Is there some monotone convergence theorem to prove that: if $f_n \uparrow f$ with $f_n$ positive and continuous, and $g\in BV([a, b])$, then $$\int_a^b f_n dg \longrightarrow \int_a^b f dg \quad ?$$
2025-01-13 02:17:16.1736734636
Monotone convergence for Stieltjes integral
69 Views Asked by mtj95 https://math.techqa.club/user/mtj95/detail AtRelated Questions in REAL-ANALYSIS
- Proving whether the limit of a sequence will always converge to 0?
- Limit of $(5n^2+2n)/(n^2-3)$ using limit definition
- If $\inf f = f(a)$, then $\exists b,c$, $f(b) = f(c)$
- Trying to prove if $S$ is a subset of $R$, every adherent point to $S$ is the limit of a sequence in $S$
- ODE existence of specific solutions
- equivalent definitions of weak topology on a topological vector space
- Bounded derivative implies uniform continuity on an open interval
- Inf and Sup question
- how to prove sup(A) where A={(n+1)/n|n∈N}?
- how to use epsilion-delta limit definition to answer the following question?
Related Questions in CALCULUS
- Derivative of Lambert W function.
- how to use epsilion-delta limit definition to answer the following question?
- Finding the equation of a Normal line
- How to Integrate the Differential Equation for the Pendulum Problem
- Help in finding error in derivative quotient rule
- How to solve the following parabolic pde?
- Finding inflection point
- How to find the absolute maximum of $f(x) = (\sin 2\theta)^2 (1+\cos 2\theta)$ for $0 \le \theta \le \frac{\pi}2$?
- Utility Maximization with a transformed min function
- Interpreting function notation?
Related Questions in MEASURE-THEORY
- Question about algebra and σ–algebra
- Invariant measure of Lorentz Group
- Set of discontinuities of bounded function has measure 0.
- To check whether a set is measurable.
- Show that ${F : F ⊆ Y, f ^{-1}[F] ∈ Σ}$ is a σ-algebra
- The exterior measure of a closed cube is equal to its volume (specific discuss)
- Does taking the limit of the inclusion-exclusion formula work here?
- Compute $\lim_{n\to\infty} \int_{0}^{1}{\frac{n^{3/2}t}{1+n^3t^3}dt}$ using Dominated Convergence Theorem
- Convergence in $L^{1}$ and integrability
- Showing continuity from above of Lebesgue-Stieltjes measure
Related Questions in LEBESGUE-INTEGRAL
- Expected value & Lebesgue integral
- Set of discontinuities of bounded function has measure 0.
- How bad can be the behavior of Lebesgue integrable functions?
- Lebesgue integral - interpretation
- Interpretation of $ \int_A f \ d\mu $ in case $\mu (A) = \infty$
- Interchanging Lebesgue integrals with limits when the limits of integration are variable
- Are upper and lower Lebesgue integrals equal for every $f$?
- Is an a.e finite function integrable?
- The Lebesgue integral of an integrable function is continuous
- Is the change of the integral order valid even when its integral diverges? - Integrability assumption in Tonelli theorem-
Related Questions in STIELTJES-INTEGRAL
- Books or Materials on $n$-dimensional Lebesgue-Stieltjes Measure
- How are the Probability Measure and Cumulative Distribution Function linked when calculating the Expectation of a RV X?
- Discontinuous Function and RS-Integral
- Stieltjes transform and distributional solution
- Lebesgue-Stieltjes measure function is a bijection
- Riemann-Stieltjes integral and function averages
- Two dimensional Riemann-Stieltjes integral.
- Under what conditions does this "two dimensional Riemann-Stieltjes integral" exist?
- Monotone convergence for Stieltjes integral
- Compute the Riemann-Stieltjes integral $\int_{-1}^1 \cos x\ \mathsf dg(x)$ without reduction.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity