Riemann-Stieltjes integralRiemannStieltjesIntegral2002-07-23 11:05:342009-05-10 20:08:02DefinitionRiemann-Stieltjes sumintegrator% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $f$ and $\alpha$ be bounded, real-valued functions defined upon a closed finite interval $I = [ a, b ]$ of $\mathbb{R} (a \neq b)$, $P = \{ x_{0}, ..., x_{n} \}$ a partition of $I$, and $t_{i}$ a point of the subinterval $[ x_{i - 1}, x_{i} ]$. A sum of the form
$$S(P, f, \alpha) = \sum_{i = 1}^{n} f(t_{i}) (\alpha(x_{i}) - \alpha(x_{i - 1}))$$
is called a \textbf{Riemann-Stieltjes sum} of $f$ with respect to $\alpha$. $f$ is said to be \textbf{Riemann Stieltjes integrable with respect to} $\alpha$ on $I$ if there exists $A \in \mathbb{R}$ such that given any $\epsilon > 0$ there exists a partition $P_{\epsilon}$ of $I$ for which, for all $P$ finer than $P_{\epsilon}$ and for every choice of points $t_{i}$, we have
$$|S(P, f, \alpha) - A| < \epsilon$$
If such an $A$ exists, then it is unique and is known as the \textbf{Riemann-Stieltjes integral of $f$ with respect to $\alpha$}. $f$ is known as the \textbf{integrand} and $\alpha$ the \textbf{integrator}. The integral is denoted by
$$\int_{a}^{b}fd\alpha \quad \textrm{or} \quad \int_{a}^{b}f(x)d\alpha(x)$$