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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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The real part of the Stieltjes transform.

Suppose that $\mu$ is a probability measure with pdf f. Consider the Stieltjes transform of $\mu$ being $$G(z) = \int\frac{\mu(d\tau)}{\tau -z}.$$ We know that the imaginary part of $G(z)$ has an ...
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Chain rule for integrator function with Riemann-Stieltjes integral

I have a random variable $X$ with distribution function $F$. I am interested in evaluating the integral $$ \int g(x) \beta'(F(x)) F(dx) $$ where $\beta$ is smooth and monotone and $g$ is such that the ...
Masanja M.'s user avatar
3 votes
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62 views

uniqueness of the inversion to Riemann-Stieltjes integral equation

I believe that if, for a Riemann-Stieltjes integral with $h(s)$ of bounded variation, $$ \int_0^1 s^\alpha dh(s) = 0 \qquad\text{for any }\alpha\in(\alpha_0,\alpha_1) , \tag{1}\label{eq1}$$ then $h$ ...
Martin Lanzendörfer's user avatar
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Riesz representation theorem with Stieltjes integral?

I am a bit new to functional analysis and I stumbled upon this problem that confuses me. Consider the space $X$ of bounded, non-decreasing, right-continuous functionals $F$ on $[0,1]$. Specifically, ...
Ruben van Beesten's user avatar
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Show that a locally finite measure is a Stieltjes measure

A locally finite measure $\mu$ is a measure such that for all $x \in \mathbb{R}$, there exists $\epsilon > 0$ such that $\mu(]x-\epsilon;x+\epsilon]) < + \infty$. My goal is to show that a ...
Alex's user avatar
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1 answer
101 views

How $|S(x_n, \xi_n) - S(x'_n, \xi'_n)|$ being arbitrary small implies existence of Stieltjes-Riemann Integral?

I am studying Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan (E-book). In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous ...
Ali's user avatar
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Is the integrator of a Riemann--Stieltjes integral necessarily of bounded variation?

According to https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral, if $g$ is a real-valued function of bounded variation and $f$ is another real-valued function, then one can define the ...
xy z's user avatar
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41 views

Integral of multivariable fuction with function differential

I want to solve for $f = f(\psi)$. $\psi = \psi(x,y)$ is a known function in $[0,2\pi] \times [0,2\pi]$. I have the following equation: $\partial_\psi f = - \frac{\nabla^2 \psi}{|\nabla \psi|^2}$ ...
Francisco Sáenz's user avatar
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Does $-\int_{[0,\infty)}y^p d(1-F)(y)=-\int_{[0,\infty)}y^p dT(y)$ hold true for $T(y)=P(Y\ge y)$?

There is a calculation step in Lemma 5.4 of Probability Theory by Varadhan I am wondering about. He defines the distribution function as $F(x):=P(X\le x)$ according to Chapter 1.6 and he also ...
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Integrability of vector-valued functions expressed in terms of partitions

Question: Let $\mathbf{f}$ be a function $[a, b] \rightarrow \mathbb{R}^{k}$, and $\alpha:[a, b] \rightarrow \mathbb{R}$ an increasing function. Show that $\mathbf{f}$ is integrable with respect to $\...
hmeng's user avatar
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2 votes
1 answer
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Repeated Riemann-Stieltjes integration.

Suppose $\alpha$, $\beta$ are increasing functions on $[a, b]$ such that $\alpha \in \mathscr{R}(\beta)$, and $f \in \mathscr{R}(\alpha) \cap \mathscr{R}(\beta)$. Show that $\int_{a}^{x} f(t) d \alpha(...
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Integral of a piecewise continuous function - is this correct?

I recently learned about the Riemann-Stieltjes integral, and I tried playing around a bit with it. I realized that you can turn sums into integrals using the floor function, and that's what led to the ...
cdog's user avatar
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Doubt about Rudin exercise 6.3 a

Define three functions $B_1, B_2, B_3$ as follows: $B_j(x) = 0$ if $x < 0$, $B_j(x) = 1$ if $x > O$ for $j = 1, 2, 3$; and $B_1(0) = 0, B_2(0) =1, B_3(0) = \frac{1}{2}$. Let $f$ be a bounded ...
pie's user avatar
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3 votes
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For an arbitrary continuous function $f$, is the Stieltjes integral $\int_0^1(df(x))^3=0$?

Suppose $f:[0,1]\to\mathbb R$ is continuous, possibly with unbounded variation. We consider sums of the form $$\sum_{i=1}^n\Big(f(x_i)-f(x_{i-1})\Big)^3$$ where $0=x_0<x_1<x_2<\cdots<x_{n-...
mr_e_man's user avatar
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For an arbitrary continuous curve $\gamma:[0,1]\to\mathbb R^n$, does the Riemann-Stieltjes integral $\int_0^1\gamma(t)\wedge d\gamma(t)$ exist?

We consider sums of the form $$\sum_{i=1}^m\gamma(t_i^*)\wedge\Big(\gamma(t_i)-\gamma(t_{i-1})\Big),$$ where $\gamma:[0,1]\to\mathbb R^n$ is a continuous function, and $$0=t_0\leq t_1^*\leq t_1\leq ...
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Lebesgue-Stieltjes Integral of Singleton

I have two functions $f,g\in\textit{BV}(K,\mathbb R)$, where $\textit{BV}(K,\mathbb R)$ is the set of all functions mapping $K\subset\mathbb R$ to $\mathbb R$ that are of bounded variation. The ...
Quertiopler's user avatar
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245 views

Hadamard differentiability of function

Let $X$ and $Y$ be Banach spaces. Definition: A function $f:X\rightarrow Y$ is called Hadamard differentiable at $x\in X$ tangentially to $U\subseteq X$ iff $x\in U$ and there exists a continuous ...
Syd Amerikaner's user avatar
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1 answer
86 views

Infimum of the Upper Sums [closed]

Let $P$ be the partition of the interval $[a,b]$. If $P$ is divided into two partitions $P_1$ and $P_2$ such that $P_1$ covers the interval $[a,c]$ and $P_2$ covers the interval $[c,b]$, then $U(P, f, ...
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3 votes
0 answers
63 views

What do the limits represent on a Riemann–Stieltjes integral?

Say I have the integral $$ \int_0^{10} 1\ d(x^2) $$ Are the limits on the range of $x$ or of $x^2$ ? Based on the wiki page I am assuming it is the limits of $x$ in which case I imagine the answer ...
gowerc's user avatar
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Riemann–Stieltjes integral with itself. [closed]

Suppose that $f:[a,b] \to \mathbb R$ is continuous everywhere and non-decreasing. I am wondering that whether $$\int_a^b f(x)^{n-1}df(x) = \frac{1}{n}[f(b)^n-f(a)^n]$$ for $n=1,2,\ldots$. Notice that $...
EconKR00's user avatar
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25 views

Riemann-Stieltjes Integral Does not Add Up

Does there exist functions $f,\phi : [a,b]\rightarrow \mathbb{R}$ and $c\in [a,b]$ such that the Riemann-Stieltjes integrals $\int_a^c f\,d\phi$ and $\int_c^b f\,d\phi$ exist but $\int_a^b f\,d\phi$ ...
Laurence PW's user avatar
1 vote
0 answers
36 views

Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?

Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals? In particular, I am interested in asymptotic behaviour of the Riemann-Stieltjes ...
Mikhail's user avatar
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A Lebesgue Stieltjes integral where the integrator is a Lebesgue Stieltjes integral

Let $f: \mathbb{R} \to \mathbb{R}$ be a Borel measurable and non-negative function; let $g: \mathbb{R} \to \mathbb{R}$ have bounded variation but not necessarily be right-continuous. Then, the ...
user551504's user avatar
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1 answer
341 views

Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform.

Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction $$ \frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$...
xyz's user avatar
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1 vote
1 answer
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Total variation of a class in $L^1 [0, 1]$ and lower semicontinuity

Consider a function $f \in \mathcal L^1 [0, 1]$: we define the total variation of $f$ as usual by $$ V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |, $$ where the supremum is taken ...
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7 votes
1 answer
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Path Signatures and Picard iterations

Recently, I've started studying path signatures and, currently, I'm reading a standard reference, namely "A Primer on the Signature Method in Machine Learning" by Ilya Chevyrev and Andrey ...
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1 answer
80 views

Riemann-Stieltjes Integral for discontinuous integrator

I was wondering if there is a good general technique for evaluating the Riemann-Stieltjes integral in cases where the integrator is discontinuous on a few points. For example: find $$\int_{0}^{2} f(x) ...
random's user avatar
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-1 votes
1 answer
163 views

Proper notation for integration on manifolds?

If I have an integral over a manifold $\mathcal{M}\subset\mathbb{R}^n$, and I have an invertible continuously differentiable map $\varphi:\mathcal{M}\to\mathcal{D}\subseteq \mathbb{R}^d$, and $\...
lightxbulb's user avatar
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5 votes
1 answer
122 views

Riemann-Stieltjes integral with respect to functions equal almost everywhere [closed]

Let $f, g$ be Lebesgue integrable, real-valued functions on $[0, 1]$ with bounded variation, and let $\phi : [0, 1] \to \mathbb R$ be continuous. Assuming that $f = g$ almost everywhere, does one have ...
KCJV's user avatar
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1 vote
1 answer
238 views

Where does the Change Of Variable of Riemann-Stieltjes integral theorem uses continuity of COV function? Isn't the assumption superfluous? Apostol 7.7

Here is theorem 7.7 from Tom Apostol, Mathematical Analysis: Let $f\in \mathfrak R (\alpha)$ on $[a,b]$ and let $g$ be a strictly monotonic continuous function defined on an interval $S$ having ...
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Is the Darboux approach / definition compatible with a bounded integrator in Riemann–Stieltjes integrals? Why these hypothesis in Rudin PMA chapter 6?

I'm reading Rudin's Principles of Mathematical Analysis $6^{th}$ chapter and am wondering why doesn't he require the weaker condition of $\alpha$ (the integrator) being bounded on $[a,b]$ ? Is he ...
niobium's user avatar
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5 votes
1 answer
181 views

(Complex Stieltjes Integral) If $f$ is integrable wrt $\alpha$, is $\overline{f}$ also integrable wrt $\alpha$?

Let $f,\alpha$ be two bounded complex functions on $[0,1]$. We say that $f$ is integrable w.r.t. $\alpha$ iff the Riemann sum $$\sum f(t_i)(\alpha(x_i)-\alpha(x_{i-1}))$$ converges to a fixed number $...
Dilemian's user avatar
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1 vote
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190 views

What is the difference between a line integral and Riemann-Stieltjes integral?

I'm particularly concerned with the visual representation of line integration. The following is provided by Wikipedia while this is shown in the Wikihow page Lastly, the Riemann-Stieltjes integral ...
ric.san's user avatar
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1 vote
1 answer
146 views

Are two Riemann integrable functions, Riemann stieltjes integrable with respect ro each other?

Let f,g be real valued Riemann integrable functions on [a,b] then does it imply, f is Riemann Stieltjes integrable with respect to g on [a,b]? If not please do provide a counter example demonstrating ...
Swag Jensen's user avatar
2 votes
0 answers
64 views

Does the $\Gamma\subset\mathbb{C}$ curve always has to be smooth in Cauchy's integral theorem?

In Cauchy's integral theorem (see: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem), does the $\gamma$ representation of the $\Gamma\subset\mathbb{C}$ curve always has to be smooth (i.e. at ...
Kapes Mate's user avatar
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1 answer
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Stieltjes Integral of Product of Gaussian Processes

Let $(\Omega,\mathcal F, P)$ be a probability space, let $\mathcal G_1$ and $\mathcal G_2$ be two (not necessarily independent) centered real-valued Gaussian processes, i.e., measurable functions $\...
Syd Amerikaner's user avatar
2 votes
1 answer
84 views

LLN and rates in terms of Stieltjes integral

I am learning about Stieltjes integrals. Written in terms of a Stieltjes, the law of large numbers, $\frac1n\sum_{i=1}^n f(X_i)\to E(X)$ as $n\to\infty$, looks like: $$ \int_{-\infty}^{\infty}f(x)dF_n(...
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5 votes
1 answer
808 views

If the Riemann-Stieltjes integral $\int_a^b{f\,dg}$ exists for every $f\in\mathcal{R}[a,b]$, then $g$ is absolutely continuous

In Remark 18.13 of Real Analysis: Foundations and Functions of One Variable, the author states that The class of Riemann integrable functions and the class of absolutely continuous functions are also ...
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1 vote
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A question on integrating a function, with respect to another function?

I've been asked to find the area described by $$A={ (x,y), y^2 \leqslant2x; and, y\geqslant 4x-1 } $$ which is effectively the following area(If I can insert a graph using mathJax, please let me ...
math and physics forever's user avatar
1 vote
1 answer
149 views

How does this theorem on continued fractions relate to the Riemann-Stieltjes integral?

I'm going to include two precursor sections here to introduce all of the content referenced in my actual question (if you're familiar with both continued fractions and the Riemann-Stieltjes integral, ...
joshuaheckroodt's user avatar
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0 answers
39 views

What is the „limit type” in the definition of Stieltjes-integral?

Let's say $f$ and $g$ are „nice enough” to $$\sum_{k}f\left(\xi_{k}\right)\left(g\left(x_{k+1}\right)-g\left(x_{k}\right)\right)\rightarrow\int_{a}^{b}f\left(x\right)dg\left(x\right),$$ where $\xi_{k}\...
Kapes Mate's user avatar
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7 votes
2 answers
489 views

Let $\alpha$ be an increasing function on $[a,b]$. Show that $\int^a_b\alpha d \alpha = \frac{1}{2}[\alpha (b)^2 - \alpha(a)^2]$

I am wanting to try to prove the question below, but there is a step that I can't get pass. I know that the proof is worthless if I assume incorrectly, and should have stopped proving from there, but ...
Reuben's user avatar
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0 answers
55 views

Is a real random variable's moment-generating function the Laplace-Stieltjes transform of its CDF?

Wikipedia gives that the moment-generating function for a real random variable $X$ with cumulative distribution function $F$ equals $M_x(t) = \int \limits_{-\infty}^\infty e^{tx}\, dF(x)$, using the ...
tparker's user avatar
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A special case of the Riemann–Stieltjes integral

A special case of the Riemann–Stieltjes integral is found here. The claim is that if $g(x)$ is continuously differentiable over $\mathbb{R}$, then $$\int_{a}^b f(x)\ \mathrm{d}g(x)=\int_a^bf(x)g^{\...
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4 votes
1 answer
95 views

Is $M_t:=\exp\left(-\int_0^tc(Y_s)\:{\rm d}s\right)$ differentiable at $0$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $E$ be a topological space $(Y_t)_{t\ge0}$ be an $E$-valued right-continuous process on $(\Omega,\mathcal A,\operatorname P)$ $c:E\to[...
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If $g$ is bounded, nondecreasing and right-continuous, how do we show $\int f(t)\:{\rm d}g(t)=\int f(t)g'(t)\:{\rm d}t$?

Let $g:\mathbb R\to\mathbb R$ be bounded, nondecreasing and right-continuous and $\mu_g$ denote the Lebesgue-Stieltjes measure on $\mathcal B([0,\infty))$ associated with $g$. Moreover, let $\lambda$ ...
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Convergence of Riemann-Stieltjes integral of square function.

I got this question which I think I know the solution to but I am not certain I am doing it right, will gladly use some help about it. So the question is like that, assume we have function $\alpha$ ...
Abzikro's user avatar
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2 answers
205 views

How can we prove $\int_a^bf\:{\rm d}g=\int_a^bf(s)g'(s)\:{\rm d}s$ if $g'$ is not continuous?

Let $a,b\in\mathbb R$ with $a<b$, $$\mathcal D_{[a,\:b]}:=\{(t_0,\ldots,t_k):k\in\mathbb N\text{ and }a=t_0<\cdots<t_k\}$$ and $$\mathcal T_\varsigma:=\{(\tau_1,\ldots,\tau_k):\tau_i\in[t_{i-...
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Lebesgue-Stieltjes Integration by parts formula

Let $A,B$ be two right-continuous functions of finite variation. Then the integration by parts formula states: $$ A_{t}B_{t} = A_0B_0+\int_{0}^{t} A_{-s}dB_s \, + \int_{0}^{t} B_{-s}dA_s + \sum_{ s\...
Smasher640's user avatar
1 vote
0 answers
50 views

Sum of a locally BV right-continuous function and a locally BV left-continuous function

Let $f:\mathbb{R}\to\mathbb{R}$ be a right-continuous function that is locally of bounded variation (i.e., of bounded variation on every $[a,b]\subset\mathbb{R}$; henceforth, "LBV function"),...
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