Products of integrable functions

From the linearity of the Riemann-Stieltjes integral in the integrand, we know that the collection of functions that are integrable with respect to a given integrator over a given interval form a real vector space. That is, we can add and subtract them and multiply by real number scalars. It turns out that if the integrator is of bounded variation, then they actually form a real algebra — we can multiply them too.

First of all, let’s show that we can square a function. Specifically, if $\alpha$ is a function of bounded variation on $\left[a,b\right]$ , and of $f$ is bounded and integrable with respect to $\alpha$ on this interval, then so is $f^2$ . We know that we can specialize right away to an increasing integrator $\alpha$ . This will work here (unlike for the order properties) because nothing in sight gets broken by subtraction.

Okay, first off we notice that $f(x)^2$ is the same thing as $|f(x)|^2$ , and so they have the same supremum in any subinterval of a partition Then the supremum of $|f(x)|^2$ is the square of the supremum of $|f(x)|$ because squaring is an increasing operation that preserves suprema (and, incidentally, infima). The upshot is that $M_i(f^2)=M_i(|f|)^2$ . Similarly we can show that $m_i(f^2)=m_i(|f|)^2$ . This lets us write

$\displaystyle M_i(f^2)-m_i(f^2)=M_i(|f|)^2-m_i(|f|)^2=\left(M_i(|f|)+m_i(|f|)\right)\left(M_i(|f|)-m_i(|f|)\right)$
$\leq2M\left(M_i(|f|)-m_i(|f|)\right)$

where $M$ is an upper bound for $|f|$ on $\left[a,b\right]$ . Riemann’s condition then tells us that $f^2$ is integrable.

Now let’s take two bounded integrable functions $f$ and $g$ . We’ll write

$f(x)g(x)=\frac{1}{2}\left(\left(f(x)+g(x)\right)^2-f(x)^2-g(x)^2\right)$

and then invoke the previous result and the linearity of integration to show that the product $fg$ is integrable.

March 20, 2008 - Posted by John Armstrong | Analysis, Calculus

8 Comments »

Typo: 2nd paragraph, 2nd line. I think you mean “…and if $f$ is bounded…”

Comment by Vishal | March 20, 2008 | Reply
I came across your blog, I dont know how to sound less strange but then again I am a strange person. Well most of my friends say that I am intresting, howvever that isnt the point. I cant tell you how much I love math I dont know why but it intrests me. I could sit through math problems for 5 hours before puting them down. I am only 18 and a senoir in high school but my math teachers dont understand that I want to learn more. I ask in Hopes that you may have time and would teach me. I will understand if you wouldn’t want to. I just want to learn “feed my mind” kind of thing. Write me either way Yes or no. So I will know.
thank you either way…. 🙂

Comment by Kasey | March 21, 2008 | Reply
Kasey, I understand what you mean, and I’m sorry that your teachers aren’t in a better position to show you where to go next. If you don’t have easy access to a nearby college or university, you’re sort of stuck until you leave high school.

But from this distance I’m not in much of a position to guide you directly either. The best I can suggest is to go through my archives. Start here, and anything you don’t understand either try to find links back, or search for it in the bar on the right. Alternatively, there are some great mathematics texts out there you could try to get ahold of from Amazon or something. Actually, I’ll make a post tomorrow about that sort of thing, but I’ve got to head off to something else right now…

Comment by John Armstrong | March 21, 2008 | Reply
[…] math books Yesterday, someone left a comment, which I’m reinterpreting a bit as a call for help in what to do for self-directed study of […]

Pingback by Classic math books « The Unapologetic Mathematician | March 22, 2008 | Reply
Vishal, I did not become interested in mathematics until I turned 21 or so.. I did not go to college, my background is in computer science (been coding since i was 9 years old). I taught myself mathematics by buying books and learning them myself.. it has been very expensive.. I spend on average about $2000 a year on mathematics/statistics books and I choose to learn from advanced research oriented material rather than the basics as I believe it is worth it in the long run.

I suggest getting a copy of Maple and TexMacs for typesetting in a GUI very mathematical and professional looking papers.

Comment by Stephen Crowley | March 22, 2008 | Reply
Stephen, I think you’re meaning that towards Kasey.

Comment by John Armstrong | March 22, 2008 | Reply
My very first mathematics book was Measure Theory and Integration by MM Rao.

Comment by Stephen Crowley | March 22, 2008 | Reply
John, you are right, my comment was directed towards Kasey.

Comment by Stephen Crowley | March 22, 2008 | Reply

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

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