Measurable Functions

Key idea

Functions compatible with measures.

In depth

f is measurable if f⁻¹((a, ∞)) is measurable for all a. Continuous functions, limits of measurable functions, and sums and products are measurable. It is the class of functions that can be integrated.

Worked example

Indicator functions of measurable sets and step functions are measurable.

Real-world uses

• Ensuring that random variables and signals can be averaged in statistics and engineering

Common mistake

Assuming every function is measurable.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.