2022–23–B

The system of the real numbers (without Dedekind cuts). The supremum axiom. Convergent sequences, subsequences, monotonic sequences, upper and lower limits. Series: partial sums, convergent and divergent series, examples, nonnegative series, the root test, the quotient test, general series, Dirichlet, Leibnitz, absolute convergence implies convergence (without a proof). Limits of functions, continuity, the continuity of the elementary functions, extrema in compact intervals. The derivative of a function, Lagrange’s Mean Value Theorem, high order derivatives, L’hospital’s rules, Taylor’s Theorem, error estimates, lots of examples. The Riemann integral: only for piecewise continuous functions (finitely many points of discontinuity). Riemann sums and the definition of the integral, The Fundamental Theorem of Calculus, the existence of primitive functions (anti-derivatives). Integration techniques: integration by parts, substitutions, partial fractions (without proofs), improper integrals, applications of integrals, estimation of series with the aid of integrals, Hardy’s symbols O, o and Omega, approximation of momenta and the Stirling formula.

1) Probability space 2) Law of total probability 3) Conditional probability, independent events 4) Bayes’ law 5) Discrete random variables. Discrete distributions: uniform, Bernoulli, binomial, geometric, Poisson 6) Continuous random variable. Continuous distributions: uniform, exponential, normal 7) Discrete two-dimensional joint random variables 8) Independence of random variables 9) Expectation 10) Variance, covariance, correlation coefficient

1. General background: sets and operations on them, Complex numbers: definition (via ordered pairs), addition and multiplication, inverses, adjoint, absolute value. Real and complex polynomials and their roots.
2. Fields: Definition, properties, examples: Rationals, reals, complex numbers, integers mod p.
3. Linear equations over fields, matrices and elementary row operations, rank of a matrix, solutions of homogeneous and non homogeneous systems of linear equations and the connections between them.
4. Vector spaces over fields, subspaces, bases and dimensions, coordinates change of coordinate matrix, row rank as rank of a subspace, sums, direct sums of subspaces and the dimension theorem.
5. Matrices multiplication, the algebra of square matrices, inverse determinants: properties, Cramer’s rule, adjoint and its use for finding the inverse.
6. Linerar transformationsbasic propertieskernel and image of a linear trasformationrepresentaion of linear transformations by matrices and the effect of change of bases.linear functionals, dual bases

• Rings. Ring of polynomials and its ideal structure. The prime factorization of a polynomial. Lagrange interpolation.
• Eigenvalues and eigenvectors of linear operators. Characteristic polynomial and Cayley–Hamilton theorem. The primary decomposition theorem. Diagonalization. Nilpotent operators. Jordan decomposition in small dimension. Jordan decomposition in general dimension- time permitting.
• Linear forms. Dual basis. Bilinear forms. Inner product spaces. Orthogonal bases. Projections. Adjoint linear transformation. Unitary and Hermitian operators. Normal operators and the spectral decomposition theorem. Singular value decomposition theorem and applications.

Optional topics:

• Quadratic forms.
• Sylvester theorem.
• Classification of quadrics in two-dimensional spaces.

1. Introduction to number theory. Intervals and segments. Concept of a function. Elementary functions. 2. Limit of a function.3. Continuity and discontinuity of functions.4. Derivative and differential. Basic derivatives. Differentiability and continuity. Linear approximation by differentials. High-order derivatives. The fundamental theorems of differentiation and their applications. L’Hopital’s theorem and its application to calculation of limits.5. Taylor’s polynom. Expansion of functions into Taylor’s and McLoran’s series. Expansions of some usage functions. Application of Taylor’s and McLoran’s polynoms a) to approximate calculations, and b) to calculation of limits.6. Investigation of a function. Extremal points. Necessary and sufficient conditions for extrema. Max. and min. of a function within a segment. Convexity and concavity, inflection point. Asymptotes. Graph construction.7. Primitive function and indefinite integral. Table integrals. Calculation of indefinite integrals by decomposition, by parts, by substitution. Integration of rational and trigonometric functions.8. Definite integrals. Reimann’s sum. The fundamental theorem. Formula of Newton-Leibnitz. Calculation of definite integrals. Integration by decomposition, by parts, by substitution.9. Use in definite integrals to calculation of areas, volumes and curve lengthes. Rectungular and polar coordinate systems.10. First-order ordinary differential equations. General definitions. Cauchy problem. Separated variables.

1. Infinite series. Tests for convergence. Taylor series and Taylor polynomials. Absolute convergence. Alternating series. Conditional convergence. Power series for functions. Convergence of power series; differentiation and integration.
2. Vectors and parametric equations. Parametric equation in analytic geometry. Space coordinates. Vectors in space. The scalar product of two vectors. The vector product of two vectors in space. Equations of lines and planes. product of three vectors and more. Catalog of the quadratic surfaces. Cylindres.
3. Vector functions and their derivatives. Vector functions. differentiation formulas. Velocity and acceleration. Tangential vectors. Curvature and normal vectors. Polar coordinates.
4. Partial differentiation. Functions of two and more variables. The directional derivative. limits and continuity. Tangent plane and normal lines. The gradient. The chain rule for partial derivatives. The total differentiation. Maxima and minima of functions of several independent variables. Higher order derivatives.
5. Multiple integrals. Double integrals. Area and volume by double integrals. Double integrals in polar coordinates. Physical applications. triple integrals. Integration in cylindrical and spherical coordinates. Surface area. Change of variable in multiple integrals.
6. Vector analysis. Vector fields. Line integrals. Independence of path. Green’s theorem. Surface integrals. The divergence theorem. Stokes’ theorem.

1. Real numbers and real line, elementary functions and graphs, some functions arising in economics. The limit of a function, calculating limits using the limit laws, continuity, the number e.2. The derivative of a function, differential rules, higher derivatives, L’Hospital rules.3. Extreme values of functions, monotonic functions, point of inflection, concavity, curve sketching, applications to economics.4. Indefinite integrals, techniques of integration, definite and improper integrals, areas between curves, applications to economics.5. Functions of several variables, economics examples, partial derivatives, linearization, the chain rile, implicit and homogeneous functions, maximum and minimum, Lagrange multipliers.6. Introduction to linear algebra, matrices, linear systems.

1. Introduction: the real and complex numbers, polynomials.
2. Systems of linear equations and Gauss elimination.
3. Vector spaces: examples (Euclidean 2-space and 3-space, function spaces, matrix spaces), basic concepts, basis and dimension of a vector space. Application to systems of linear equations.
4. Inverse matrices, the determinant, scalar products.
5. Linear transformations: kernel and image, the matrix representation of a transformation, change of basis.
6. Eigenvalues, eigenvectors and diagonalization.

1. The real numbers, inequalities in real numbers, the complex numbers, the Cartesian representation, the polar representation, the exponential representation, the Theorem of de Moivre, root computations.
2. Systems of linear equations over the real or complex numbers, the solution set and its parametric representation, echelon form and the reduced echelon form of a matrix, backwards substitution, forward substitution and their complexity, the Gauss elimination algorithm and its complexity, the reduction algorithm and its complexity.
3. Vector spaces, sub-spaces of vector spaces, linear combinations of vectors, the span of a set of vectors, linear dependence and linear independence, the dimension of a vector space, row spaces and column spaces of matrices, the rank of a matrix.
4. Linear mappings between vector spaces, invertible mappings and isomorphisms, the matrix representation of finite dimensional linear mappings, inversion of a square matrix, composition of mappings, multiplication of matrices, the algebra of matrices, the kernel and the image of a linear mapping and the computation of bases, changing of a basis, the dimension theorem for linear mappings.
5. Inner product spaces, orthogonality, the norm of a vector, orthonormal sets of vectors, the Cauchy-Schwarz inequality, the orthogonal complement of a sub-space, orthogonal sequences of vectors, the Gram-Schmidt algorithm, orthogonal transformations and orthogonal matrices.
6. The determinant of a square matrix, minors and cofactors, Laplace expansions of the determinant, the adjoint matrix and Laplace theorem, conjugation of a square matrix, similarity transformations and their invariants (the determinant and the trace).
7. Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral theorem for Hermitian matrices.

1. Lines and planes. Cross product. Vector valued functions of a single variable, curves in the plane, tangents, motion on a curve.
2. Functions of several variables: open and closed sets, limits, continuity, differentiability, directional derivatives, partial derivatives, the gradient, scalar and vector fields, the chain rule, the Jacobian. Implicit differentiation and the implicit function theorem. Extremum problems in the plane and in space: the Hessian and the second derivatives test, Lagrange multipliers.
3. Line integrals in the plane and in space, definition and basic properties, work, independence from the path, connection to the gradient, conservative vector field, construction of potential functions. Applications to ODEs: exact equations and integrating factors. Line integral of second kind and arclength.
4. Double and triple integrals: definition and basic properties, Fubini theorem. Change of variable and the Jacobian, polar coordinates in the plane and cylindrical and spherical coordinates in space. Green’s theorem in the plane.
5. Parametric representation of surfaces in space, normals, the area of a parametrized surface, surface integrals including reparametrizations
6. Curl and divergence of vector fields. The theorems of Gauss and Stokes.

Sets. Set operations and the laws of set theory. Power set. Cartesian product of sets.The rules of sum and product. Permutations, combination, distributions. The Binomial Theorem. The well-ordering principle: mathematical induction. The principle of inclusion and exclusion. The pigeonhole principle. Recurrence relations. Generating functions.Relations and functions. Properties of relations. Equivalence relations and their properties. Partial order. Functions and their properties. Injective, surjective functions. Function composition and inverse functions.Graph, subgraph, complements. Graph isomorphism. Euler`s formula. Planar graph. Euler trails and circuits. Trees.Propositional logic. Syntax of propositional logic. Logical equivalence. The laws of logic. Logical implication. Equivalence and disjunctive normal form. Predicate logic. Syntax of predicate logic. Models. Equivalence of formulas. Normal form.Algebraic structures. Rings, groups, fields. The integer modulo n. Boolean algebra and its structure.

1. Real numbers. Supremum and Infimum of a set. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Limit superior and limit inferior. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Conditional and absolute convergence. The Leibnitz test for series with alternating signs. Rearrangements of series (without proof) 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval: boundedness and attainment of extrema. Uniform continuity, Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem. Lagrange remainder formula.

In this course the basic concepts of one-dimensional analysis (a limit, a derivative, an integral) are introduced and explored in different applications: graphing functions, approximations, calculating areas etc.

1. Limit of a function, continuity.
2. Derivative, basic derivative formulas.
3. Derivative of an inverse function; derivative of a composite function, the chain rule; derivative of an implicit function.
4. Derivatives of high order.
5. The mean value problem theorem. Indeterminate forms and l’Hopital’s rule.
6. Rise and fall of a function; local minimal and maximal values of a function.
7. Concavity and points of inflection. Asymptotes. Graphing functions.
8. Linear approximations and differentials. Teylor’s theorem and approximations of an arbitrary order.
9. Indefinite integrals: definition and properties.
10. Integration methods: the substitution method, integration by parts.
11. Definite integrals. The fundamental theorem of integral calculus (Newton-Leibniz’s theorem).
12. Calculating areas.

Bibliography

Thomas & Finney, Calculus and Analytic Geometry, 8th Edition, Addison-Wesley (World Student Series).

The aim of the course is to study main principles of probability theory. Such themes as probability spaces, random variables, probability distributions are given in details.Some applications are also considered.1. Probability space: sample space, probability function, finite symmetric probability space, combinatorial methods, and geometrical probabilities.2. Conditional probability, independent events, total probability formula, Bayes formula. 3. Discrete random variable, special distributions: uniform, binomial, geometric, negative binomial, hypergeometric and Poisson distribution. Poisson process.4. Continuous random variable, density function, cummulative distribution function. Special distributions: uniform, exponential, gamma and normal. Transformations of random variables. Distribution of maximum and minimum. Random variable of mixed type.5. Moments of random variable. Expectation and variance. Chebyshev inequality.6. Random vector, joint probability function, joint density function, marginal distributions. Conditional density, covariance and correlation coefficient.7. Central Limit Theorem. Normal approximation. Law of Large Numbers.

1. Complex numbers, open sets in the plane.
2. Continuity of functions of a complex variable
3. Derivative at a point and Cauchy–Riemann equations
4. Analytic functions; example of power series and elementary functions
5. Cauchy’s theorem and applications.
6. Cauchy’s formula and power series expansions
7. Morera’s theorem
8. Existence of a logarithm and of a square root
9. Liouville’s theorem and the fundamental theorem of algebra
10. Laurent series and classification of isolated singular points. The residue theorem
11. Harmonic functions
12. Schwarz’ lemma and applications
13. Some ideas on conformal mappings
14. Computations of integrals

1. Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
2. Sturm-Liouville theory.
3. String or wave equation. Initial and boundary value conditions (fixed and free boundary conditions). The d’Alembert method for an infinitely long string. Characteristics. Wave problems for half-infinite and finite strings. A solution of a problem for a finite string with fixed and free boundary conditions by the method of separation of variables. The uniqueness proof by the energy method. Well-posedness of the vibrating string problem.
4. Laplace and Poisson equations. Maximum principle. Well-posedness of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson formula. An ill-posed problem - the Cauchy problem. Uniqueness of a solution of the Dirichlet problem. Green formula in the plane and its application to Neumann problems.
5. Heat equation. The method of separation of variables for the one-dimensional heat equation. Maximum principle. Uniqueness for the one-dimensional heat equation. The Cauchy problem for heat equations. Green?s function in one dimension. If time permits: Green?s function in the two dimensional case.
6. Non-homogeneous heat equations, Poisson equations in a circle and non-homogeneous wave equations.
7. If time permits: free vibrations in circular membranes. Bessel equations.

1. Integral calculus in one variable and its application: the integral, Riemann sums, integrability of bounded functions with countably many discontinuity points (the proofs only for continuous functions and monotone functions), antiderivatives and the Fundamental Theorem of Calculus, change of variables and integrations by parts, partial fractions (without proofs). Applications of integral calculus: computation of areas, volume of the solid of revolution, the length of a smooth curve. Improper integral, and convergence tests for positive functions, application to series.
2. Functions of several variables: open, closed, and compact sets, level curves and surfaces, vector valued functions, paths and path-connectedness.
3. Limits and continuity in several variables: arithmetic of limits, Weierstrass theorem, intermediate value theorem.
4. Multivariable differential calculus: partial and directional derivatives, differentiability and the tangent plane, the chain rule, the orthogonality of the gradient to the level surfaces, implicit function theorem for a curve in the plane and a surface in the space (without a proof), the Hessian, Taylor approximation of order 2, critical points (classification only in dimension 2), Extremum problem, including Lagrange multipliers and gradient descent.
5. Integration in dimension 2: Reimann integral in dimension 2, change of variables and Fubini theorem (without proofs), changing the order of integration, polar coordinates, computation of volumes. If time permits: integration in dimension 3.

First-Order PDE Cauchy Problem Method of Characteristics The Wave Equation: Vibrations of an Elastic String D’Alembert’s Solution Fourier Series Fourier Sine Series Initial-Boundary Value Problems The Wave Equation .Separation of Variables Fourier Series Solution of the Heat Equation The Heat Equation. Duhamel’s Principle. Laplace’s Equation. Dirichlet Problem for a Disc

Ordinary Differential EquationsBasic concepts: ordinary differential equations, differential equations of the first order, general solution, initial value problems, partial solutions. Linear differential equations with separable variables, exact equations, integration factor, homogeneous equations. Existence and Uniqueness theorem (without proof). System of differential equation of first order, solution by matrixes. Linear differential equations of second order, non- homogeneous equations, Wronskian. Linear differential equations of n-th order.Integral TransformsLaplace transform, properties of the Laplace transform. Convolution of two functions and convolution theorem. Heavyside (unit step) function, ?-function (Dirac), particularly continuous functions, their Laplace transform. Solution of non-homogeneous differential equations by Laplace transform.Fourier transform, properties of the Fourier transform. Convolution of two functions and convolution theorem. Cosines and Sine Fourier transform. Solution of integral equations by Fourier transform..

Analytic Geometry: planes and lines, quadric surfaces, cylinders.Vector functions: derivatives and integrals.Partial derivatives: functions of two or more arguments, chain rules, gradient, directional derivatives, tangent planes, higher order derivatives, linear approximation, differential of the first and higher order, maxima, minima and saddle points, Lagrange multipliers.Multiple integrals: double integrals, area, changing to polar coordinates, triple integrals in rectangular coordinates, physical applications.Vector analysis: vector and scalar fields, surface integrals, line integrals and work, Green’s theorem, the divergence theorem, Stokes’s theorem.Infinite series: tests for convergence of series with nonnegative terms, absolute convergence, Alternating series, conditional convergence, arbitrary series.Power series: power series for functions, Taylor’s theorem with remainder: sine, cosine and e , logarithm, arctangent, convergence of power series, integration, differentiation.

1. Basic notions: equations of the first order, general solution, initial value problem, particular solution. Linear equations, separable equations, exact equations, homogeneous equations, integrating factor. The existence and uniqueness theorem (without proof). The Riccatti equations, the Bernoulli equations. Linear systems of the first order differential equations. Solution via the matrix calculus. The second order linear equations. Non-linear equations and the Wronskian. The Euler equations. Linear equations of the first order. 2. The Laplace transformation, properties of the Laplace transformation, solutions of the linear non-homogeneous equations via the Laplace transformation, the Heaviside functions, the delta functions.3. The Fourier transformation, properties of the Fourier transformation. Cosine and sine Fourier transformation. Solution of the integral equations via the Fourier transformation.

The aim of the course is to learn basis of Calculus of functions of two and more variables. It includes: a) short study of vector algebra and analytic geometry in plane and space; b) differential calculus of two and more variables and its applications to local and global extremum problems, directional derivatives, Teylor’s formula, etc.;c) Integral calculus (line, double and triple integrals) and its applications; d) vector field theory and in particular its applications for studying potential vector fields.

1. The Riemann integral: Riemann sums, the fundamental theorem of calculus and the indefinite integral. Methods for computing integrals: integration by parts, substitution, partial fractions. Improper integrals and application to series. 2. Uniform and pointwise convergence. Cauchy criterion and the Weierstrass M-test. Power series. Taylor series. 3. First order ODE’s: initial value problem, local uniqueness and existence theorem. Explicit solutions: linear, separable and homogeneous equations, Bernoulli equations. 4. Systems of ODE’s. Uniqueness and existence (without proof). Homogeneous systems of linear ODE’s with constant coefficients. 5. Higher order ODE’s: uniqueness and existence theorem (without proof), basic theory. The method of undetermined coefficients for inhomogeneous second order linear equations with constant coefficients. The harmonic oscillator and/or RLC circuits. If time permits: variation of parameters, Wronskian theory.

1. Infinite series of nonnegative terms and general series. Absolute and conditional convergence. Power series.
2. Vector algebra. Dot product, cross product and box product.
3. Analytic geometry of a line and a plane. Parametric equations for a line. Canonic equations for a plane. Points, lines and planes in space.
4. Vector-valued functions. Derivative. Parametrized curves. Tangent lines. Velocity and acceleration. Integration of the equation of motion.
5. Surfaces in space. Quadric rotation surfaces. Cylindrical and spherical coordinates.
6. Scalar functions of several variables. Scalar field. Level surfaces. Limit and continuity. Partial derivatives. Directional derivative. Gradient vector. Differential. Tangent plane and normal line. Chain rules. Implicit function and its derivative. Taylor and MacLaurin formulas. Local extreme values. Absolute maxima and minima on closed bounded regions.
7. Vector-valued functions of several variables. Vector field. Field curves. Divergence and curl.
8. Line and path integrals. Work, circulation. Conservative fields. Potential function.
9. Double integral and its applications. Green’s theorem.
10. Parametrized surfaces. Tangent plane and normal line. Surface integrals. Flux. Stokes’s theorem.
11. Triple integral and its applications. Divergence theorem.

1. Analytic geometry in space. Vector algebra in R3. Scalar, cross and triple product and their geometric meaning. Lines, planes and quadric surfaces in space including the standard equations for cones, ellipsoids, paraboloids and hyperboloids.
2. Functions of several variables.Graphs and level curves and surfaces. Limits and continuity. Properties of the continuous functions on a closed bounded domain. Partial derivatives. The plane tangent to graph of the function. Differentiability, the total differential and the linear approximation. Differentiability implies continuity. The chain rule. The gradient vector and the directional derivative. Tangent plane and the normal line to a surface at a point. 201.1.9761
3. Maxima and minima for functions of several variables. Higher-order partial derivatives and differentials. Taylor’s formula. Local extrema and saddle points. Necessary conditions for local maxima and minima of a differentiable function. Sufficient conditions for local maxima and minima via the Hessian. Global extrema in closed bounded sets. Lagrange Multipliers.
4. Double integrals . Double integrals on rectangles. Connection with the volume. Properties and evaluation of double integrals in non-rectangular domains. Iterated integrals and change of order of integration. Change of variables formula for the double integral and the Jacobian. Double integrals in polar coordinates. Applications of the change of variables formula to the computation of area.

1. Descriptive statistics: organizing, processing and displaying data. 2. Sampling distributions: Normal distribution, the student t-distribution, Chi-Square distribution and Fisher’s F-distribution 3. Estimation: A point estimate and Confidence Interval of population parameters: Mean variance and proportion. Tolerance interval. 4. Testing hypothesis about a population’s parameters: Mean, variance and proportion. 5. Evaluating the properties of a statistical test: errors, significance level and power of a test. 6. Testing hypothesis about equality of variances, equality of means and equality of proportions of two populations. 7. Testing for independence of factors: Normal and Chi-Square methods. 8. Testing for goodness of fit of data to a probabilistic model: Chi-Square test. 9. Linear regression: Inference about the utility of a linear regression model. Covariance and correlation coefficient. Confidence and prediction intervals. 10. Weibull distribution: estimating the distribution’s parameters

1. Fields: the definition of a field, complex numbers.

2. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous equations, parametrization of solutions.

3. Vector spaces: examplex, subspaces, linear independence, bases, dimension.

4. Matrix algebra: matrix addition and multiplication, elementary operations, the inverse matrix, the determinant and Cramer’s law. Linear transformations: examples, kernel and image, matrix representation.

Part A: Logic and set-theory. Propositional calculus, Boolean operations. Truth tables, the truth-value of a propositional formula (without induction at this stage), logical implication and logical equivalence, tautologies and contradictions, the useful tautologies, distributivity and de-Morgan’s Law. Sets: the notion of a set, membership and equality, operations: union, intersection, set-difference and power-set. Ordered pairs and Cartesian products. Equivalence relations, quotient spaces and partitions.Partial orders. Functions, injective and surjective functions, invertibility of a function. The ordered set of natural numbers.The axiom of induction in different forms.

Part B: Finite and infinite sets. The notion of cardinality. Countable sets. Cantor’s theorem on the power set of a set.

Part C: Combinatorics. Basic counting formulas. Binomials. Inclusion-exclusion technique. Recursive definition and formulas.

Part D: Graph Theory. Graphs, examples, basic facts, vertex degrees, representing a graph, neighborhood matrices, connected components, Euler graphs, bipartite graphs, matching in bipartite graphs, Hall’s marriage theorem, graph colorings.

1. Fields: the definition of a field, complex numbers.

2. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous equations, parametrization of solutions.

3. Vector spaces: examplex, subspaces, linear independence, bases, dimension.

4. Matrix algebra: matrix addition and multiplication, elementary operations, the inverse matrix, the determinant and Cramer’s law. Linear transformations: examples, kernel and image, matrix representation.

Metric spaces:

closed sets, open sets, Cauchy sequences, completeness, compactness, Theorem of Heine–Borel, continuity and uniform continuity of functions, uniform convergence of sequences of functions.

Measure theory:

algebras, measures and outer measures, measurable sets, discrete measure spaces, Lebesgue measure on the real line, measurable functions, Lebesgue integral, dominated convergence theorem, $L_p$-spaces as complete normed spaces. Time-permitting: signed measures and absolute continuity of measures and the Radon-Nykodim theorem.