# 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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### $\lim_{n\to\infty} \sqrt[n]{n!}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $\sup_{\Gamma}\sum |\phi(x_i)-\phi(x_{i-1})|=\infty$ 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 ...
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### 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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### 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 ...
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}...