MathProblemsBank Math Problems Bank
  • Home
  • Forum
  • About Us
  • Contact Us
  • Login
  • Register
  • language
 MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Mathematics sections
  • Algebra
    • Vector Algebra
    • Determinant calculation
    • Permutation group
    • Matrix transformations
    • Linear transformations
    • Quadratic forms
    • Fields, Groups, Rings
    • Systems of algebraic equations
    • Linear spaces
    • Polynomials
    • Tensor calculus
    • Vector analysis
  • Analytic geometry
    • Curves of the 2-nd order
    • Surfaces of the 2-nd order
    • Lines on a plane
    • Line and plane in space
    • Tangents and normals
  • Complex analysis
    • Operations with complex numbers
    • Singular points and residues
    • Integral of a complex variable
    • Laplace transform
    • Conformal mappings
    • Analytic functions
    • Series with complex terms
    • Calculating integrals of a real variable using residues
  • Differential equations
    • Ordinary differential equations
      • First order differential equations
      • Second order differential equations
      • Higher order differential equations
      • Geometric and physical applications
    • Systems of ordinary differential equations
    • Stability
      • Stability of the equations
      • Stability of the systems of equations
    • Operating method
      • Differential equations
      • Systems of differential equations
  • Differential geometry
  • Discret mathematics
    • Boolean algebra
    • Set theory
    • Combinatorics
    • Graph theory
    • Binary relations
    • Propositional algebra
      • Propositional calculus
      • Sequent calculus
    • Predicate calculus
    • Theory of algorithms and formal languages
    • Automata theory
    • Recursive functions
  • Functional analysis
    • Metric spaces
      • Properties of metric spaces
      • Orthogonal systems
      • Convergence in metric spaces
    • Normed spaces
      • Properties of normed spaces
      • Convergence in normed spaces
    • Measure theory
      • Lebesgue measure and integration
      • Measurable functions and sets
      • Convergence (in measure, almost everywhere)
    • Compactness
    • Linear operators
    • Integral equations
    • Properties of sets
    • Generalized derivatives
    • Riemann-Stieltjes integral
  • Geometry
    • Planimetry
      • Transformations on the plane
      • Construction problems
      • Complex numbers in geometry
      • Various problems on the plane
      • Locus of points
    • Stereometry
      • Construction of sections
      • Various problems in the space
    • Affine transformations
  • Mathematical analysis
    • Gradient and directional derivative
    • Graphing functions using derivatives
    • Plotting functions
    • Fourier series
      • Trigonometric Fourier series
      • Fourier integral
    • Number series
    • Function extrema
    • Power series
    • Function properties
    • Derivatives and differentials
    • Functional sequences and series
    • Calculation of limits
    • Asymptotic analysis
  • Mathematical methods and models in economics
  • Mathematical physics
    • First order partial differential equations
    • Second order partial differential equations
      • d'Alembert method
      • Fourier method
      • With constant coefficients
      • With variable coefficients
      • Mixed problems
    • Convolution of functions
    • Nonlinear equations
    • Sturm-Liouville problem
    • Systems of equations in partial derivatives of the first order
  • Mathematical statistics
  • Numerical methods
    • Golden section search method
    • Least square method
    • Sweep method
    • Simple-Iteration method
    • Approximate calculation of integrals
    • Approximate solution of differential equations
    • Approximate numbers
    • Function Interpolation
    • Approximate solution of algebraic equations
  • Olympiad problems
    • Olympic geometry
    • Number theory
    • Olympic algebra
    • Various Olympiad problems
    • Inequalities
      • Algebraic
      • Geometric
    • Higher mathematics
  • Probability theory
    • One dimensional random variables and their characteristics
    • Theory of random processes
    • Markov chains
    • Queuing systems
    • Two-dimensional random variables and their characteristics
    • Definition and properties of probability
    • Limit theorems
  • Real integrals
    • Integrals of functions of a single variable
      • Indefinite integrals
      • Definite Integrals
      • Improper integrals
    • Double integrals
    • Triple integrals
    • The area of a region
    • Volume of a solid
    • Volume of a solid of revolution
    • Flux of the vector field
    • Surface integrals
    • Curvilinear integrals
    • Potential and solenoidal fields
    • Vector field circulation
    • Integrals depending on a parameter
  • Topology
  • USE problems
  • Variational calculus
Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Let \( \mu(A)<\infty \). Prove that the non-negative function \( f \) is integrable with respect to \( A \) only when the following series converges: \[ \sum_{n=1}^{\infty} 2^{n} \mu\left(A \cap\left\{x: f(x) \geq 2^{n}\right\}\right) . \]

19.6.1.1 Lebesgue measure and integration

7.71 $

Problem: Calculate the integral: \[ I=\int_{0}^{1} k(x) d k^{2}(x) \text {, } \] where \( k(x) \) is the Cantor function.

19.6.1.3 Lebesgue measure and integration

1.8 $

Problem: The signed measure of \( v_{F} \), constructed from the function \( F(x)=\left\{\begin{array}{ll}e^{x}, & 0 \leq x \leq 2 \\ x^{2}, & 2

19.6.1.4 Lebesgue measure and integration

5.14 $

Problem: 1) Find the Radon-Nikodym density with respect to the Lebesgue measure for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) 2) Find the Jordan expansion for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) What sets can be taken as \( X^{+}, X^{-} \)in Hahn decomposition??

19.6.1.2 Lebesgue measure and integration

7.71 $

Problem: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \). Prove that \[ \{x \in \mathbb{R}: f(x) \geq c\}=\bigcap_{n=1}^{\infty}\left\{x \in \mathbb{R}: f(x) \geq c-\frac{1}{n}\right\} . \]

19.6.1.5 Lebesgue measure and integration

2.57 $

Problem: For the function \( f \), given on the segment \( [a, b] \) 1) prove that it's simple; 2) calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it does exist. \[ a=0, \quad b=1, \quad f(t)=\left[\frac{1}{\sqrt{t}}\right] . \]

19.6.1.9 Lebesgue measure and integration

2.57 $

Problem: By definition, calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \). \[ a=0, \quad b=3, \quad f(t)=|t-1| . \]

19.6.1.10 Lebesgue measure and integration

2.57 $

Problem: Let \( f \) be an integrable function on the set \( E, g, h \) are measurable on \( E \). What can be said about the integrability of \( g, h \), if: \( f=g \cdot h, g, h>0 \).

19.6.1.11 Lebesgue measure and integration

2.57 $

Problem: Calculate the Lebesgue integral \( \int_{[0,1]} f(x) d \mu \) from the function \[ f(x)=\left\{\begin{array}{c} x \cdot e^{x}, \quad x \in[0,1] \backslash Q \\ \tan ^{-1} \frac{1}{x+1}, \quad x \in Q \cap[0,1] \end{array}\right. \]

19.6.1.12 Lebesgue measure and integration

3.85 $

Problem: Let \( f \) be an integrable function on \( X \). Is it true that if \( \int_{X} f(t) d \mu=0 \), then \( f(t)=0 \) almost everywhere?

19.6.1.13 Lebesgue measure and integration

3.08 $

Problem: For the function \( f:[a, b] \rightarrow R \) a) find out if it's bounded; b) find the measure of the set of discontinuity points; c) find out if there is a proper or improper Riemann integral for it; d) find out if it's measurable \( f \); e) find the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it exists. \( a=-1, \quad b=1, \quad K \) is the Cantor set, \( f(t)=\left\{\begin{array}{cc}n, \quad t \in\left(\frac{1}{3^{n+1}}, \frac{1}{3^{n}}\right) \backslash K, n \in \mathbb{N}, \\ & {\left[e^{t^{2}}\right], \quad t \in K,} \\ \frac{1}{\sqrt{1+t}}, \quad & t \in\left([-1,0) \cup\left(\frac{1}{3}, 1\right)\right) \backslash K .\end{array}\right. \)

19.6.1.6 Lebesgue measure and integration

5.14 $

mathproblemsbank.net

Terms of use Privacy policy

© Copyright 2025, MathProblemsBank

Trustpilot
Order a solution
Order a solution to a problem?
Order a solution
Order a solution to a problem?
home.button.login