3
$\begingroup$

Lebesgue's criterion for Riemann-integrability:

Let $f$ be defined and bounded on $[a,b]$ and let $D$ denote the set of discontinuities of $f$ in $[a,b]$.Then $f\in R$ on $[a,b]$ if,and only if,$D$ has measure zero.

My question is,does the Lebesgue's criterion for Riemann-integrability also hold for Riemann-Stieltjes integral?That is,does the following hold?

Let $f$ be defined and bounded on $[a,b]$ and let $D$ denote the set of discontinuities of $f$ in $[a,b]$.Then $f\in \mathcal{R}(\alpha)$ on $[a,b]$ if,and only if,$D$ has measure zero.($\alpha$ is a monotonically increasing function on $[a,b]$).

  • 0
    In the case of R.-S., you want to replace *measure zero* with *countable*. For details, see here: http://en.wikipedia.org/wiki/Riemann_integral.2012-06-22
  • 0
    @William Are you sure that *countable* gives "if and only if"? Certainly not "only if", because $\alpha(x)$ could be $x$, for example. And not "if" either, because if $f$ is discontinuous just at one point which happens to have positive mass, the integral does not exist.2012-06-22
  • 0
    @LeonidKovalev: Perhaps a better conjecture would be that $f \in \mathcal{R}(\alpha)$ iff $D$ has $\alpha$-measure zero. Of course, one must correct this based on your discussion below about one-sided continuity.2012-06-22

1 Answers 1