Lebesgue Integral

Key idea

Integrate by partitioning the range.

In depth

Instead of slicing the domain, slice the range: ∫f dμ = sup of integrals of simple functions below f. It handles more functions than Riemann, and has better convergence theorems.

Worked example

The Dirichlet function (1 on Q, 0 elsewhere) has Lebesgue integral 0 on [0, 1].

Real-world uses

• Computing expectations of random variables and integrals of rough signals in probability

Common mistake

Using the Lebesgue integral and expecting that improper conditional integrals converge.

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.