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 ↓


Sigma Algebras


Introduction to sigma algebra

Sigma algebras are a fundamental concept in measure theory, which is an integral part of real analysis and probability theory. They provide a rigorous way to handle collections of "measurable" sets. Instead of dealing with all possible subgroups of a given set, sigma algebras focus on subgroups that behave well with respect to a measure, such as length, area, probability, etc.

Basics and definitions

The sigma algebra (or σ-algebra) on a set X is a collection Σ of subsets of X that satisfy the following properties:

  1. Non-nullity: X is itself in Σ.
  2. Closure under complement: If a set A is in Σ, then its complement Ac = X A is also in Σ. This means that if we have a subset in our collection, then we must also have everything outside that subset.
  3. Closure under countable unions: If we have a countable collection {A1, A2, A3, ...} of sets that all belong to Σ, then the union of all these sets A = A1 ∪ A2 ∪ A3 ∪ ... is also in Σ.

These properties ensure that sigma algebras are well behaved and suitable for measuring "size" in a consistent way. Let's explore some examples to illustrate this idea.

Visual example of sigma algebra

Consider a simplicial set X = {1, 2, 3}. A sigma algebra on X would be a collection of subsets of X. For example, such a sigma algebra could be:

{ {}, {1, 2, 3}, {1, 2}, {3} }

This selection of sets satisfies all the conditions necessary for a sigma algebra:

  • Nonemptiness: {1, 2, 3} is in the collection.
  • Closed complement: For every subset of a collection, there exists a closed complement of the set.
  • Closed under countable unions: any union of sets from a collection is also a subset of the collection or of X itself.

Mathematical example

An example of a more complicated sigma algebra concerns the real line ℝ. The Borel sigma algebra on ℝ is generated by the open intervals:

B(ℝ) = σ({ (a, b) : a, b ∈ ℝ })

In this case, the Borel sigma algebra consists of all kinds of subsets of ℝ that can be constructed using open intervals, closed intervals, and the many operations of taking complements and unions. Borel sets are useful in many areas of analysis, such as defining Lebesgue measurable functions.

Properties of the sigma algebra

Let's look at more properties and characteristics of the sigma algebra:

Closed down intersection

While sigma algebras are known to be closed under countable unions, they are also closed under countable intersections. This is because the intersection of countably many sets can be expressed using complements and unions. For example, if A1, A2, ... are in Σ, then:

A1 ∩ A2 ∩ ... = (A1C ∪ A2C ∪ ... )C

Minimal and maximum sigma algebra

Every set X has at least two sigma algebras. The trivial sigma algebra contains only the empty set and the set itself: { {}, X }. This is the smallest possible sigma algebra over any set.

The largest sigma algebra is the power set of X, denoted by P(X), which contains every subset of X. Although this is rarely of interest for measurement purposes, it serves as an upper bound.

Generated sigma algebra

We often "generate" a sigma algebra from a particular collection of subsets. The sigma algebra σ(G) generated by G is the smallest sigma algebra containing all the sets in G. It contains the subsets that can be formed by taking the complements and countable unions of sets in G.

Working with sigma algebra

Consider applications and their interpretations in real-world contexts such as probability.

Sigma algebra in probability

In probability theory, the sigma algebra represents the collection of all events that can be measured. For example, if you roll a dice, the sample space is {1, 2, 3, 4, 5, 6}. The sigma algebra can define which combinations of outcomes (events) you like, such as "getting an even number" or "rolling a number greater than 3".

More formally, in the probability space (Ω, Σ, P):

  • Ω is the sample space.
  • Σ is the sigma algebra of subsets of Ω (events).
  • P is a probability measure that assigns probabilities to events in Σ.

Applications in measure theory

In measure theory, the sigma algebra provides a way to handle "measurable" sets, which are needed to define integrals and measures. For example, the Lebesgue measure uses the sigma algebra to define which subsets of the real numbers are measurable, allowing areas and volumes to be calculated precisely.

Construction of the sigma algebra: an example

Let's consider a simple example with X = {a, b, c}:

  1. Start with a basic collection: {{}, {a, b, c}}.
  2. Add singletons and their complements to ensure closure under complement: {{}, {a, b, c}, {a}, {b, c}}
  3. Add the pair-wise unions needed to reach the closure under countable unions: {{}, {a, b, c}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}}

This is now a sigma algebra, satisfying all the required properties.

Why is sigma algebra important

The beauty of sigma algebras lies in their structure, which provides a consistent framework for analyzing various mathematical and real-world phenomena. They are important in the following:

  • Defining measurable functions and sets.
  • Formulating probability spaces and expectations.
  • To construct integration theories such as Lebesgue integration.
  • Allowing advanced proofs and applications in areas such as functional analysis.

Conclusion

Sigma algebras embody the principles of selection and combination in set theory, which serve as the cornerstone for more advanced analytical constructions. Their well-defined rules and properties allow mathematicians to generalize beyond simple set operations, enabling deeper explorations into probability, analysis, and beyond.


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Sigma Algebras

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