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Semester 2: Real Analysis II
Measure on the Real line - Lebesgue Outer Measure - Measurable sets and functions
Lebesgue Outer Measure
Lebesgue outer measure is a way to assign a measure to subsets of real numbers, extending the concept of length. It is defined using coverings of a set by open intervals, and it plays a crucial role in the development of measure theory.
Definition and Properties
The Lebesgue outer measure of a set A is defined as the infimum of the sums of lengths of open intervals that cover A. Important properties include monotonicity, translation invariance, and countable subadditivity.
Measurable Sets
A set is Lebesgue measurable if it can be approximated well by open sets in terms of measure. Formally, a set A is measurable if for every set E, the outer measure satisfies the condition: m*(E) = m*(E ∩ A) + m*(E ∩ A^c).
Measurable Functions
A function is said to be measurable if the preimage of every Borel set is measurable. Measurable functions are essential in integration and probability, allowing the extension of integration to a broader class of functions.
Relationship between Measurable Sets and Functions
Measurable functions are intimately related to measurable sets as they often arise from transformations of measurable sets. This relationship is a foundational concept in real analysis and probability theory.
Applications of Lebesgue Measure
Lebesgue measure is not only a theoretical construct but is widely applied in various fields such as probability theory, statistical mechanics, and real analysis, facilitating the rigorous treatment of concepts like integration and differentiation.
Integration of functions of a real variable - Riemann and Lebesgue integrals
Integration of functions of a real variable
Introduction to Integration
Integration is the process of finding the integral of a function, representing the accumulation of quantities. It is fundamental in mathematics, connecting the concepts of area, volume, and total accumulation.
Riemann Integral
The Riemann integral defines integration using partitions of the interval and upper and lower sums. A function is Riemann integrable if it is bounded and the set of its discontinuities has measure zero. The Riemann integral is suitable for continuous functions on closed intervals.
Properties of the Riemann Integral
Key properties include linearity, additivity over intervals, and continuity. The Fundamental Theorem of Calculus links differentiation and integration, establishing that the integral of a function's derivative recovers the original function up to a constant.
Lebesgue Integral
The Lebesgue integral extends integration to a broader class of functions. It measures the size of sets rather than intervals, integrating functions by summing values over the measure of their sets. This makes it well-suited for handling functions with many discontinuities.
Difference between Riemann and Lebesgue Integrals
The primary difference lies in the handling of discontinuities and convergence. The Lebesgue integral is more powerful, allowing integration of functions not Riemann integrable and processes like Fatou's Lemma and the Dominated Convergence Theorem apply.
Applications of Lebesgue Integral
Lebesgue integration is critical in probability theory, functional analysis, and areas requiring measure theory. It provides a foundation for understanding convergence of integrals and the structure of various function spaces.
Fourier Series and Fourier Integrals - Orthogonal systems - Convergence problems
Introduction to Fourier Series
Fourier series represent periodic functions as the sum of sine and cosine terms. Fundamental to connecting the realms of trigonometry and calculus, Fourier series decompose complex waveforms into simpler components. Mathematically, for a function f defined on the interval [-L, L], the Fourier series is given by: f(x) = a0/2 + Σ [an * cos(n * π * x / L) + bn * sin(n * π * x / L)], where an and bn are the Fourier coefficients defined as: an = (1/L) * integral from -L to L of f(x) * cos(n * π * x / L) dx and bn = (1/L) * integral from -L to L of f(x) * sin(n * π * x / L) dx.
Fourier Integrals
Fourier integrals extend the concept of Fourier series to non-periodic functions. They express a function as an integral of sine and cosine terms, broadening the applicability of Fourier analysis. A common representation is: f(x) = (1/2π) * integral from -∞ to ∞ of F(ω) * e^(iωx) dω, where F(ω) is the Fourier transform of f(x). This formulation is crucial for analyzing signals and systems in a variety of fields.
Orthogonal Systems
In the context of Fourier series and integrals, orthogonality refers to the property of functions being perpendicular in a functional space. The inner product of two functions φ and ψ is defined as: <φ, ψ> = integral of φ(x) * ψ(x) dx over a specified interval. For a set of functions to form an orthogonal system, the inner product must yield zero for distinct functions in the set. This concept is foundational in ensuring the uniqueness of Fourier coefficients in decomposing functions.
Convergence Problems
Convergence issues arise when considering the summation of Fourier series. Various types of convergence exist, such as pointwise convergence and uniform convergence. Some key results include the Dirichlet conditions, which specify conditions under which the Fourier series converges to the function at points where the function is continuous. At points of discontinuity, the series converges to the average of the left-hand and right-hand limits. The Riemann-Lebesgue lemma follows, asserting that the Fourier coefficients tend to zero as n approaches infinity, emphasizing the diminishing influence of higher frequency components.
Multivariable Differential Calculus - Directional derivatives - Taylor's theorem
Multivariable Differential Calculus
Directional Derivatives
Directional derivatives measure how a function changes as we move in a specific direction. Given a function f defined in R^n, the directional derivative of f at a point a in the direction of a unit vector u is defined as the limit of the difference quotient as follows: D_u f(a) = lim(h -> 0) [f(a + hu) - f(a)] / h. This derivative can be computed using the gradient of f if it exists, such that D_u f(a) = ∇f(a) · u, where ∇f(a) is the gradient vector at point a.
Taylor's Theorem in Multiple Variables
Taylor's theorem extends the concept of Taylor series to functions of multiple variables. For a function f that is sufficiently smooth, the Taylor expansion about a point a is given by: f(x) = f(a) + ∇f(a) · (x - a) + (1/2!)(x - a)^T H_f(a) (x - a) + R(x), where H_f(a) is the Hessian matrix at point a, and R(x) is the remainder term which accounts for the error in this approximation. This theorem is particularly useful for approximating functions near a specific point.
Implicit Functions and Extremum Problems - Inverse function theorem - Extremum problems with side conditions
Implicit Functions and Extremum Problems
Implicit Functions
Implicit functions are defined by equations in which the dependent variable cannot be isolated on one side. For instance, in the equation F(x,y) = 0, y is defined implicitly as a function of x.
Inverse Function Theorem
The inverse function theorem states that if a function is continuously differentiable and its derivative is non-zero at a point, then it has a locally defined inverse near that point. This is significant for solving equations where variables are interdependent.
Extremum Problems
Extremum problems involve finding the maximum or minimum values of functions subject to certain constraints. This is often solved using methods such as Lagrange multipliers.
Side Conditions in Extremum Problems
When dealing with side conditions, one must ensure that the constraints do not contradict the possible solutions of the original problem, thereby altering the feasible region for optimization.
