Lebesgueintegral

Lebesgue integral is a construction in measure theory that defines the integral of a function with respect to a measure on a measure space. Let (X, A, μ) be a measure space. A function f: X → [−∞, ∞] is measurable if for every α ∈ R the set {x ∈ X : f(x) > α} belongs to A. For nonnegative measurable f, the integral ∫ f dμ is defined as the supremum of ∫ s dμ over all simple functions s ≤ f, where simple functions have the form s = ∑ α_k 1_{A_k} with A_k ∈ A and α_k ≥ 0. For general f, write f = f^+ − f^− with f^+ = max(f, 0) and f^− = max(−f, 0). If ∫ f^+ dμ and ∫ f^− dμ are not both infinite, then ∫ f dμ := ∫ f^+ dμ − ∫ f^− dμ; otherwise the integral is undefined (or infinite with sign).

Core properties include linearity on integrable functions and monotonicity: if 0 ≤ f ≤ g then ∫ f ≤ ∫ g. Major theorems such as the Monotone Convergence Theorem, the Dominated Convergence Theorem, and Fatou’s lemma provide conditions for exchanging limits and integrals. Fubini’s and Tonelli’s theorems give ways to integrate functions on product spaces.

The Lebesgue integral extends the Riemann integral and is central to modern analysis. It underpins L^p spaces, probability theory, and many applications by handling limits and unbounded or highly oscillatory functions more robustly than the Riemann approach.