Questions tagged [stieltjes-integral]

For questions about Stieltjes integrals. Use with other tags as needed, such as [riemann-integration], to specify Riemann–Stieltjes, Lebesgue–Stieltjes, etc.

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17 views

Associativity of convolution using Lebesgue-Stieltjes integral.

A First Course in Stochastic Processes (Karlin, S.) defines, for increasing, right-continous functions $A,B$ with $A(0)=B(0)=0$, the convolution $A*B$ as $$A*B(t)=\int_0^t B(t-y)dA(y).$$ In addition, ...
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1answer
71 views

Riemann-Stieltjes integral question

Let $f.g:[a,b] \to \mathbb{R}$, $g(x)=\begin{cases} 0, & x=a \\ 1, & x\in (a,b] \end{cases}$. Prove that $f$ is Riemann-Stieltjes integrable with respect to $g$ if and only if $f$ is ...
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1answer
78 views

Is it true that $ \int_{\mathbb R} f(t)g(t) \, dt = \int_{\mathbb R} f'(t) \, dg(t) $?

Assume that $f : \mathbb R \to \mathbb C$ is a $C^\infty$ function. Further assume that $g$ is continuous and of bounded variation. Is it true that $$ \int_{\mathbb R} f(t)g(t) \, dt = \int_{\mathbb R}...
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1answer
23 views

How to find the bounded variation function for a Riemann-Stieltjes representation of a functional in $C \,[a,b]$?

The Riesz's representation theorem says that every bounded linear functional $L$ on $C \,[a,b]$ can be representated by a Riemann - Stieltjes integral: $$L(f) = \int_{a}^{b}f(x)d(\alpha(t))$$ where $\...
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38 views

Bounds on Integral $\int_{0}^{t}\sin(2\pi\omega \tau)d\eta(\tau)$, where the function $\eta(\tau)$ is of bounded variation

I am trying to find an upper bound on the integral: $$\int_{0}^{t}\sin(2\pi\omega \tau)~d\eta(\tau),$$ where the function $\eta(\cdot)$ is continuous and of bounded variation, hopefully in terms of ...
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1answer
22 views

Move a function from the integrand into the differential in a Stieltjes-Integral

If I have an integral like this $$\int_{0}^{\infty} e^{-st}f(t)d(\alpha(t)),$$ then is it possible to transform it into a "classic" Laplace-Stieltjes-Integral of the form $$\int_{0}^{\infty}...
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1answer
40 views

Calculation of a Stieltjes-Integral

I've been studying Stieltjes-Integrals a bit, and came upon an integral where I don't see why the result given should be correct: The integral is $$\int_{1/2}^{n} \frac{d\lfloor t \rfloor}{t} = \frac{\...
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1answer
40 views

Suggestion for proof on the integral Riemman - Stieltjes, impropie integral

Suppose $f$ is real function on $(0,1]$ and $f\in \mathcal{R}(\alpha)$ on $[c,1]$ for every $c>0$. If $f\in \mathcal{R}(\alpha)$ on $[0,1]$, show that $\displaystyle\int_{0}^{1}fd\alpha=\...
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1answer
30 views

Lebesgue-Stieltjes measure on $\mathbb R$ vs on $[a,b]$ and weak convergence

Let $F : \mathbb R \rightarrow \mathbb R$ be non-decreasing and right-continuous. Then there exists a unique Borel measure $\mu_F$ on $\mathbb R$ such that for any interval $J \subset \mathbb R$ : $$\...
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1answer
64 views

Inequality about functions of bounded variation and Riemann-Stieltjes integrals

Let $\alpha$ be a functions of bounded variations on $[a, b]$ and $f$ a buonded function such that $f \in R(\alpha)$ (i.e. integrable with $\alpha$ as integrator).$V(x)$ is the total variation of $\...
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79 views

Relate $\pi(x)$ and $\vartheta(x)$ using the Riemann-Stieltjes integral

From "Mathematical Analysis" of T.M. Apostol. a) If $x \ge 2$, prove that $\pi(x)$ and $\vartheta(x)$ can be expressed as the following Riemann-Stieltjes integrals: \begin{gather*} \...
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84 views

Prove by definition that $ \int_a^b fdf = {f^2(b)-f^2(a) \over{2}}$ when $f$ is continuous

Let $f :[a,b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is Riemann Stieltjes integral with respect to itself that is: $f\in RS_a^b(f)$ by definition and $ \int_a^b fdf = {f^2(b)-...
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3answers
67 views

Evaluating $\int x f(x)\ \mathrm{d}f(x)$

$$ \int xf(x)\ \mathrm{d}f(x) $$ I firstly thought we can assume $x$ as a constant but it isn't $f(x)$ depends on $x$ therefore $x$ depends on $f(x)$ . I just couldn't find a way out. If we say $f(x)=...
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23 views

Characterization of all functions which are Riemann-Stieltjes-integrable with respect to $g$

Pretty simple question. I know that when it comes to Riemann-integrable functions, we know that a function $f$ on a compact interval $[a, b]$ is Riemann-integrable iff $f$ is bounded and the set of ...
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42 views

(Proof) $g\in BV[a,b]$ and $f\in\mathcal{R}_g[a,b]$ and $f$ bounded $\Rightarrow 1/f\in\mathcal{R}_g[a,b]$

As presented in the title, $g\in BV[a,b]$, $f\in\mathcal{R}_g[a,b]$ with $1/f$ bounded by some positive $M$. How do I imply that $1/f$ is also Riemann-Stieltjes integrable with respect to the ...
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1answer
34 views

Proof verification: Possibly false argument in the proof of $\int_{I}g(x)dF(u)=\int_{I}g(u)F'(u)du$ (Riemann-Stieltjes)

In this paper, I am questioning the proof of the following lemma (Lemma 2, page 5): Assume $F$ is differentiable with $F'=f$ continuous. Then if $g$ is integrable, $$\int_{I}g(x)dF(u)=\int_{I}g(u)...
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36 views

Integrability of absolute value in the Riemann-Stieltjes sense

In Principles of Mathematical Analysis (Rudin) it is proved in Theorem 6.13 that if $f \in \mathcal R(\alpha)$, then $|f| \in \mathcal R(\alpha)$ and $$ \left|\int_a^b fd\alpha \right| \le \int_a^b |...
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42 views

Steps to solve a Stieltjes integral with a step function?

I am trying to understand the steps for solving the equation: $$\epsilon(t)=\int_0^tJ(t,t')\mathrm{d}\sigma(t')$$ assuming $σ(t)=H(t-t')$, which I understand is known as a Heaviside step function. A ...
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2answers
35 views

Are increasing functions Riemann-Stieltjes integrable?

It is easy to prove that if $f$ is increasing on $[a,b]$ then it is Riemann integrable. If $f$ is increasing is it necessarily Riemann-Stieltjes integrable where $\alpha$ is increasing on $[a,b]$?
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22 views

If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

I can figure out how to prove this, assuming C is positive, but I'm not sure how to take into consideration if C < 0. The sup of any partition of $Cf$ will simply be Csupf(x). This makes the proof ...
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1answer
56 views

Computing probability as a Lebesgue-Stieltjes integral

I came across an expression for a probability in terms of a Lebesgue-Stieltjes integral and I'd like to understand why it is true. Let $T$ and $C$ be independent random variables. Let $F$ be the ...
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1answer
40 views

Show that f is Riemann-Stieltjes Integrable. Is my solution correct?

Proof: Let g be increasing on [0,1] and continuous on (0,1]. Show that $f \in \mathbf{R}([a,b],g)$, where f is defined on [0,1] as follows. $f(x) = \frac{1}{k}$ if $x=\frac{1}{k}$ for some natural ...
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31 views

Stieltjes integral and dominated convergence

Let $a<b$. Let $(f_n)_n\subseteq C^1([a, b])$ and $g\in \operatorname{BV}([a, b])$. By Stanislaw Hartman and Jan Mikusinski, "The Theory of Lebesgue Measure and Integration" (p. 165), we can ...
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29 views

Compute the Riemann-Stieltjes integral

I'm fairly sure I have solved it correctly, but certainty in these matters is most appreciated. $\int_{-1}^{1} e^{x}d(4g(x)),$ where $g(x)=-1_{[-1,0]}(x) + 3\cdot_{(0,1]}(x) = [e^{0}][4 \cdot 3 - 4 \...
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1answer
59 views

Compute the Riemann-Stieltjes integral $\int_{-1}^1 \cos x\ \mathsf dg(x)$ without reduction.

Compute the Riemann-Stieltjes integral $$\int_{-1}^{1}\cos x \ \mathsf dg(x),$$ where $g(x) = -\mathsf 1_{[-1,0]}(x)+3\cdot \mathsf 1_{(0,1]}(x)$. Solution: $$\int_{-1}^{1}\cos x\ \mathsf dg(x) = f(...
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21 views

Monotone convergence for Stieltjes integral

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 $$\int_a^b f_n dg \longrightarrow \int_a^b f dg \quad ?$$
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59 views

Under what conditions does this “two dimensional Riemann-Stieltjes integral” exist?

Assume you have two functions $f,g: [a,b]\times [c,d]\rightarrow \mathbb{R}$. Let $\Pi_1^n=(a=x_0,x_1,x_2,\ldots,x_n=b)$ be a partition of $[a,b]$ and let $\Pi_2^n$ be a partition of $[c,d]$. Define ...
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67 views

Two dimensional Riemann-Stieltjes integral.

In an old text I am reading I have encountered something that looks like a two-dimensional Riemann-Stieltjes integral. I am wondering if there are any good books or texts on this subject? Update: It ...
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40 views

Riemann-Stieltjes integral and function averages

$x,y:[0,1]\rightarrow\mathbb{R}$ are positive continuous functions. $y\left(p\right)$ is also monotone increasing (weakly). What are the conditions on $y$ so that there is $p \in [0, 1]$ for which $$ ...
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171 views

Lebesgue-Stieltjes measure function is a bijection

I came across this exercise in my Measure Theory workbook and I've been stuck on it. This is the question : Let F be the set of all non-decreasing right-continuous functions $f : \mathbb{ R} \...
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21 views

Stieltjes transform and distributional solution

let be the Stieltjes transform $$ g(s)= \int_{0}^{\infty} \frac{f(t)}{t+s} $$ then let be $ m_{n} = \int_{0}^{\infty}t^{n}f(t) $ then i get the distributional solution $$ f(t) = \sum_{n=0}^{\...
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1answer
105 views

What does $\int_0^1 x^n d\alpha(x)$ converge to?

Let $\alpha$ be a monotonically increasing function on $[0,1]$. What does the sequence of Riemann-Stieltjes integrals \begin{equation*} \int_0^1 x^n d\alpha(x) \end{equation*} converge to as $n \to \...
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2answers
67 views

Discontinuous Function and RS-Integral

I saw the following exercise in Bartle’s Elements of Real Analysis: If f is a function from the interval [a,b] to the reals such that f is discontinuous at some point of the interval, then there ...
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1answer
31 views

How are the Probability Measure and Cumulative Distribution Function linked when calculating the Expectation of a RV X?

Given a probability space $(\Omega, \Sigma, P)$ and a measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, let $X: \Omega\rightarrow\mathbb{R}$ be a RV which is $(\Sigma, \mathcal{B}(\mathbb{R}))$-...
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21 views

Books or Materials on $n$-dimensional Lebesgue-Stieltjes Measure

I am looking for a book dealing with $n$-dimensional Lebesgue-Stieltjes Measure, especially dealing with its construction and its extension to Borel $\sigma$-algebra on $R^d$ rigorously. For example,...
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29 views

Understanding Lebesgue Stieltjes Integral

Let $f:[a,b]\rightarrow \mathbb{R}$ be a Borel measurable and bounded and $g:[a,b]\rightarrow \mathbb{R}$ be of bounded variation and right-continuous with $g(b) = 0 $ and $g(a)=0$. I assume that $f(b)...
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28 views

Representation of Stieltjes integration by part.

I'm wondering if the following holds: $\forall g\in c[0,1]$, $\exists g'\in L_\infty[0,1]$ such that $\forall f\in BV[0,1], \int gdf=f(1)g(1)-f(0)g(0)-\int fg'dx$. And in general, is there something ...
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1answer
145 views

Baby Rudin Chapter 6, Problem 15 : Strict inequality

Problem 15 in Chapter 6 of Principles of Mathematical Analysis by Walter Rudin: Suppose $f$ is real, continuously differentiable on $[a,b]$, $f(a)=f(b)=0$, and $\int_a^b f^2(x)dx = 1$. Prove that $\...
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1answer
49 views

Reduction step in Theorem 1.9 of Conway's Complex Analysis

Theorem 1.9 If $\gamma$ is piecewise smooth and $f : [a,b] \to \Bbb{C}$ is continuous, then $$\int_{a}^{b} f d \gamma = \int_{a}^{b} f(t) \gamma '(t) dt$$ Here's part of the proof that confuses me: ...
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1answer
118 views

Is writing $dx^2$ same as writing $d(x^2)$ in calculus

When we write $dx^2$, do we actually mean $$d(x^2)$$ (the change in respect to $x^2$), or $$(dx)^2$$ ($dx$ squared)?
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23 views

Riemann Stieltjes Integration by Parts on RCLL functions

I have the following integral that I am trying to evaluate with integration by parts. $\int_{[0,T]} P_{t^{-}} d\gamma_t$ Where $P_{t^{-}}$ is the left limit of $P_t$ for a simple random walk, $dP_t=...
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1answer
54 views

How do I evaluate Stieltjes-integral with sgn(\sin x)?

How do I calculate the integral Stieltjes integral of: $$\int_{-\pi}^{\pi} (x+2) d(x*sign(\sin x))$$ I know that $\int f(x) dg(x) = \int f(x)g(x)'dx$ But does the derivative $[sign(\sin x)]'=0$? ...
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164 views

Existence of $\int_a^bf\,dg$ when $f,g \not\in BV([a,b])$

This question without satisfactory answer asks about necessary and sufficient conditions for existence of the Riemann-Stieltjes integral $\int_a^b fdg$ when both $f$ and $g$ are continuous. Related ...
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1answer
117 views

Riemann Stieltjes Mean Value theorem - result

This is a well known result of Riemann Stieltjes integration: All the proofs I found use the fact that $f$ is bounded and apply one the Mean Value Theorem Riemann-Stieltjes Integrals (this one). I ...
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1answer
62 views

Riemann Stieltjes integration

Given $f(x)=k $ where k is a constant and $g(x)=x^2 \in [a,b]$, how do I find the the Riemann Stieltjes integrals, $\int fdg$ and $\int gdf$ taken over the intervals [a,b]?
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1answer
79 views

Show that if $f \in R_\alpha$,and g increasing and continuous then $ f(g(x)) \in R_{\alpha(g(x))}$

Let $f \in R_\alpha[a,b]$ and $g:[c,d] \rightarrow \mathbb{R}$ continuous and strictly increasing, such that $g(c) = a$ and $g(d) = b$. Prove that $f(g(x)) \in R_{\alpha(g(x))}$. In calculus I ...
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1answer
144 views

Change of variables - Lebesgue-Stieltjes integral

I am trying to find a proof of a result as follows: Let $\rho(\lambda)$ be a real function. Suppose that $\rho(\lambda)$ is monotone increasing and bounded. Suppose that $f(\lambda)$ is measurable ...
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1answer
57 views

Showing that $f \in R_\alpha[a,b]$ and $\lim_{n\rightarrow \infty}\int_a^b f d\alpha_n = \int_a^b f d\alpha $

Let $\left(\alpha_n \right)_{n\in \mathbb{N}}$ a succesion in $BV[a,b]$ and $f:[a,b] \rightarrow \mathbb{R}$ such that $f \in R_{\alpha_n} [a,b]$. If $\alpha \in BV[a,b]$ and $V_a^b(\alpha_n - \alpha) ...
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1answer
88 views

If $f \in BV[0,2\pi], f(0)= f(2\pi)$ Show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural.

This is from Carothers 14.38 If $f \in BV[0,2\pi], f(0)= f(2\pi)$ show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural and $$\left|\int_0^{2\pi}f(x)\sin(nx)dx\right| \leq \frac{V_0^{2\...
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1answer
81 views

Show that $\sum_{{i = 1}}^{n} f(i) = \lfloor n \rfloor f(n)- \int_{1}^{n}f'(x)\lfloor x\rfloor\, dx$

Where $f$ is a function defined in $\mathbb{R}$ with countinuos derivative in all $\mathbb{R}$, for each $n\in \mathbb{N}$ and the function $\lfloor x \rfloor$ is the floor function. I tried using ...