Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis → Integration ↓


Riemann and Lebesgue Integration


Integration is a fundamental concept in mathematics, especially in the field of real analysis. It is a process that allows us to find area, volume, central point, and many useful things. In real analysis, two important types of integration are Riemann integration and Lebesgue integration. Each has its own specific approach and is used depending on the nature of the function and the context in which it is applied.

Riemann integration

Riemann integration, named after German mathematician Bernhard Riemann, is one of the earliest and simplest methods of integration. It works well for piecewise continuous functions and provides a way to perform integration by approximating the area under a curve. The idea behind Riemann integration is to approximate the area under a curve by dividing it into simpler shapes whose areas we can calculate.

Definition of Riemann integration

The Riemann integral of a function is defined based on the concept of partition. The partition P of the interval [a, b] is a finite sequence of numbers such that:

a = x 0 < x 1 < x 2 < ... < x n = b

Using this partition, we draw a rectangle between x values. For each subinterval [x i-1, x i], we can choose any point c i and the height of the rectangle is f(c i), where f is the function being integrated.

The Riemann sum for a function f over a partition is defined as:

S(P) = Σ f(c i)(x i - x i-1)

The limit of this sum when the width of the subintervals is zero gives us the Riemann integral:

∫ a b f(x) dx = lim |P|→0 Σ f(c i)(x i - x i-1)

Here |P| is the net or width of the largest subinterval in the partition.

Example of Riemann integration

Consider a simple example: the integration of f(x) = x^2 over the interval [0, 1]. To find the Riemann integral, we divide the interval [0,1] into n equal parts, with each part having width Δx = 1/n.

1 f(x) = x²

The height on each subinterval can be chosen as (i/n)^2, where i = 1, 2, ..., n. The Riemann sum will be:

S(P) = Σ (i/n)^2 * 1/n = 1/n^3 Σ i²

This approximates the integral as n → ∞, giving:

∫ 0 1 x² dx = lim n→∞ 1/n³ Σ i² = 1/3

Lebesgue integration

Lebesgue integration, named after Henri Lebesgue, extends the ideas of Riemann integration to a much broader class of functions. It is particularly useful for handling more complicated functions, including those that are not necessarily continuous.

Definition of Lebesgue integration

Instead of dividing the domain (as is the case with the Riemann integral), Lebesgue integration divides the range of the function. This involves measuring the size of regions in the domain where the function takes certain values.

For a non-negative function f defined on a measure space, the Lebesgue integral is defined by examining the distribution function F(y) = m({x : f(x) ≥ y}), where m is the measure (usually the Lebesgue measure).

The formal Lebesgue integral is expressed as:

∫ f(x) dm = ∫ y dF(y)

where the right-hand side integral is the Riemann–Stieltjes integral.

Example of Lebesgue integration

Consider the characteristic function of the rational numbers on the interval [0, 1], which is a function that is 1 on the rational numbers within [0,1] and 0 on the irrational numbers. In Riemann integration, such a function would be nearly impossible to integrate due to its discontinuity at every rational point. However, using Lebesgue integration:

The rational numbers have Lebesgue measure zero, and so the integral evaluates to

∫ 0 1 χ Q (x) dx = 0

This is a simple example demonstrating the power of Lebesgue's method. It naturally handles infinite discontinuities because it focuses on the measure of a set of points rather than their discrete locations.

Comparison of Riemann and Lebesgue integrals

The purpose of both types of integration is to find the area under the curve, but they do so in very different ways.

  • Riemann integration splits the domain of the function, while Lebesgue integration splits the range.
  • Riemann integration is limited to functions that are finite and have a finite number of discontinuities on a closed interval. Lebesgue integration can handle more complicated functions, including those with an infinite number of discontinuities.
  • Lebesgue integration is more adaptable to limit operations, such as taking the limit of a sequence of functions.
  • If a function is Riemann integrable, it will also be Lebesgue integrable, and both integrals will have the same value. However, the opposite is not true.

Despite their differences, both play essential roles in integral analysis. Riemann integration is generally more intuitive and useful for elementary calculus, while Lebesgue integration is powerful in more advanced analysis and applications, especially in areas such as probability and functional analysis.

Visual example of Lebesgue integration

X Y y = 0.25 y = 0.5 y = 0.75

This visual example shows how Lebesgue integration proceeds toward integration by determining equal regions horizontally, known as integrated slices, of the range of the function, which parallels the construction of the measure vertically.

Both integration methods are important tools in mathematics, with Riemann's method serving as an entry point for the study of calculus, and Lebesgue's method providing greater flexibility and power for advanced applications.


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