Alphabeta MathArticle items · 01 — Additivity of the nonnegative Lebesgue integral
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Monotone convergence for the integral
- Every nonnegative measurable function is the increasing limit of simple measurable functions
- Closure properties of measurable functions used by the integral
- The simple integral is monotone, homogeneous, and additive
- The nonnegative integral agrees with the simple integral on simple functions
- Every complete ordered field is Archimedean
- Inverses of positives are positive, and reciprocation reverses order
- Complete ordered field (least-upper-bound property)
- Ordered field
- Field
- The set A^B of functions B → A between finite sets is finite, with | A^B| = | A|^| B|
- Exponentiation of natural numbers, mⁿ, and its agreement with the integer power in ℝ
- The cardinality | A| of a finite set
- Cantor's theorem: A ≺ P(A)
- The pigeonhole principle on ℕ
- Trichotomy of the order on ℕ
- Order on the natural numbers
- Injection, surjection, bijection
- Equinumerous sets, A ≈ B and A ⪯ B
- The natural numbers ℕ (von Neumann)
- Borel measurable and Lebesgue measurable functions on ℝⁿ
- A continuous map has Borel preimages of Borel sets
- Compactly supported smooth vector fields are complete
- Smooth sections, local sections, and support
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- The exponential addition formula exp(x+y)=exp(x)exp(y)
- The real exponential function and the number e by a power series
- Squares of nonzero elements are positive
- The integral exponential E is the published exponential function
- The integral exponential E:ℝ→(0,∞) as the inverse of L
- The natural logarithm as the inverse of the exponential function
- The integral logarithm satisfies L'(x)=1/x for x>0 and L(1)=0
- A function differentiable at c is continuous at c
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Quarter-turn values and shifts by pi/2 and pi
- Pi as twice the smallest positive zero of cosine
- Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose
- A real normal endomorphism is orthogonally block-diagonalisable with 1x1 real blocks and 2x2 rotation-scaling blocks
- The addition formulas for sine and cosine
- The derivatives of sine and cosine are cosine and minus sine
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- A function continuous on an interval I whose derivative vanishes at every interior point of I is constant on I; consequently two such functions with the same derivative differ by a constant
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- Finite rectangular matrices over a commutative ring, their entries, rows and columns
- The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Factor theorem over a commutative ring
- For every field F, F[x] is a unique factorisation domain
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Choice function
- Real random variables on finite probability spaces and their finite distributions
- Pairwise and mutual independence of finite-valued random variables
- Finite sums and finite products, by recursion
- A subset of a finite set is finite, with | B| ≤ | A|, and equality holds if and only if B = A
- On ℕ the order is membership: m < n ⟺ m ∈ n
- If a ∈ A and b ∈ B then (a,b) ∈ P(P(A ∪ B))
- The Axiom Schema of Separation: for each formula φ, ∀ p̄ ∀ x ∃ y ∀ z (z ∈ y ↔ (z ∈ x ∧ φ(z,p̄)))
- The Kuratowski ordered pair (a,b) := {{a},{a,b}}
- The power set P(x) = { z : z ⊆ x }
- The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ
- Integer powers aᵐ
- Continuously differentiable maps, local inverses, and local diffeomorphisms
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Through each point there is a unique maximal integral curve
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- The Cauchy-sequence reals have the least-upper-bound property
- The complex exponential by its power series
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Unless stated otherwise, graph means finite, simple and undirected; orders, sizes and empty-set conventions are fixed here
- Subgraphs, induced subgraphs and spanning subgraphs
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
- Every walk between two vertices contains a path between the same endpoints
- A convex subset of ℝᵐ contains every line segment between two of its points
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Finite, countably infinite, countable, uncountable
- The well-ordering principle
- A nonempty set is at most countable iff it is a surjective image of ℕ
- The finite product of finite probability spaces
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
CorollaryLibrary prerequisite
Additivity of the nonnegative Lebesgue integral
CorollaryLibrary prerequisite
For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
CorollaryLibrary prerequisite
lvertP(A)| = 2^| A| for finite A
CorollaryLibrary prerequisite
Continuous functions on Euclidean spaces are Borel measurable
CorollaryLibrary prerequisite
Every smooth vector field on a compact manifold is complete
CorollaryLibrary prerequisite
The exponential is positive and satisfies exp(-x)=1/exp(x)
CorollaryLibrary prerequisite
The integral logarithm L is the published natural logarithm
CorollaryLibrary prerequisite
The integral logarithm is continuous and strictly increasing on (0,∞)
CorollaryLibrary prerequisite
Pi is the first positive zero of sine
CorollaryLibrary prerequisite
Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
CorollaryLibrary prerequisite
Parity and the Pythagorean identity for sine and cosine
DefinitionLibrary prerequisite
Absolute value in an ordered field
DefinitionLibrary prerequisite
The adjacency matrix of a finite simple graph
DefinitionLibrary prerequisite
Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T)
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
DefinitionLibrary prerequisite
The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|
DefinitionLibrary prerequisite
The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }
DefinitionLibrary prerequisite
Cᵏ maps and multi-index derivative notation in Euclidean space
DefinitionLibrary prerequisite
Cᵏ Euclidean maps and diffeomorphisms
DefinitionLibrary prerequisite
Complete vector fields
DefinitionLibrary prerequisite
Real and imaginary parts, complex conjugation, and modulus
DefinitionLibrary prerequisite
Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
DefinitionLibrary prerequisite
Connected components, quasicomponents, and totally disconnected spaces
DefinitionLibrary prerequisite
Connected graphs and connected components defined by the existence of vertex paths
DefinitionLibrary prerequisite
Convex and strictly convex functions on Euclidean convex sets
DefinitionLibrary prerequisite
The Axiom of Countable Choice (AC_ω)
DefinitionLibrary prerequisite