The Unapologetic Mathematician

Mathematics for the interested outsider

Increasing Integrators and Order

For what we’re about to do, I’m going to need a couple results about increasing integrators, and how Riemann-Stieltjes integrals with respect to them play nicely with order properties of the real numbers.

When we consider an increasing integrator we have a certain positivity result: if the integrand is nonnegative and the integral exists, then it is nonnegative as well. That is, for \alpha increasing and f(x)\geq0 on \left[a,b\right] we have \int_{\left[a,b\right]}f\,d\alpha\geq0 as long as it exists. This should be clear, since every Riemann-Stieltjes sum takes the form

\displaystyle f_{\alpha,x}=\sum\limits_{i=1}^nf(t_i)\left(\alpha(x_i)-\alpha(x_{i-1})\right)\geq0

where the inequality follows because each value f(t_i) and each difference \alpha(x_i)-\alpha(x_{i-1}) is nonnegative. Thus the limit of the sums must be nonnegative as well. From this and the linearity of the integral we see that if \alpha is increasing and f(x)\geq g(x) on \left[a,b\right], then we have the inequality

\displaystyle\int\limits_{\left[a,b\right]}fd\alpha\geq\int\limits_{\left[a,b\right]}gd\alpha

as long as both integrals exist.

Now, when we talked about absolute values — the metric for the real numbers — we saw that the absolute value of a sum was always less than the sum of the absolute values. That is, |x+y|\leq|x|+|y|. And since an integral is just a limit of sums, it stands to reason that a similar result would hold here. Specifically, if \alpha is increasing and f is integrable with respect to \alpha on \left[a,b\right], then so is the function |f|, and further we have the inequality

\displaystyle\left|\int\limits_{\left[a,b\right]}fd\alpha\right|\leq\int\limits_{\left[a,b\right]}|f|d\alpha

Indeed, given a partition of \left[a,b\right] the difference M_i(f)-m_i(f) between the supremum and infimum of f on the ith subinterval is the supremum of f(x)-f(y), where x and y range across \left[x_{i-1},x_i\right]. Then, adapting the above inequality we see that

||f(x)|-|f(y)||\leq|f(x)-f(y)|

and so we conclude that

M_i(|f|)-m_i(|f|)\leq M_i(f)-m_i(f)

Then we can multiply by \alpha(x_i)-\alpha(x_{i-1}) and sum over a partition to find

U_{\alpha,x}(|f|)-L_{\alpha,x}(|f|)\leq U_{\alpha,x}(f)-L_{\alpha,x}(f)

Riemann’s condition then tells us that |f| is integrable, and the inequality follows by the previous result.

We might hope to extend these results to integrators of bounded variation, but it won’t work right. This is because we go from increasing functions to functions of bounded variation by subtracting, and this operation will break the order properties.

March 18, 2008 Posted by | Analysis, Calculus | 1 Comment

Plans for tomorrow

I’ve been going over notes in preparation for tomorrow’s talk at the University of Pennsylvania (scroll down a bit).

For anyone who happens to be there (Isabel, Charles…) I’ll be heading out from a little south of Baltimore early enough to (hopefully) compensate for the fact that I-95 is closed a little north of the exit for UPenn. I should be there in plenty of time for lunch with Jim Stasheff, and dinner later on. Drop an email (if you remember that I teach at Tulane it’s not too hard to find the address) with any contact information you want to pass along.

March 18, 2008 Posted by | Uncategorized | 1 Comment

Category Theory Isn’t Useless After All!

Today on the arXiv, we find a posting of an old paper, which uses spans of “reflexive graphs” to give an algebraic framework for describing partita doppia — double-entry bookkeeping.

Now I need to find a follow-on to this paper and start applying to those financial math jobs.

March 18, 2008 Posted by | Algebra, Category theory | Leave a comment

   

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