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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52 views

$\lim_{n\to\infty} \sqrt[n]{n!}$ 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 ...
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0answers
41 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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32 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-...
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18 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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2answers
77 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 ...
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0answers
17 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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47 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) = ...
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1answer
38 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 ...
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1answer
58 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 ...
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1answer
22 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 ...
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25 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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43 views

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
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1answer
96 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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1answer
137 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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1answer
62 views

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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1answer
43 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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1answer
67 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 ...
2
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1answer
70 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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21 views

Riemann-Stieltjes integral of a function integer part

Let $g(t)=\lfloor t\rfloor$, and $f(t)=\frac1{t^\alpha+1}$ with $\alpha\neq1$ and $\alpha\in\Bbb R$. Find the value of the Riemann-Stieltjes integral $\int^n_1f(t)\,\mathrm dg(t)$ if it exists, where $...
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1answer
83 views

A monotonically increasing function $g$ as a measure.

I have three simple questions. I'm working with a problem in an old qualification exam, which asks me to express $\mu(E)$ explicitly, within the settings. Settings. Let $g:[0,1]\to \mathbb R$ be be a ...
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15 views

Partial derivative of Ito integral w.r.t. initial value of Ito diffusion

Question: Let $W_t$ be a Brownian motion, $x\in\mathbb{R}$ and $Z_{t,x}$ denote the solution to the SDE $$ dZ_{t,x} = b(Z_{t,x})dt + \sigma(Z_{t,x})dW_t, \qquad Z_{0,x}=x. $$ Under appropriate ...
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0answers
74 views

Calculation of a Expectation

I am not very familiar with the properties of Riemann-Stieltjes Integral and I would like your opinion if I have correctly solved the following exercise. Let $X$ a random variable such that $F(u)=\...
1
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1answer
42 views

Stieltjes integral of product of two functions is zero

Let $\alpha$ be a monotonically increasing continuous function. Suppose $f \in C[a,b]$ satisfies the property $$\int fg \; d\alpha=0, \forall g \in C[a,b]$$ Then show that $f=0$ This is how I tried: ...
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0answers
24 views

Simple Riemann - Stieltjes integral

Calculate $\int\limits_{0}^{\pi}(x-1)d(xsgn(\cos x))$. I used fact that if $f$ is continuous and $g'$ is Riemann integrable over the specified interval, then: $\int f(x)dg(x)=\int f(x)g'(x)dx.$ So for ...
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0answers
31 views

Computing the measure induced by a Riemann-Stieltjes integral

Summary: (1). Is the Statement below true? (2). If so, then how to complete my proof sketched below? (3). Is there any way to compute the integral term in the Statement more explicity? I'm trying to ...
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16 views

$BV$-function and Riemann-Stieltjes integral

Let $\alpha,\beta\in BV([0,1])$. If $\alpha(0)=\beta(0)=0$ and $\int_0^1 f\,d\alpha=\int_0^1 f\,d\beta,\,\forall f\in C([0,1])$, then show that $\alpha=\beta$ a.e. Hint: If $\alpha$ is non-decreasing, ...
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0answers
26 views

How to prove $f(b)g(b)-f(a)g(a)=\int_a^b f(s-)dg(s) + \int_a^b g(s-) df(s) + \sum_{a<s\le b} \Delta f(s)\Delta g(s)$?

The Riemann-Stieltjes integral formula $\int_a^b f(s)dg(s) + \int_a^b g(s)df(s) = f(b)g(b)-f(a)g(a)$, holds only at the case that $f,g$ are continuous and of finite variation. However, if $f,g$ are ...
2
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1answer
33 views

Partition of sum of two bounded functions

Following my previous question about Riemann-Stieltjes integration, I'm asking this problem. Let $f_1$ and $f_2$ be bounded on $[a,b]$ and $\alpha$ is increasing on $[a,b]$. Define $f = f_1 +f_2$. ...
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1answer
31 views

Partition and Riemann-Stieltjes integration

Let $f$ and $g$ be bounded on $[a , b]$ and $g \in \mathcal{R}(\alpha)$ on $[a ,b]$. Also $P$ is an arbitrary partition. If $f\le g$ and $\alpha$ is increasing on $[a , b]$ determine whether the ...
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22 views

Check if the product is completely monotonic function on the interval?

I know that Stieltjes function, say f, is completely monotonic on $(0, \infty)$. I f there is a polynomial, say g, with positive finite derivatives multiply with Stieltjes function. Then, can we say ...
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0answers
51 views

Step Functions and Riemann Stieltjes Integrals

Let $f:[a,b] \to \mathbb{R}$ be bounded, $a < s < b$ and $\alpha :[a,b] \to \mathbb{R}$ be $\alpha(x) = I(x-s)$. Prove $f$ is integrable with respect to $\alpha$,$ f \in \mathcal{R}(\alpha) \iff ...
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0answers
53 views

Sources for multiple Stieltjes integral

I'd like to study more on multiple Stieltjes integral and want to know which sources (books or papers) provide a detailed discussion of multiple Riemann–Stieltjes integral or multiple Lebesgue-...
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1answer
62 views

Integration by Parts proof for Lebesgue-Stieltjes Measure

I'm trying to complete exercise 35 b) on from chapter 3.5 from Folland's Real Analysis that goes like if $F,G$ are NBV and $- \infty < a< b< \infty $ and there are not points in $[a,b]$ ...
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1answer
46 views

$\sup_{\Gamma}\sum |\phi(x_i)-\phi(x_{i-1})|=\infty$ but $\lim_{|\Gamma|\to 0}\sum (\phi(x_i)-\phi(x_{i-1}))$ exists

Let $\Gamma=\{a=x_0,\cdots,b=x_m\}$ a partition of $[a,b]$, is it possible to find a function $\phi$ (any) such that $$\sup_{\Gamma}=\sum_{i=1}^m |\phi(x_i)-\phi(x_{i-1})|=\infty$$ but such that the ...
3
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1answer
79 views

Prove that there is a continuous $f$ not Riemann-Stieltjes integrable with respect to $\phi$

Let $V[\phi;a,b]=\infty$, show there is a continuous function $f$ such that $$\int_a^b f\ d\phi$$ not exists. Info: This result seems easy to follow from the results already proved. But I couldn't ...
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326 views

Riemann-Stieltjes integral of a continuous function w.r.t. a step function

$ \newcommand{\para}[1]{\left( #1 \right)} \newcommand{\abs}[1]{\left| \, #1 \, \right|} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\ds}{\displaystyle} \newcommand{\ol}[1]{\overline{#1}} \...
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1answer
59 views

Locally finite Borel measure on $\mathbb{R}^2$ that is not a product measure.

In the construction of Lebesgue-Stieltjes measures on $\mathbb{R}$, I have learned that a Borel measure that is finite on bounded intervals corresponds to a right-continuous increasing real-valued ...
3
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0answers
99 views

$\int_{-\infty}^{+\infty}f \ d\phi$ exists if $f(x)$ is continuous, $\phi$ of bounded variation

Let $f$ continuous on $(-\infty,\infty)$, $\phi$ of bounded variation on $(-\infty,\infty)$, and $\lim_{|x|\to\infty}\ f(x)=0$ then the following Riemann-Stieltjes integral exists: $$\int_{-\infty}^{...
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0answers
61 views

Riemann–Stieltjes integral in Apostol: is the integrating function monotone?

By defining the Riemann–Stieltjes integral, Apostol in Mathematical analysis (1974), require for the integrand and integrator to be bounded on the definition interval, but I do not see anywhere that ...
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0answers
68 views

Using Riemann–Stieltjes to compute an integral

I am calculating the following Riemann–Stieltjes integral $$\int_0^bx \ d\left(\sum_{k=0}^{[x]}\frac{2^{k}}{k!}\right)$$ where $x>0$ and $[\,\cdot\,]$ is the floor function. My idea is using the ...
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0answers
167 views

Show that $\lim \frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right)=\int_0^1f$ [duplicate]

I want to show that if $f$ is integrable on the interval $[0,1]$ then $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right)=\int_0^1f$. I am using the definition of integrability in the ...
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0answers
21 views

Duhamel's formula for a discontinuous dynamic

Let $f^+,f^- \in \mathcal{C}^1(\mathbb{R}^n,[0,1])$ and we define the following differetial equation, with a dynamic with one discontinuity: $$ \dot x(t)=\begin{cases} f^-(x(t),t), & t\in A\\ f^+(...
1
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1answer
65 views

Proving $E\left[G(X)\right]=\int_0^\infty P(X\ge t)\,dG(t)$ where $X\ge 0$ and $G(\ge 0)$ is an increasing, right-continuous function

Suppose $G$ is an increasing, right-continuous function such that $G(x)\ge 0$ for all $x\ge 0$ and $G(0)=0$. If $X$ is an arbitrary non-negative random variable, then I am trying to show that $$E\...
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0answers
66 views

Integration with Signed Lebesgue-Stieltjes Measure

Let $g$ be a continuous (not need to be monotone) function of bounded variation and $f$ be a measurable function. $m$ is Lebesgue measure on $\mathbb{R}$, and $m_g$ is the signed Lebesgue-Stieltjes ...
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0answers
50 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, ...
1
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1answer
121 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 ...
2
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1answer
94 views

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

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}...
2
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1answer
37 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 $\...
1
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0answers
39 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 ...
0
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1answer
29 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}...