Riemann Integral Formal Construction and Definite Integrals

Practical examples illustrate integrability criteria: The function f(x) = x is continuous and integrable. The greatest integer function [x] is discontinuous at countably many integers but remains integrable. However, the Dirichlet function (1 at rationals, 0 at irrationals) is discontinuous everywhere (uncountable discontinuities) and thus not integrable. These examples demonstrate how the nature and countability of discontinuities determine integrability.

A bounded function f on a closed interval [a,b] is Riemann integrable if and only if the set of points where f is discontinuous is countable. If a bounded function has a finite number of discontinuity points on a closed interval, then it is Riemann integrable. This is a key criterion for Riemann integrability. The set of discontinuity points must be countable for the function to be Riemann integrable.

A function f is integrable on a set X = ∪X_n (disjoint union) if and only if ∑∫|f|_Xn converges. A function is integrable if and only if |f| is integrable. A function is integrable if and only if there exists an integrable g such that |f(x)| ≤ g(x) almost everywhere. Three main methods are used to check integrability: direct comparison with known integrable functions, the series criterion for functions on disjoint sets, and the comparison test using a dominating function.

A measurable function f is integrable if and only if its absolute value |f| is integrable, since f is integrable iff its positive and negative parts are integrable, and |f| = f^+ + f^-. The dominated convergence theorem states that if f is measurable and |f| ≤ g almost everywhere for some integrable function g, then f is integrable. A special case is that bounded measurable functions on sets of finite measure are integrable, as the constant bound serves as an integrable majorant.

Three equivalent criteria for Riemann integrability exist: (1) The Darboux criterion: a function is integrable if and only if it is bounded and the upper Darboux integral equals the lower Darboux integral. (2) The Riemann criterion: a function is integrable if and only if for any ε > 0, there exists a partition such that the difference between upper and lower Darboux sums is less than ε. (3) The oscillation criterion: a function is integrable if and only if the sum of oscillations on each subinterval, multiplied by subinterval lengths, approaches zero as mesh approaches zero. The oscillation of a function on a set E is the supremum of |f(x₁) - f(x₂)| for all x₁, x₂ in E. For continuous functions on closed intervals, this equals the difference between the maximum and minimum values of f on E. The Riemann criterion is often more practical as it requires finding at least one partition rather than checking all partitions with small mesh.

An improper integral occurs when the interval of integration is unbounded (extending to infinity) or the function has an infinite discontinuity; to evaluate it, replace the infinite limit with a variable (such as T), compute the definite integral normally, and then take the limit as the variable approaches infinity; if the resulting limit is a finite number, the integral is convergent, but if it evaluates to positive or negative infinity, the integral is divergent.

A proper integral has numerator power less than denominator power (e.g., ∫x^2/x^3 dx). An improper integral has numerator power greater than or equal to denominator power (e.g., ∫x^3/x^2 dx). Only proper integrals can be integrated directly using the power rule. For improper integrals, multiply numerator and denominator by the denominator to convert to proper form. For example, ∫x^3/x^2 dx = ∫x^3·x^2/(x^2·x^2) dx = ∫x^5/x^4 dx = ∫x dx = x^2/2 + C.

An improper integral is defined when either the function f(x) becomes infinite at some point within the closed interval [a, b], or when the limits of integration are infinite (a = ∞, b = ∞, or both). There are two types: Type 1 (First Kind) occurs when the limits of integration are infinite but the function remains bounded, and Type 2 (Second Kind) occurs when the limits are finite but the function becomes infinite at some point within the interval.

Improper integrals are calculated by introducing a limit variable in place of an infinite bound or a discontinuity, performing standard integration with antiderivatives, and then evaluating the limit; if the resulting value is finite, the integral converges, otherwise it diverges.

Improper integrals are integrals where either the limits of integration are infinite or the function becomes infinite at some point within the interval. Unlike proper integrals, which are always convergent and yield finite values, improper integrals can be convergent (yielding a finite value), divergent (yielding infinity or negative infinity), or oscillatory (not settling to a single value). For an integral to be improper, it must violate at least one of the proper integral conditions: having finite limits and a bounded function throughout the interval.

Lebesgue integration, developed by French mathematician Henri Lebesgue, provides a more powerful alternative to Riemann integration by measuring the size of sets in the range (y-axis) rather than partitioning the domain (x-axis); this approach enables integration of highly discontinuous functions like the Dirichlet function (which equals 1 for irrationals and 0 for rationals) by summing the product of function values and the measure of their corresponding sets, and forms the foundation for modern probability theory and expected value calculations.

Lebesgue integration is a university-level mathematical theory that measures the size of objects by dividing them into measurable sets, with the key advantage being its superior handling of function sequence convergence compared to Riemann integration; it expands the range of integrable functions and provides the theoretical foundation for probability theory, statistics, stochastic processes, and functional analysis, making it essential for understanding continuous and infinite phenomena in modern mathematics.

Lebesgue integration requires measure theory, which defines a measure space with three components: a set X, a sigma-algebra (collection of measurable subsets closed under countable operations), and a measure function assigning non-negative values to sets. The measure must satisfy countable additivity. Characteristic functions equal 1 on sets and 0 elsewhere; their integral equals the set's measure. Simple functions are finite linear combinations of characteristic functions. The integral of a simple function is the sum of constants multiplied by the measures of their pre-image sets.

This lecture introduces the Lebesgue measure as a generalization of length that extends the Riemann integral to a broader class of functions. The course will cover Lebesgue measure, Lebesgue integral, differentiation, and LP spaces. The motivation for this extension is that the space of continuous functions with the integral norm is not complete, meaning Cauchy sequences may not converge within the space. The Lebesgue measure is constructed using outer measure, defined as the infimum of the sum of lengths of open intervals covering a set. This measure satisfies key properties: it extends the notion of length for intervals, is countably additive, and is translation invariant. The domain of the Lebesgue measure is restricted to Lebesgue measurable sets, which form a sigma algebra.

Lebesgue integration is defined through a four-step process: (1) For characteristic functions of measurable sets, the integral equals the measure of the set multiplied by the function value; (2) For simple functions (finite linear combinations of characteristic functions), the integral is the sum of function values multiplied by the measures of their corresponding sets; (3) For non-negative measurable functions, the integral is defined as the limit of integrals of simple functions that monotonically increase to the function; (4) For general real-valued measurable functions, the integral is defined as the difference between the integrals of the positive and negative parts of the function. This systematic approach allows Lebesgue integration to handle a broader class of functions than Riemann integration, with powerful convergence theorems like the Dominated Convergence Theorem and Fubini's Theorem making it essential for modern analysis.

This section covers numerical integration techniques. Trapezoidal Rule approximates integrals using trapezoids: ∫_a^b f(x)dx ≈ (h/2)[f(x_0) + 2f(x_1) + ... + f(x_n)]. Simpson's 1/3 Rule uses parabolic arcs: ∫_a^b f(x)dx ≈ (h/3)[f(x_0) + 4f(x_1) + 2f(x_2) + ... + f(x_n)], requiring even n. Romberg Integration combines Trapezoidal Rule results with Richardson extrapolation: I(h/2) = (4I(h/2) - I(h))/3, achieving higher accuracy. For ∫_0^1 1/(1+x)dx, Romberg extrapolation yields 0.6931, matching the exact value ln(2) ≈ 0.6931. Gaussian Quadrature approximates integrals using weighted sums at specific Gauss points: ∫_a^b f(x)dx ≈ (b-a)/2 * Σ w_i f(x_i). For [-1,1], the formula is ∫_{-1}^1 f(x)dx ≈ Σ w_i f(x_i), where Gauss points are roots of Legendre polynomials. For n = 2: x = ±1/√3, w = 1. For n = 3: x = -√(3/5), 0, √(3/5), w = 5/9, 8/9, 5/9. For n = 4: x = ±√((3 - 2√(6/5))/7), ±√((3 + 2√(6/5))/7), w = (18 - √(30))/36, (18 + √(30))/36.

Numerical integration approximates integrals using quadrature rules: ∫f(x)dx ≈ Σw_i × f(x_i), where w_i are weights and x_i are abscissae. The midpoint rule uses one point at the interval midpoint with weight equal to interval length. The trapezoidal rule uses endpoints with equal weights. Simpson's rule uses three points with specific weights. Higher-order methods use more points. These techniques approximate integrals by evaluating functions at strategic locations and weighting their contributions.

Numerical integration evaluates definite integrals from tabulated function values. Three main methods exist: (1) Trapezoidal Rule - easy to apply but less accurate, uses h/2[y₀ + 2y₁ + ... + yₙ]. (2) Simpson's 1/3 Rule - more accurate, requires even intervals, uses h/3[y₀ + 4y₁ + 2y₂ + ... + yₙ]. (3) Simpson's 3/8 Rule - most accurate but requires most effort, requires multiple of 3 intervals, uses 3h/8[y₀ + 3y₁ + 3y₂ + 2y₃ + ... + yₙ].

Numerical integration approximates definite integrals when antiderivatives cannot be found. The trapezoidal rule uses: ∫[a,b]f(x)dx ≈ (b-a)/2n [f(x₀) + 2f(x₁) + ... + f(xₙ)]. For ∫[0,1]x²dx with n = 4: (1/8)[0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1] = 0.34375 (actual: 0.3333). Simpson's rule uses quadratic polynomials: ∫[a,b]f(x)dx ≈ (b-a)/6n [f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)], where n must be even. For ∫[0,1]x³dx with n = 2: (1/12)[0 + 4(0.125) + 1] = 0.125 (actual: 0.25). For approximation, use the bisection method: find where f(x) changes sign, then repeatedly bisect the interval. For √2, start with [1,2], f(1) = -1, f(2) = 2, so root is in [1,2]. Midpoint 1.5: f(1.5) = 0.25 > 0, so root is in [1,1.5]. Midpoint 1.25: f(1.25) = -0.4375 < 0, so root is in [1.25,1.5]. Continue until desired precision. This method is essential for finding roots of equations that cannot be solved algebraically.

This segment covers two fundamental numerical integration methods. The Trapezoidal Rule approximates definite integrals by dividing the area under the curve into trapezoids: ∫ₐᵇf(x)dx ≈ (b-a)/2n [f(x₀) + 2f(x₁) + ... + f(xₙ)]. The video demonstrates this with ∫₀¹ 1/(1+x²)dx, showing table setup and formula application. The Runge-Kutta Method (RK4) solves differential equations using four intermediate slopes: k₁ = h·f(x₀,y₀), k₂ = h·f(x₀+h/2, y₀+k₁/2), k₃ = h·f(x₀+h/2, y₀+k₂/2), k₄ = h·f(x₀+h, y₀+k₃), then y₁ = y₀ + (k₁ + 2k₂ + 2k₃ + k₄)/6. The video demonstrates this with dy/dx = √(x+y), showing step-by-step calculation of each k value.