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 ↓


Fubini's Theorem


Fubini's theorem is a fundamental result in measure theory that provides the ability to change the order of integration in a multiple integral. This theorem is particularly useful in mathematics when dealing with integrals over multidimensional spaces. This theorem is named after the Italian mathematician Guido Fubini.

Introduction to measurement theory

Before diving into Fubini's theorem, it is important to understand some basics about measure theory. Measure theory extends the concept of integration beyond the simple summation of areas under curves and allows for a more rigorous and generalized notion of integration.

In measure theory, we deal with measurements, which can be thought of as generalized notions of length, area, or volume. Measurements help us measure the size or volume of spaces that cannot be easily understood using elementary geometry.

Understanding Fubini's theorem

We start with the essential statement of Fubini's theorem. Consider two σ-finite measure spaces ( (X, mathcal{A}, mu) ) and ( (Y, mathcal{B}, nu) ). Fubini's theorem states that if ( f: X times Y rightarrow mathbb{R} ) is a measurable function that is absolutely integrable with respect to the product measure, then we can compute the double integral of ( f ) on the product space by iterated integration:

(int_{X times Y} f(x, y) , d(mu times nu)(x, y) = int_X left(int_Y f(x, y) , dnu(y)right) , dmu(x) = int_Y left(int_X f(x, y) , dmu(x)right) , dnu(y))

The equation essentially tells us that the integral of a function over a product space can be reduced to an integral over one variable and then an integral over the other variable.

Prerequisites

To fully understand Fubini's theorem it is necessary to understand the following concepts:

  • Measure space: A measure space is a triplet ((X, mathcal{A}, mu)) consisting of a set (X), a σ-algebra (mathcal{A}) on (X), and a measure (mu) that assigns to every set in (mathcal{A}) a non-negative extended real number.
  • σ-finite measure: A measure (mu) is called σ-finite if the space (X) can be decomposed into a countable union of measurable sets, each of which has a finite measure.
  • Product measure: Product measure (mu times nu) is defined on a product σ-algebra (mathcal{A} times mathcal{B}) such that for any rectangular region (A times B) where (A in mathcal{A}) and (B in mathcal{B}), ((mu times nu)(A times B) = mu(A)nu(B)).

Visual example of integration across product locations

X Y A × B

Consider a rectangular region (A times B) in the plane where (A) lies in the horizontal axis space (X) and (B) lies in the vertical axis space (Y). Fubini’s theorem helps us perform integration over this 2D region, breaking it into integration first along one axis and then along the other.

Example: Calculating double integrals

For a more concrete example, consider a function ( f(x, y) = e^{-(x^2 + y^2)} ). We want to integrate this function over all of ( mathbb{R}^2 ).

According to Fubini's theorem, we can first integrate with respect to (y) keeping (x) fixed, and then with respect to (x):

(int_{-infty}^{infty} left(int_{-infty}^{infty} e^{-(x^2 + y^2)} , dy right) dx)

Alternatively, we can integrate first with respect to (x) and then with respect to (y):

(int_{-infty}^{infty} left(int_{-infty}^{infty} e^{-(x^2 + y^2)} , dx right) dy)

Regardless of the order of integration, the result will be ( pi ). This commutativity is the power provided by Fubini's theorem, which is especially important for complex or computationally intensive functions.

Applications in probability and statistics

Fubini's theorem has many applications in probability theory and statistics, often used to simplify integrals in the expected values and variances of random variables defined over joint distributions.

Suppose we have two continuous random variables (X) and (Y) with joint probability density function (f_{XY}(x, y)). The expected value of a function (g(X, Y)) is given by:

(mathbb{E}[g(X, Y)] = int_{-infty}^{infty} int_{-infty}^{infty} g(x, y) f_{XY}(x, y) , dy , dx)

According to Fubini's theorem, we can change the order of integration if it simplifies the calculations:

(mathbb{E}[g(X, Y)] = int_{-infty}^{infty} int_{-infty}^{infty} g(x, y) f_{XY}(x, y) , dx , dy)

The case with a product of functions

An interesting scenario arises when the function (f(x, y)) can be expressed as the product of two different functions of (x) and (y), i.e., (f(x, y) = h(x)k(y)). The Fubini theorem states:

(int_{X times Y} h(x)k(y) , d(mu times nu)(x, y) = left(int_X h(x) , dmu(x)right)left(int_Y k(y) , dnu(y))

The product measure of such functions separates clearly, which demonstrates the independence of variables in integrals.

Properties and conditions

Fubini’s theorem applies under specific conditions that need to be satisfied; primarily, the function (f(x, y)) must be measurable and absolutely integrable on (X times Y). Measurability ensures that the function respects the structure induced by the σ-algebra, and absolutely integrability assures the finiteness of the integral.

Full integrability is much more important than mere integrability. It ensures:

  • ( int_{X times Y} |f(x, y)| , d(mu times nu)(x, y) < infty )

This condition prevents changing the order of integration in cases where the integral exists, but making the change may lead to different sums.

Conclusion

Fubini's theorem is a powerful tool in analysis, particularly within measure theory, which facilitates a simpler approach to evaluating complex multidimensional integrals. Whether used in pure mathematics or in applied disciplines such as statistics and physics, this theorem proves crucial in facilitating seemingly impossible integrations. The ability to change the order of integration without changing the result dramatically simplifies the analytical and computational complexity associated with dealing with multidimensional datasets.


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