Rigorous foundation for integration, probability, and functional analysis. Assigns 'size' to sets via σ-additive set functions. Lebesgue measure generalizes length/area/volume to a much broader class of sets; the Lebesgue integral handles…
measure-theory
Lebesgue measure
The unique translation-invariant complete measure on ℝⁿ assigning interval [a,b] measure b−a. Constructed via the Carathéodory extension…
Lebesgue integral
∫ f dμ defined via simple functions approximating f from below; agrees with Riemann where both are defined, but handles a strictly larger…
Dominated convergence theorem
If f_n → f pointwise a.e. and |f_n| ≤ g for some integrable g, then ∫ f_n → ∫ f. Lets us swap limit and integral without uniform…
Fubini theorem
For f integrable on a product space X × Y, ∫∫ f d(μ×ν) = ∫(∫ f dν) dμ = ∫(∫ f dμ) dν. Justifies iterated integration.
Lp space
L^p(μ) = {f : ∫|f|^p dμ < ∞} modulo a.e. equality, with norm ||f||_p = (∫|f|^p)^{1/p}. L² is a Hilbert space (quantum mechanics); L^∞ is…
Radon-Nikodym theorem
If ν ≪ μ (ν absolutely continuous w.r.t. μ), there exists a measurable density dν/dμ with ν(E) = ∫_E (dν/dμ) dμ. Underpins…
Borel set
An element of the σ-algebra ℬ(X) generated by the open sets of a topological space X. Tame by most measure-theoretic standards; strictly…
Measurable function
A map f : (X, ℳ) → (Y, 𝒩) with f^{-1}(A) ∈ ℳ for every A ∈ 𝒩. Foundational for Lebesgue integration.
Monotone convergence theorem
If 0 ≤ f_n ↑ f a.e., then ∫ f_n ↑ ∫ f. Lebesgue's first major interchange theorem.
Fatou's lemma
For non-negative measurable f_n: ∫ liminf f_n ≤ liminf ∫ f_n. Converts pointwise liminfs to measure-liminfs; used widely to construct…
Signed measure
A σ-additive set function μ : ℳ → ℝ (allowing at most one infinite value). Hahn decomposition ⇒ μ = μ^+ − μ^− uniquely; Jordan…
Complex measure
A σ-additive set function μ : ℳ → ℂ; equivalently μ = μ_r + iμ_i with μ_r, μ_i finite signed measures.
Product measure
On (X×Y, ℳ⊗𝒩), the unique σ-finite measure μ⊗ν with (μ⊗ν)(A×B) = μ(A) ν(B). Underlies Fubini/Tonelli.
Haar measure
On any locally-compact Hausdorff topological group G, there exists a left-invariant Radon measure, unique up to positive scalar. …
Radon measure
A Borel measure on a locally-compact Hausdorff space that is finite on compacts, inner-regular on opens, and outer-regular on all Borels. …
Weak convergence of measures
Probability measures μ_n → μ weakly (or in distribution) iff ∫ f dμ_n → ∫ f dμ for every bounded continuous f. Characterised by…
Carathéodory extension theorem
Pre-measure on ring extends uniquely to measure on generated σ-algebra (σ-finite case). Construction of Lebesgue, product measures.
Lebesgue decomposition theorem
σ-finite μ = μ_{ac} + μ_s w.r.t. ν, with μ_{ac} << ν, μ_s ⊥ ν. Absolutely continuous / singular parts.
Fubini-Tonelli theorem
σ-finite product measure: ∫∫ f d(μ×ν) = ∫(∫ f dν) dμ. Tonelli for ≥ 0 functions; Fubini for integrable.
Monotone convergence theorem
0 ≤ f_n ↑ f a.e. ⟹ ∫ f_n ↑ ∫ f. Extends linearity to positive measurable limits; foundational.
Dominated convergence theorem
f_n → f a.e., |f_n| ≤ g integrable ⟹ ∫ f_n → ∫ f and ∫ |f_n − f| → 0. Lebesgue's theorem; interchange limit/integral.
Lᵖ duality
1 ≤ p < ∞, 1/p + 1/q = 1: (Lᵖ)* ≅ Lᵠ via ⟨f,g⟩ = ∫ fg dμ. L^∞ dual larger (ba = finitely-additive signed measures).
Hölder / Minkowski inequalities
||fg||_1 ≤ ||f||_p ||g||_q (Hölder); ||f+g||_p ≤ ||f||_p + ||g||_p (Minkowski). Make Lᵖ a normed space (p ≥ 1).
Riesz representation (C_c, C_0)
Positive linear functional on C_c(X) (LCH) = integration against unique Radon measure. (C_0(X))* = finite signed regular Borel measures.
Egorov's theorem
On a finite-measure set, pointwise a.e. convergence is almost uniform convergence: for every δ > 0 there is a subset F of E with μ(E\F) < δ…
Lusin's theorem
Every measurable function on [a, b] is continuous after restriction to a closed set whose complement has arbitrarily small measure.…
Vitali convergence theorem
Strengthens the dominated-convergence theorem: the dominating hypothesis can be relaxed to uniform integrability. With convergence in…
Hahn decomposition theorem
Every finite signed measure ν on (X, Σ) admits a partition of X into a positive set P (ν(A∩P) ≥ 0 ∀A) and a negative set N. The…
Jordan decomposition of a signed measure
Given a Hahn decomposition X = P ⊔ N, set ν⁺(A) = ν(A∩P), ν⁻(A) = −ν(A∩N). Then ν⁺, ν⁻ are mutually singular finite positive measures with…
Riesz-Fischer theorem (L^p completeness)
L^p(μ) is a Banach space for all 1 ≤ p ≤ ∞. Equivalently: every absolutely summable series Σ‖f_n‖_p < ∞ converges in L^p. On L²([0,2π])…
Vitali covering theorem
Every Vitali cover of E ⊆ ℝⁿ (a family of closed balls containing arbitrarily small balls around each x ∈ E) has a countable disjoint…
Lebesgue differentiation theorem
For every locally integrable f on ℝⁿ, the spherical averages of f converge to f(x) at almost every point. Corollary of the Vitali covering…