Why is the Lebesgue integral preferred in advanced analysis?

It integrates more functions and, crucially, allows limits and integrals to be interchanged under mild conditions, which makes function spaces complete and is indispensable in probability and Fourier analysis.

Do the two integrals ever disagree?

For functions that are Riemann integrable on a bounded interval the two integrals give the same value; the Lebesgue integral simply applies to a larger class of functions where the Riemann integral is undefined.

Riemann and Lebesgue Integration

Integration assigns a rigorous value to the area under a curve; the Riemann integral does so by partitioning the domain, while the Lebesgue integral partitions the range and integrates a far wider class of functions.

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Definition

The Riemann integral is the common limit of upper and lower sums over finer partitions of the domain. The Lebesgue integral, defined by approximating functions with simple functions measured by a measure, extends integration to a broader class and behaves well under limits.

Scope

This topic covers the construction of the Riemann integral via upper and lower sums, the criterion for Riemann integrability, the fundamental theorem of calculus, the limitations of Riemann integration under limits, and the Lebesgue integral built on measure with its monotone, Fatou, and dominated convergence theorems.

Core questions

Exactly which functions are Riemann integrable, and what characterizes them?
How does the fundamental theorem of calculus link integration and differentiation?
Why does the Riemann integral fail to commute with many limits?
How does the Lebesgue integral overcome these limitations?

Key theories

Lebesgue criterion for Riemann integrability: A bounded function on a closed interval is Riemann integrable if and only if its set of discontinuities has measure zero, precisely delimiting the reach of the Riemann theory.
Fundamental theorem of calculus: Differentiation and integration are inverse operations: the integral of a derivative recovers the function, and the derivative of an integral recovers the integrand, linking the two central operations of analysis.
Monotone and dominated convergence: For the Lebesgue integral, monotone increasing sequences and dominated sequences of functions permit the interchange of limit and integral, the convergence power the Riemann integral lacks.

Clinical relevance

Integration theory underlies the computation of areas, probabilities, expectations, and accumulated quantities throughout science. The Lebesgue integral's robust limit behavior is essential for probability theory, Fourier analysis, the completeness of function spaces, and the rigorous treatment of solutions to differential equations.

History

Riemann gave the first rigorous definition of the integral in 1854. Its inability to handle many limits and discontinuous functions motivated Lebesgue's 1902 measure-based integral, which became the standard tool of modern analysis and probability.

Key figures

Bernhard Riemann
Henri Lebesgue
Emile Borel

Seminal works

rudin1976
stein2005real

Frequently asked questions

Why is the Lebesgue integral preferred in advanced analysis?: It integrates more functions and, crucially, allows limits and integrals to be interchanged under mild conditions, which makes function spaces complete and is indispensable in probability and Fourier analysis.
Do the two integrals ever disagree?: For functions that are Riemann integrable on a bounded interval the two integrals give the same value; the Lebesgue integral simply applies to a larger class of functions where the Riemann integral is undefined.