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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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 ...
1 vote
1 answer
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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, ...
0 votes
0 answers
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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}\...
7 votes
2 answers
147 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 ...
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1 vote
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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 ...
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Weak conditions ensuring $\int_0^tm(s)c(x(s))\:{\rm d}s=1-m(t)$

Let $$m(t):=\exp\left(-\int_0^tc(x(s))\:{\rm d}s\right)\;\;\;\text{for }t\ge0.$$ What do we need to assume in order to show $$\int_0^tm(s)c(x(s))\:{\rm d}s=1-m(t)$$ for all $t\ge0$? The claim is ...
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Can we show that if $m$ is differentiable at $0$, then $G(t):=\int_0^th(s)\:{\rm d}m(s)$ is differentiable at $0$ with $G'(t)=h(0)m'(0)$?

If $a,b\in\mathbb R$ with $a<b$, $f\in C([a,b])$ and $g\in C^1([a,b])$, then $$F(t):=\int_a^tf(s)\:{\rm d}g(s)=\int_a^tf(s)g'(s)\:{\rm d}s\tag1\;\;\;\text{for all }t\in[a,b].$$ In particular, $F$ ...
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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
88 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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1 vote
1 answer
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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 $\alpha$ ...
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5 votes
2 answers
148 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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0 answers
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How can we show $\int_0^tM_sc(X_s)\:{\rm d}s=1-M_t$, where $M_t:=\exp\left(-\int_0^tc(X_s)\:{\rm d}s\right)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(E,\mathcal E)$ be a measurable space; $(X_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued process on $(\Omega,\mathcal A,\operatorname P)$...
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0 votes
1 answer
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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\...
0 votes
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23 views

$L^{\infty}([a,b])$ and the Lebesgue-Stieltjes integral

Let $[a,b]$ be a compact interval in $\mathbb{R}$, and $L^{\infty}([a,b])$ the space of all lebesgue measurable functions on $[a,b]$ essentially bounded; my question is whether these functions are ...
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Complex valued Stieltjes integrals : If $f\in\mathcal{R}(\alpha)$, do we have $\mathfrak{Re}(f), \mathfrak{Im}(f)\in\mathcal{R}(\alpha)$?

Given a bounded complex function $\alpha:[a,b]\to\mathbb{C}$, we can define the Riemann-Stieltjes integral of $f:[a,b]\to\mathbb{C}$ (also bounded) in a way that is very much analogous to the usual ...
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Stieltjes integral and Riesz representation theorem

Would you please explain how I can derive eq(1.2) from eq (1.1) by using Stieltjes integral in the Riesz representation theorem?? And is it necessary to suppose that the time interval begins from -$\...
0 votes
0 answers
23 views

Total variation of $F:[a,b]\to\mathbb{R}^{N}$ where each component of $F$ is given by a definite integral

Let $\alpha:[a,b]\to\mathbb{R}$ be monotone increasing and let $\mathcal{R}(\alpha)$ be the set of all functions $f:[a,b]\to\mathbb{R}$ that are Riemann-Stieltjes integrable with respect to $\alpha$. ...
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1 vote
0 answers
29 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"),...
0 votes
0 answers
35 views

Stieltjes integral - IMC

Why in the solution of exercise 10 https://www.imc-math.org.uk/imc2018/imc2018-day2-solutions.pdf, we have the following part: Denote by $N(r)$ the number of lattice points in the open disk $x^ 2 + y^ ...
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2 votes
1 answer
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Lebesgue-Stieltjes integral and Dynkin $\pi-\lambda$ theorem

I am studying the Lebesgue-Stieltjes integral from this PDF: https://www.math.utah.edu/~li/L-S%20integral.pdf. In Theorem 8 the authors claim to use Dynkin's theorem in a way that I do not understand. ...
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1 vote
0 answers
40 views

Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals?

I am wondering whether the following result is true: Let $F: [0, m] \rightarrow [0, 1]$ be a non-decreasing right-continuous function and $q: [0, m] \rightarrow \mathbb{R}_+$ be measuable. Suppose ...
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32 views

Lebesgue-Stieltjes Integration

Question Let $\alpha:[a,b]\rightarrow\mathbb{R}$ be a function with finite total variation. Let $F:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable on [a,b]. Denote $\Delta\alpha(s):=\alpha(s)-\...
2 votes
0 answers
23 views

If $M,N$ are martingales, show that $\operatorname E\left[M_tN_t\right]=\operatorname E\left[\sum_{s\in(0,\:t]}\Delta M_s\Delta N_s\right]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $M,N\in\mathcal V$ (see definition below$^2$) be càdlàg $\...
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0 votes
1 answer
89 views

Total variation of Lebesgue-Stieltjes integral

Setting We work on a filtered probability space. Let $A$ be a process of finite variation (FV) started from $0$ and $H$ be predictable, positive and such that $H\cdot A$ is defined $\omega$-wise as a ...
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1 vote
0 answers
45 views

Lebesgue–Stieltjes integration with discontinuous function

My question comes from the book Credit risk modeling: Valuation and hedging by Tomasz R. Bielecki and Marek Rutkowski, Page 137 at the bottom. It is easily described as follows: Consider a probability ...
1 vote
0 answers
33 views

How do I integrate a not necessarily differentiable multivariate (Riemann)-Stieltjes integral?

I am having a bit of trouble grasping the way of calculating a Stieltjes integral where the differential term might not be at least once differentiable with respect to every variable. I have a good ...
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0 votes
1 answer
74 views

Regarding Theorem 1.9, Conway, complex Analysis

I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable. I know that the estimation of the integral in equation 1.10, comes from the previous ...
1 vote
0 answers
42 views

Convert probability into an integral with respect to a distribution function

I'm studying a proof and have problems with a step where a probability is converted into a Lebesgue Stieltjes integral with respect to a distribution function. Let $(X_i)$ be a sequence of independent,...
0 votes
1 answer
48 views

Resources for Riemann-Stieltjes or Lebesgue-Stieltjes integrals

I'm an engineer, who works with retarded delay differential equations. So far I've encountered systems with single or multiple point-wise delays, and systems with distributed delays. Someone on a ...
1 vote
0 answers
34 views

Decomposition of propagation function

In the book of Jean Prüss (Evolutionary Integral Equations and Applications, 2012, pg. 111) it says in the Proposition 4.10... Let $dc$ be completely positive, let $w(t;\tau)$ denote the associated ...
-3 votes
1 answer
97 views

Let $f(x) = x^2$, and define $\alpha$ as follows, find $100\int_{-1}^{100}f\ d\alpha$.

Making edits to this question. I think I may have figured out how to do the first part of the question Based on this website (http://www.stat.rice.edu/~dobelman/notes_papers/math/Riemann.Stiltjes.pdf)...
0 votes
2 answers
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$\lim_{n\to\infty} \sqrt[n]{n!}$ and $\lim_{n\to\infty}\frac{1}{n} \sqrt[n]{n!}$ with differentiation and integration tools.

Find the following limits: $$\lim_{n\to\infty} \sqrt[n]{n!} \hspace{3cm} \lim_{n\to\infty}\frac{1}{n} \sqrt[n]{n!}$$ This exercise are in my homework of Real Analysis about Riemann-Stietljes ...
2 votes
0 answers
50 views

Law of Iterated Expectation for RVs with Stieltjes Integral.

I want to show that the law of iterated expectations $E[E[X|Y]] = E[X]$ holds for RVs that are not discretely or continuously distributed. In specific in our class we have defined the expectation of X ...
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0 votes
0 answers
36 views

Calculate $\int_{(-\infty,1)} xdm_f(x)$ for certain $f$

I've been solving some problems from my Functional Analysis course, and I want to check if my approach to this exercise is correct (and how to properly end it). It goes like this: Define the Lebesgue-...
0 votes
0 answers
21 views

Question about the Lebesgue-Stieltjes integration over subintervals: $\int_J f(s) dZ(s)$

-Proposition: Let $I \subset \mathbb{R} $ be an interval and $Z: I \to \mathbb{R}$ monotonically increasing and rightside continuous. If $I=(a,b)$ or $I=(a,b]$ then there exist a unique measure $m_Z$ ...
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0 votes
2 answers
112 views

A visual intuition of a proof of Riemann integrals.

Consider the following theorem: Let $f:[a,b]\to \mathbb R$ be a bounded function continuous at all but finitely many points,then $f$ is Riemann integrable. The proof I found in textbooks is not very ...
0 votes
0 answers
32 views

Lebesgue-Stieltjes integral for discrete function $Z$

Let $Z(t):= \sum_{i=0}^\infty z_i \mathbb{1}_{[i,\infty)}(t)$ with $z_i\geq 0$ and $g:[0,\infty)\to [0,\infty)$ be a (Borel-) measurable function. Why do we obtain that $$\int_{[0,t]} g(s) \; dZ(s)= \...
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7 votes
0 answers
62 views

Why is the measure determined by the Stieltjes inversion formula?

$f$ is a Herglotz-Nevanlinna function if it is analytic in the open upper half plane and the imaginary part $\Im f (z) \geq 0$ for $\Im z >0$. $f$ has the following integral representation $$f(z) = ...
1 vote
1 answer
49 views

Integrable w.r.t every bounded monotone increasing function, then continuous

I'm working on a problem, stated as follows: If $f$ is integrable with respect to every bounded, monotone increasing function $g$ on $[a, b]$, then is $f$ continuous on $[a, b]$? I have proved that ...
1 vote
1 answer
66 views

Using Bernstein's Theorem to conclude this equality

in a part of book of Prüss (Evolutionary integral equations and applications pg. 99) it say..."Since $b$, $c \in \mathcal{BF}$ (Bernstein functions) by Bernstein's Theorem exist a function $\beta ...
1 vote
1 answer
27 views

Function completly monotonic

I am reading the book of Püss (Evolutionary Integral Equations and Applications pg. 101) about subordination principle and it have one thing that i dont understand. If $c$ is a completly positive ...
0 votes
0 answers
30 views

Why the weight function should be nonzero when the interval is $[0,\infty)$?

In Riemann-Stieltjes integral, $ \int_{0}^{\infty} f(x) w(x)dx$ why the weight function should be nontrivial when the interval is $[0,\infty)$?
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0 answers
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Lebesgue-Stieltjes integral for the expected value of a die

I am learning Lebesgue–Stieltjes integration and decided to test myself computing the expected value of a 6-sided die roll "from scratch". I know to obtain the expected value I need to ...
5 votes
1 answer
131 views

Confused about the derivation of the identity: $\ln(\zeta(s))=s\int_{0} ^ {\infty} J(x)x^{-s-1}dx$ and it's relationship to Stieltjes integrals

In the book Prime Obsession by John Derbyshire the following identity is explained: $$\ln(\zeta(s))=s\int_{0} ^ {\infty} J(x)x^{-s-1}dx\tag{1}$$ Where J(x) is the Riemann Prime Counting Function. For ...
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2 votes
1 answer
152 views

How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation?

How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation? I read this in many sources but I can not find any explanation of this term. I know ...
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1 answer
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Assume that $\int_a^b f(t)dα(t)$ exists for every increasing $\alpha$. Show that $f$ is continuous on $[a, b]$. [closed]

Let $f$ be a real function defined on $[a, b]$. Assume that the Riemann-Stieltjes integral $\int_a^b f(t) d \alpha(t)$ exists for every increasing function $\alpha$. Show that $f$ is continuous on $[a,...
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1 vote
1 answer
51 views

Riemann/Stieltjes sum - $dt^2$?

Let $0=t_0<t_1<\cdots<t_n=T$ be a partition of the interval $[0,T]$. Denote $\Delta t_k\equiv t_{k+1}-t_k$ for every $0\leq k<n$. Assume $\lim_{n\to\infty}\Delta t_k=0$. Find the limit of $...
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0 votes
1 answer
68 views

Calculate $\sup_{f \in \mathcal{F}} \int_{-1}^1 f dg$ for $g=x^2$.

The following is from Bruckner's Real Analysis book (Ex.12:8.3): Let $g(x)=x^2$ on $[−1, 1]$, and let $\mathcal{F} = {\{f \in \mathcal{C}[−1,1] : |f(x)| ≤ 5 \ \text{for all} \ x∈[−1,1]}\}$. Calculate ...
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0 votes
1 answer
72 views

Show that $|\int_a^b f dg | \le \|f\|_{\infty} \|g\|$

The following is from Bruckner's Real Analysis : I could make a rigorous argument for items (a) and (b), but although intuitively item (c) seems to be reasonable but I cannot make a rigorous proof ...
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