({1})=A,1A−1({0})=Ac\mathbf{1}_A^{-1}(\{1\}) = A,\quad \mathbf{1}_A^{-1}(\{0\}) = A^c

  3. 1_A is measurable iff both A and Aᶜ are in ℱ, which holds iff A ∈ ℱ (since ℱ is closed under complements).

    1A measurable  ⟺  A∈F\mathbf{1}_A \text{ measurable} \iff A \in \mathcal{F}

Answer: 1_A is measurable if and only if A ∈ ℱ.

Practice Problems

Difficulty 6/10

Show that if f is measurable, then |f| is measurable.

Difficulty 7/10

Prove that the pointwise supremum of a countable family of measurable functions is measurable.

Difficulty 8/10

Give an example of a function f : ℝ → ℝ that is Lebesgue measurable but not Borel measurable.

Common Mistakes

Common Mistake

Measurability is the same as continuity

Continuity is much stronger. Every continuous function is Borel (hence Lebesgue) measurable, but measurable functions can be wildly discontinuous — e.g., the Dirichlet function 1_ℚ is measurable but nowhere continuous.

Common Mistake

If f² is measurable, then f is measurable

This is false. f² = (|f|)² and |f| being measurable does not determine the sign of f. Counterexamples exist using non-measurable sign functions.

Common Mistake

The image f(A) of a measurable set is measurable

The pre-image f⁻¹(B) of a measurable set is measurable, but images need not be. Images of Borel sets under continuous functions can be non-Borel (though they are still analytic/Suslin sets).

Quiz

f : (X, ℱ) → (ℝ, ℬ(ℝ)) is measurable if and only if:
A simple function is defined as:
The composition g ∘ f of measurable functions is:

Summary

  • f : (X,ℱ) → (Y,𝒢) is measurable if f⁻¹(B) ∈ ℱ for every B ∈ 𝒢.
  • For real-valued functions it suffices to check {f > a} ∈ ℱ for all a ∈ ℝ.
  • Compositions, pointwise limits, sup, inf, limsup, and liminf of measurable functions are measurable.
  • Every non-negative measurable function is a pointwise increasing limit of simple functions.
  • Measurable functions are the correct class for Lebesgue integration and serve as random variables in probability.

References

  1. BookFolland — Real Analysis, 2nd ed. (1999), §1.3
  2. BookRudin — Real and Complex Analysis, 3rd ed. (1987), Chapter 1
  3. WebsiteWikipedia — Measurable function