PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis ↓


Measure Theory


Measure theory is a branch of mathematical analysis that is concerned with the theoretical basis of integration rather than focusing on the techniques of integration. It is helpful in many fields such as probability, statistics, and functional analysis. Developed for problems where the classical definitions of volume and area do not apply, measure theory has wide applications in modern analysis.

Basic concepts

At the most basic level, measure theory starts with the concept of a measure space. A measure space is a triplet (X, mathcal{A}, mu) where:

  • X is a set.
  • mathcal{A} is a σ-algebra over X.
  • mu: mathcal{A} to [0, infty] is a measure.

The goal of the measure is to provide a systematic way of assigning a number that can represent the size or volume to every subset of X in mathcal{A}. This number is non-negative and can be infinite.

σ-algebra

A σ-algebra mathcal{A} on a set X is a collection of subsets of X satisfying three properties:

  1. X itself is in mathcal{A}.
  2. mathcal{A} is closed under complementation: if a set A is in mathcal{A}, then X setminus A is also in mathcal{A}.
  3. mathcal{A} is closed under countable unions: if A_1, A_2, ldots are in mathcal{A}, then bigcup_{i=1}^{infty} A_i is also in mathcal{A}.

Measures

The measure mu is a function from a σ-algebra mathcal{A} to the extended real numbers [0, infty] that satisfies:

  1. Nonnegativity: mu(A) geq 0 for every A in mathcal{A}.
  2. Zero Empty Set: mu(emptyset) = 0.
  3. Countable additivity/σ-additivity: If A_1, A_2, ldots are in mathcal{A} and are disjoint, then mu(bigcup_{i=1}^infty A_i) = sum_{i=1}^infty mu(A_i).

Examples of measures

Let us look at some examples to clarify these definitions:

1. Counting measure

For any set X, we can define a counting measure mu such that for every subset A of X, mu(A) is the number of elements in A. This measure is easy to understand: it counts the elements directly.

2. Lebesgue measure

On the real line mathbb{R}, the Lebesgue measure is the most important measure, which extends the notion of length from intervals to a wider collection of sets. The Lebesgue measure of the interval [a, b] is b - a.

A B

It denotes the interval [a, b] on the real number line that has Lebesgue measure of length b - a.

Integration via measure theory

One of the most powerful tools offered by measure theory is a very general principle of integration. It is called the Lebesgue integral. Unlike the Riemann integral, which is limited to functions on a finite interval with some discontinuity, the Lebesgue integral can handle much more complicated cases.

Riemann versus Lebesgue integration

The primary difference between the two is the way in which the integration adds function values. The Riemann integral divides the domain of the function, while the Lebesgue integral divides the range of the function.

To express this notion, consider a function f defined on an interval [a, b]. The Riemann sum approximates its integral by raising a partition of the interval:

Riemann integral: int_a^b f(x) , dx = lim_{n to infty} sum_{i=1}^n f(x_i^*) Delta x_i,

while the Lebesgue integral splits the codomain using level sets.

The power of the Lebesgue integral arises from its ability to integrate a wide class of functions. Consider a function that takes a value on a set of variables of measure zero. The Lebesgue integral can handle this easily because it emphasizes the measure of the sets where the functions take particular values.

Application

1. Probability theory

In probability, the measure theoretic framework allows us to define probability measures, which generalize the traditional Kolmogorov axioms of probability. Mathematically, a probability measure is a measure in which the entire space has measure 1. Events then become measurable sets, and probability functions become measures.

2. Functional analysis

In this area, measure theory helps define spaces such as L^p space, where functions are studied under the lens of limits and transformations, defining convergence such as almost-every space and convergence in measure.

Conclusion

Measure theory is a robust area of mathematics that allows for careful investigation of size and integration. By understanding measure, σ-algebra and the Lebesgue integral, we provide a foundation for rigorously handling fields, probability and functions. Its deep reach in both pure and applied contexts underscores its importance in the mathematical landscape.


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Measure Theory

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