Lebesgue integration constructs an integral from the measure of level sets or from approximation by simple functions. It is linear on integrable functions and is stable under the principal convergence theorems of measure theory.
If and nonnegative measurable functions satisfy for all , then their Lebesgue integrals obeyThe quantile derivative identity and monotone substitution inequality prove the one-dimensional case; Tonelli theorem extends it by induction. It implies the Brunn–Minkowski inequality.
If , then for every there is such that whenever the measure of is below . In particular, integrals over a sequence of shrinking measurable sets tend to zero.
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Lebesgue integration is a mathematical concept that extends the notion of integration beyond the traditional Riemann integral. It is a fundamental tool in real analysis and measure theory, named after the French mathematician Henri Léon Lebesgue. ### Key Concepts of Lebesgue Integration: 1. **Measure Theory**: At the core of Lebesgue integration is measure theory, which provides a rigorous way to define "size" or "measure" of sets.