# 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 ...
• 1
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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 ...
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### 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 $h$ ...
1 vote
51 views

### 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, ...
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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 ...
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### 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 ...
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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 ...
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### 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: $\partial_\psi f = - \frac{\nabla^2 \psi}{|\nabla \psi|^2}$ ...
1 vote
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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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### 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 ...
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1 vote
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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 ...
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### 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 ...
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1 vote
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### 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 ...
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181 views