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 ↓


Lebesgue Integration


Lebesgue integration is a powerful and essential concept in mathematics, especially in measure theory and analysis. This method of integration extends the concept of integrating functions, making it applicable to a wider class of functions and giving more robust results. Let us take a deeper look into this interesting topic and understand its components, importance, and applications.

Understanding the basics of integration

Before diving into Lebesgue integration, let's recall the foundations of integration. Integration, in general calculus, refers to the process of finding the accumulated value of a function. This can be the area under a curve in a graphical representation.

Riemann integration

Traditionally, integration is understood through Riemann integration. In this approach, the integral of a function over a defined interval is the limit of the sum of rectangular areas. These rectangles are constructed under the curve of the function being integrated. Mathematically, for a function f(x) defined on an interval [a, b], its Riemann integral is calculated as:

∫ f(x) dx = lim (Σ f(xᵢ) Δxᵢ) as n → ∞

where Δxᵢ is a subpartition of the interval [a, b], and xᵢ represents a point in each subinterval.

Limitations of Riemann integration

Although Riemann integration works well for functions that are continuous on a closed interval, it has limitations when dealing with more complex situations. These limitations arise with functions that are not finite, including those with discontinuities or that are defined in complex spaces.

Introduction to measurement theory

To overcome these challenges, Lebesgue developed a more generalized approach - Lebesgue integration, which is based on measure theory. Measure theory provides a systematic way of assigning size or measure to sets, which may be extremely irregular or fragmented. Can be done.

An important concept in measure theory is "measure", which, in intuitive terms, is an extension of the notion of length. Traditional definitions of length do not work if we consider more complicated sets, so we need to define a measure. A new method requires knowing how much space a set "occupies".

Basic concepts of Lebesgue integration

The main idea behind Lebesgue integration is to measure functions with respect to the measure of the set on which they are defined, rather than based on a partition of the domain of the function as in Riemann integration.

Consider a function f(x) defined with measure μ on a measure space X In Lebesgue integration, we focus on the set of x values for which the function outputs a particular value y, and we calculate the sum of these output values. , weighted by their measure. The Lebesgue integral is written as:

∫_Xfdμ

Formal definition

More formally, if f: X → [0, +∞] is a measurable function, then the Lebesgue integral of f on X is:

∫_X f dμ = sup{∫_X s dμ | 0 ≤ s ≤ f, s is simple}

A "simple function" is s that takes a limited number of values, making it easy to calculate and understand.

Visual example

Measure the space Integration focus: output values

The measurement space (rectangle) is not divided by domain, but rather classified by the output values of the function.

Difference between Riemann and Lebesgue integration

The main difference between Riemann and Lebesgue integration is their way of decomposing the space. Riemann divides it into sub-intervals of the domain, while Lebesgue focuses on the value ranges of the function and decomposes based on those values.

An intuitive motivation is to understand how the Riemann process integrates vertically (considering columns of function values) and Lebesgue does so horizontally (comparing slices of the range). This difference depends on the types of functions being integrated. affects the types of data each method can handle effectively.

Algebraic examples

Consider the Dirichlet function, which takes the value 1 for rational numbers and 0 for irrational numbers. This function is not Riemann integrable on any interval because it is not bounded by simple measurable set partitions. However, it takes the integer value 0 is Lebesgue integrable with, since the set of rational numbers has Lebesgue measure 0.

More information on simple tasks

A key concept within Lebesgue integration is "simple functions." These are functions that can take a limited number of different values. Simple functions are easier to integrate and can be used to accurately approximate more complicated functions. It is done.

For a simple function s(x), defined as s(x) = Σ aᵢ χ_{Aᵢ}(x) (where χ_{Aᵢ} is the characteristic function of the set Aᵢ), the Lebesgue integral is simply a function The weighted sum is:

∫_X s dμ = Σ aᵢ μ(Aᵢ)

Here, aᵢ are the distinct values that s takes, and μ(Aᵢ) are the measures of the corresponding sets Aᵢ.

How Lebesgue handles difficult integrals

Lebesgue integration excels at dealing with pathological cases. For instances of functions that cannot be managed via Riemann methods, Lebesgue provides a framework for solution. A classic example involves functions with discontinuities everywhere (such as the Cantor function), which is difficult if not impossible for Riemann, but can be managed with Lebesgue.

Cantor function example

The Cantor function is a strange beast, constant over the overwhelming majority of its domain, yet with a different overall measure behavior. Through Lebesgue integration, we can make sense of this through measured set analysis:

∫_{[0, 1]}f(x) dμ = 1

However the function remains constant over a large portion of its domain.

Advantages of Lebesgue integration

Lebesgue integration makes it possible to analyze functions that are not possible from the Riemann approach and offers the following advantages:

  • Consistency with limits: functions of corresponding sequences preserve integral limits.
  • Seamless handling of point-wise boundaries.
  • Adaptability to different measurement locations.

These improvements come from its fundamental reliance on measure theory, which enables it to handle more complex problems in analysis.

Conclusion

Lebesgue integration provides a comprehensive method for integrating a broad class of functions beyond the capabilities of Riemann integration. With its basis in measure theory, it handles discontinuities and pathological cases with ease, solving previously challenging problems into accessible functions. Its advantages in transformational and applied mathematics are undeniable, making it a cornerstone of analytical mathematics.


PHD → 2.4.3


U
username
0%
completed in PHD

← Prev (2.4.2)
Measures and Integrals
Next (2.4.4) →
Convergence Theorems

Comments 



Lebesgue Integration

https://www.buddymath.com/phd/2.4.3?bmal=en