1

Question:

Not all integrable functions are bounded, yet in the book I am studying, integration is defined with respect to bounded functions. Why is this?

Background:

I have been studying the Riemann-Stieltjes integral in Foundations of Mathematical Analysis by Richard Johnsonbaugh.

He gives the following definition for the Riemann-Stieltjes integral:

Let $f$ and $\alpha$ be bounded functions on $[a,b]$. We say that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a,b]$, and write $f \in \mathscr{R}_{\alpha} [a,b]$, if there exists a number $I$ having the property that for every $\epsilon>0$, there exists a partition $P$ of $[a,b]$ such that if $P^*$ is a refinement of $P$, then $$|S(f,P^*,T)-I|<\epsilon$$ for any points $T$.

EssentialAnonymity
  • 1,489
  • 15
  • 25
  • 2
    That definition of the Riemann-Stieltjes integral can never be satisfied if $f$ is unbounded. This is proved here. A function might be Lebesgue integrable or improperly Riemann integrable, but never Riemann integrable if it is unbounded. You need the broader theory of integration to handle unbounded functions and/or unbounded intervals – RRL Nov 26 '17 at 00:42
  • RRL to the rescue again! I see now that my question is a duplicate, so I'll leave it to the mods to delete or not. – EssentialAnonymity Nov 26 '17 at 00:58

0 Answers0