Colloquium: Everything I knew about the Stieltjes integral was wrong

Abstract:  

The Stieltjes integral,

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\[ \int_a^b f(x)\,d\alpha, \]

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is a generalization of the Riemann and Darboux

integrals, using a function $\alpha$ of bounded variation to define

the ``length'' of intervals. It was first introduced by Stieltjes to

study continued fractions, and was then used by Riesz to characterize

the dual of $C[a,b]$. Since then it has found application in a number

of areas. Throughout the 20th century it was a staple of

undergraduate analysis courses (see, for instance, Apostol, Bartle,

Burkill \& Burkill,

Protter \& Morrey, Rudin) but more recently has become less common.

 

The original definition, modeled on the

Riemann integral, had the problem that it was undefined if $f$ and

$\alpha$ shared a common discontinuity, and several alternative

definitions were proposed to overcome this, none fully successfully. We will discuss these various

definitions, the criterion for being integrable with respect to each

definition, and their relationship to one another. I will include a

definition I recently introduced, partly for

pedagogical reasons. The talk will conclude with a

new definition that I claim is the ``right'' definition of the

Stieltjes integral. I will justify this by reviewing its key properties.

 

This talk is based on joint work with my former students Greg

Convertito and Jacob Glidewell, and is a

commercial (only slightly abashed) for my recent book, {\em The Stieltjes Integral}, joint

with Greg Convertito.

Host: Brandon Sweeting

 

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00 - 3:00 pm.