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# Riemann-Stieltjes integral

Let $f$ 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 Riemann-Stieltjes sum of $f$ with respect to $\alpha$. $f$ is said to be 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 Riemann-Stieltjes integral of $f$ with respect to $\alpha$. $f$ is known as the integrand and $\alpha$ the integrator. The integral is denoted by

$\int_{{a}}^{{b}}fd\alpha\quad\textrm{or}\quad\int_{{a}}^{{b}}f(x)d\alpha(x)$ |

## Mathematics Subject Classification

26A42*no label found*

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## Corrections

minor changes by CWoo ✓

One more by CWoo ✓

Riemann--Stieltjes integrable by pahio ✓

defines by pahio ✓