# 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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### 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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### 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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### 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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### 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 ...
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 ...
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{...