Lebesgueintegration

Lebesgue integration is the construction of the integral with respect to a measure, developed by Henri Lebesgue in the early 20th century. It generalizes the Riemann integral by focusing on the measure of sets where the function takes values. Given a measure space (X, Σ, μ) and a measurable function f: X → [−∞, ∞], the integral is defined first for nonnegative f as the supremum of the integrals of simple functions s ≤ f, where simple functions take only finitely many values. For a general f, write f = f+ − f− with f+, f− the positive and negative parts; f is integrable if ∫ f+ and ∫ f− are finite, in which case ∫ f = ∫ f+ − ∫ f−.

Lebesgue integration has several advantages over the Riemann approach. It handles a wider class of functions and interacts well with limits. For example, the characteristic function of the rationals on [0,1] equals 0 almost everywhere and is Lebesgue integrable with integral 0, even though it is not Riemann integrable.

Key results include the Monotone Convergence Theorem, which allows interchange of limit and integral for monotone sequences of nonnegative functions; the Dominated Convergence Theorem, which gives interchange under domination by an integrable function; and Fubini's/Tonelli's theorems for iterated integrals. Hölder's and Minkowski's inequalities underpin Lp spaces and many convergence phenomena in analysis.

Lebesgue integration underpins modern analysis. It leads to the Lp spaces, enables duality results, and is essential in probability theory, Fourier analysis, and partial differential equations. The integral is defined for measurable functions on a measure space, with linearity, monotonicity, and a robust notion of convergence.