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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Expression of Riemann Stieltjes Integral

Suppose I have a function $\gamma_t$ which is right continuous with left limits. Allow $\gamma_{t^-}$ to define the left limit at $t$ I have the following integral $I = \int_{0}^T \gamma_{t^-} dt$ ...
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40 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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96 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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20 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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50 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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114 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
42 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
23 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
58 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
34 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
39 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
52 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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74 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 ...
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47 views

If $f$ is nondecreasing and $h$ is of bounded variation with $|h(t)-h(s)|≤C\sqrt{f(t)-f(s)}$, then $\int|X||{\rm d}h|≤C\sqrt{\int|X|^2\:{\rm d}f}$

Let $f,g:\mathbb R\to\mathbb[0,\infty)$ be nondecreasing and right-continuous and $h:\mathbb R\to\mathbb R$ be right-continuous and of bounded variation with $$\left|h(t)-h(s)\right|\le\sqrt{f(t)-f(s)}...
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82 views

What is the usual definition of the spectral measure for a nonnegative self-adjoint operator on a Hilbert space?

Let $H$ be a $\mathbb R$-Hilbert space and $(H_\lambda)_{\lambda\ge0}$ be a spectral decomposition of $H$ (see below). Now, let $$\mathcal D\left(A_\varphi\right):=\left\{x\in H:\int_0^\infty\varphi(\...
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45 views

Riemann-Stieltjes integral with respect to a product of functions.

Let $f, g_1, g_2:[a,b] \rightarrow \mathbb{R}$ be continuous and of bounded variation functions on $[a,b]$. Show that $$\displaystyle \int_a^b fd(g_1g_2) = \displaystyle \int_a^b fg_1dg_2 + \...
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55 views

Extensions of Riemann-Stieltjes without continuity problems

Are there any extensions of Riemann-Stieltjes integration that are able to handle the following integral? $\int_0^1 \alpha \space d\alpha$ where $ \alpha(x) = \left\{ \begin{array}{lr} 0 & ...
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1answer
26 views

Riemann-Stieltjes integrable and Right-continouos function

Let $\alpha \colon [0,1] \to \mathbb{R}$ with $\alpha (x)=0$, if $0 \leq x \leq 1/2$ and $\alpha (x)=1$, if $1/2 < x \leq 1$. I have not been able to prove that if $f$ is Riemann-Stieltjes ...
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1answer
47 views

Asymptotics of integrals with respect to asymptotically equivalent distribution functions

Let us assume that $f\colon[0,\infty)\to[0,\infty)$ is a non-decreasing continuous function with $$ \frac{f(t)}{t}\xrightarrow{t\to\infty}1. \tag{A}\label{A}$$ We then have the following elementary ...
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142 views

why is the Lebesgue-Stieltjes integral well-defined?

A function $g: [a,b] \rightarrow \mathbb{R}$ a said to be of bounded variation on the interval $[a,b]$ if $$ \sup_{P: a=x_0 < x_1 \ldots < x_i < \ldots < x_{n_P}=b} \sum_{i=1}^{n_P}...
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253 views

Riesz representation theorem for $C([0,1])$

i’m trying to prove the special case of Riesz representation theorem: Every positive (non-negative on non-negative functions) linear continuous functional $\phi$ on the normed space $C([0,1])$ is ...
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1answer
169 views

Continuity of Lebesgue Stieltjes integral

I am trying to prove that Lebesgue-Stieltjes integral defines a cadlag function (i.e. right continuous with left limits) when its integrator is a cadlag function. Assume that $A(s)$, $s\in \mathbb{R}...
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1answer
37 views

Problem 3. Page 240. Barry Simon. (Associated Lebesgue-Stieltjes measure )

In $[0,1]$ let $M$ consists of all finite unions of sets of the form $[c,d)$ where $c,d$ are rational, or $[c,1]$ where $c$ is rational. ($M$ is a algebra). Given a monotone function $\bar{\alpha}$, ...
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2answers
146 views

(Baby Rudin) ch6 Theorem 6.11 (The Riemann-Stieltjes Integral)

The theorem and the proof above is from Rudin. I have a question about the last part of the proof. For the inequality $\sum_{i \in A}(M_i^*-m^*_i)\Delta\alpha_i+\sum_{i \in B}(M_i^*-m^*_i)\Delta\...
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1answer
32 views

Stieltjes integral of proportion function

Given a finite set of real numbers $x_1\leq x_2\leq ... \leq x_n$, define the function: $F(x):= \frac{1}{n} \underset{k=1}{\overset{n}{\sum}} \chi_{(-\infty, x_k]}(x)$ Considering it's induced ...
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1answer
89 views

Undersanding definition of Riemann-Stieltjes Integral used in Edwards book

I'm trying to find an explaination of the definition of Riemann-Stieltjes Integral used on page 22 of Edwards book [RZ]: This can also be accessed hopefully legally here: Zeta 1) The question is ...
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1answer
67 views

Help understanding Riemann sum to integral method in proof.

Given Abel's Identity according to Apostol [1] it follows that, $\sum_{y <n\leq y} a(n) f(n) = A(x) f(x) - A(y) f(y) - \int\limits_y^{x} A(t) f'(t) dt\quad\quad\quad$ where $A(x) = \sum_{n\leq y}...
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3answers
60 views

How do you compute the following integral $\int\limits_{-\infty}^{+\infty} x^2 e^{-x^2/2} dx$?

I have a problem in the proof for variance of normally distributed random variable and my notes just report this: \begin{align} & \frac{\sigma^2}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty x^2e^{-\...
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1answer
169 views

What is a jump function, and which measure does it induce?

I'm facing now for the first time the topic in the title, and found myself having hard times to figure out what a jump function is. If you look for anything on google you'll always find stuff about C++...
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1answer
96 views

prove $\sum_p \frac{\log p}{p^s} = \int_{1}^{\infty}\frac{d\theta(x)}{x^s}$

I am currently studying Zagier's paper, Newman's Short Proof of the Prime Number Theorem, found here https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf . In proof (V) of his ...
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193 views

Calculate a Riemann Stieltjes Integral $\int_0^3 x d([x] - x) = 3/2$.

I need to prove that $$\int_0^3 x d([x] - x) = \dfrac32$$ However I can not think of a change of variable that can be used so I tried to approach it using Riemann Sums $\sum_{i=0}^3(\alpha(i)-\...
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1answer
82 views

How to solve Stieltjes integral $\int_0^n f(x) d \lfloor x \rfloor$?

I'm trying to solve the following $$ \int_0^n f(x) d \lfloor x \rfloor $$ where $\lfloor x \rfloor$ is the floor function, $f : [0,n] \rightarrow \mathbb{R}$ is continuous, and $n \in \mathbb{N}$. ...
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1answer
100 views

Convergence of Riemann-Stieltjes sums

Let $f$ be Riemann-Stieltjes integrable with respect $G$ (increasing function). My definition for Riemann-Stieltjes integration is: for every $\epsilon$ there is a partition $\mathcal{P}_\epsilon$ ...
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0answers
93 views

Extending laws for Riemann integral to Riemann-Stieltjes integral

I was reading Terence Tao's notes on Analysis. He says Theorem 13(g) cannot be extended from Riemann integral to Riemann-Stieltjes integral: Most (but not all) of the remaining theory from Week 9 ...
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1answer
78 views

How to calculate this Riemann-Stieltjes integral?

How to calculate this Riemann-Stieltjes integral? \begin{equation} \int_{1}^{3}e^{x}d\left\lfloor x\right\rfloor \end{equation} If $x\in\left[0,\,3\right]$, then $\left\lfloor x\right\rfloor =I\...
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1answer
36 views

Calculating expectation w.r.t. the empirical dist. fcn

I am trying to evaluate the plug-in estimator $$\hat{\theta} = T(F_n)$$ where $F_n (t) = \frac{1}{n} \sum_{i = 1}^{n}1\{t_i \leq t\}$ and $T : \eta \to \int_{}^{}x \ d\eta(x) $, (Riemann-Stieltjes ...
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1answer
121 views

What to do first: Riemann integral or Riemann-Stieltjes integral?

I will soon (self)study the topic of integration in depth, for which I'm planning to use Rudin. However, Rudin seems to treat the Riemann-Stieltjes integral, which seems more advanced and general than ...
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123 views

Riemann-Stieltjes integral $\int\limits_a^b f(x) \, \mathrm{d}g = \frac{\log ^2(n)}{2 \log (10)}|_a^b$

I'm having trouble with this integral: $\int\limits_a^b f(x) \, \mathrm{d}g = \frac{\log ^2(n)}{2 \log (10)}|_a^b$ If I want to find the product such: $10^m.10^{m+1}...\leq n$ so for example for $...
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1answer
59 views

Uniqueness of Riesz Representation of $C^{*}[a,b]$

I have seen this statement: The dual space $C^{*}\left[a,b\right]$ of $C\left[a,b\right]$ is isometrically isomorphic to $BV_{0}\left[a,b\right]$. where $$BV_{0}\left[a,b\right]=\left\{ \alpha\in BV\...
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20 views

How to obtain the weight function p(t) of a Stieltjes integral F(x)?

What is the procedure to find the weight function $p(t)$ of a Stieltjes integral $F(x)=\int_0^\infty p(t)/(1+xt)dt$ such that $F(x)=\sum_{n=0}^\infty a_n (-x)^n$ and $a_n=\int_0^\infty t^n p(t)dt$? ...
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118 views

Prove that Riemann-Stieltjes integrability transfers to subintervals, using the partition norm definition of the RS-integral

I posted this question about a month ago: If $f$ is $g$-Riemann-Stieltjes integrable on $[a,b]$, prove that it's $g$-RS-integrable on $[a,c] \subset [a,b]$, and RRL explained to me that there are ...
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1answer
40 views

Riemann - Stieltjes ingrability proof with a monotone creasing function

If $f:[a,b] \to \mathbb R$ is discontinuous in $c \in [a, b]$, c an arbitrary element of the interval then, exist a $g$ function monotone creasing such that $f$ is not $g$-integrable. I'm trying to ...
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1answer
185 views

If $f$ is $g$-Riemann-Stieltjes integrable on $[a,b]$, prove that it's $g$-RS-integrable on $[a,c] \subset [a,b]$

So, the problem, as in the title, is: If $f$ is $g$-Riemann-Stieltjes integrable on $[a,b]$, $g \in BV[a,b]$, prove that it's $g$-RS-integrable on every subinterval $[a,c] \subset [a,b]$, where $a &...
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63 views

Commutativity and associativity in Stieltjes convolution algebra

I've been trying to prove the commutativity and associativity within Stieltjes convolution algebra but haven't succeeded. In the literature it says that the convolution should be both commutative and ...
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1answer
32 views

Understanding the derivation of $\sum_{n\leq N}\frac{1}{n^\alpha} = \int_1^N\frac{1}{x^\alpha}d[x]+1.$

Use Riemann-Stieltjes Integral to derive $$\sum_{n\leq N}\frac{1}{n^\alpha} = \int_1^N\frac{1}{x^\alpha}d[x]+1.$$ Using this, show that if $\alpha>1$, the series $\displaystyle \sum_{n=1}^{\infty}...
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23 views

Estimating a difference with a Riemann-Stieltjes integral.

I'm having trouble with this problem Let $X\in \mathcal{C}^{\alpha}[0,T]$ and $Y\in\mathcal{C}^{\beta}[0,T]$ with $\alpha+\beta>1$. (Here $\mathcal{C}^{\gamma}[0,T]$ is the space of Hölder ...
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0answers
50 views

Showing that an operator between Hölder spaces is a contraction

I'm having trouble with the following problem: Consider $\beta\in(\frac{1}{2},1]$, $\xi\in\mathbb{R}$, $f:\mathbb{R}\longrightarrow \mathbb{R}$ bounded with first and second derivatives bounded, $X\...
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1answer
166 views

Show this function is Riemann-Stieltjes Integrable (RS-I)

If $f(x) = x- \left \lfloor{x}\right \rfloor $ and $\alpha(x) = x^2$, show that f is Riemann-Stieltjes Integrable on [-1,2]. Every proof in my textbook seems to show that $\alpha$ is increasing on ...
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1answer
402 views

Derivative of a Riemann–Stieltjes integral

Suppose we have smooth real functions $f,g$ such that the Riemann–Stieltjes integral $\int_0^t f(s) dg(s)$ is defined for all $t>0$ and is smooth as a function of $t$. Is their an analytic formula ...
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70 views

Differentiability of Conditional Expectation

In short, I want to examine whether the expression $$E\left[g(x,y)|y\right]=\int_Xg(x,y)dF(x,y)$$ is differentiable in $y$. Given an event space $X\subset\mathbb{R}$, the function $g:X\times \mathbb{...