Lebesgue integral
Objective. Define and calculate integrals of measurable functions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The Lebesgue integral extends the idea of area under a curve by first integrating simple functions, then taking suprema over simple functions below a nonnegative measurable f, and finally splitting general f into positive and negative parts. For a simple function s=Σcᵢ𝟙_{Aᵢ}, define ∫s dμ = Σcᵢμ(Aᵢ); then ∫f dμ = sup{∫s dμ : 0≤s≤f, s simple}. This construction handles functions too wild for Riemann integration, such as the Dirichlet function (1 on rationals, 0 on irrationals), which has Lebesgue integral 0 since the rationals have measure zero. General f is integrable when ∫f⁺dμ and ∫f⁻dμ are both finite, giving ∫f dμ = ∫f⁺dμ − ∫f⁻dμ.
Worked example
Compute the Lebesgue integral of f = 3·𝟙_{[0,2]} + 5·𝟙_{(2,4]} over [0,4].
- 1Write f as a simple function with values 3 and 5
- 2∫f dμ = 3·μ([0,2]) + 5·μ((2,4])
- 3μ([0,2])=2 and μ((2,4])=2
- 4∫f dμ = 3(2)+5(2) = 16
Answer: 16
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Practice
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Start practisingCommon mistake. Students confuse Lebesgue integrability with boundedness, but an unbounded function like 1/√x on (0,1) can still be Lebesgue integrable since ∫₀¹x^{-1/2}dx=2 is finite despite the function blowing up near 0.
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