# content zero

• Jul 30th 2010, 02:10 PM
Danneedshelp
content zero
Let $\displaystyle f$ be bounded on $\displaystyle [a,b]$. Show that if the set of discontinuous points of $\displaystyle f$ has content zero, then $\displaystyle f$ is integrable.

I am trying do a second semester analysis self study and I am pretty stuck on a few major concepts of integration. I would appriciate some help with this one. If any thing, I just need a place to get started.

Here is what I have set up so far. I am given:

$\displaystyle D=\{d_{1}, d_{2},...,d_{N}\}$ has content zero and $\displaystyle f$ is bounded on $\displaystyle [a,b]$. Thus, $\displaystyle D\subseteq\bigcup_{n=1}^{N}O_{n}$ and $\displaystyle \sum_{n=1}^{N}|O_{n}|\leq\epsilon$ (where, $\displaystyle O_{n}$ represents open sets). Moreover, there exists an $\displaystyle M>0$ satistying $\displaystyle |f(x)|\leq\\M$ for all $\displaystyle x\in[a,b]$.

My goal is to show there exists a partition $\displaystyle P_{\epsilopn}$ such that $\displaystyle U(f,P_{\epsilopn})-L(f,P_{\epsilopn})<\epsilon$ for any choice of
$\displaystyle \epsilon>0$.

Since the goal begins with a universal quantifier I let $\displaystyle \epsilon>0$ be arbitrary. Now, I need to show the existance of such a partition.

As you can see I do not have much. I am not sure how to get started.

Q1: Does the set of open intervals whos union contain the set D have to be nested?
Q2: Can I assume $\displaystyle [a,b]$ is compact to create a finite subcover and use that somehow?

Thanks
• Jul 30th 2010, 02:38 PM
TheEmptySet
If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

Hint: squeeze the one point of discontinuity into a set of length $\displaystyle \frac{\epsilon}{2M}$ then the function is continuous on $\displaystyle [a,c-\frac{\epsilon}{4M}]$ and $\displaystyle [c+\frac{\epsilon}{4M},b]$
• Jul 30th 2010, 03:29 PM
Plato
The concept of content is a bit dated.
If you are using the text material by T.H. Hildebrand then a subset $\displaystyle E\subseteq [a,b]$ has content if and only if its characteristic function $\displaystyle \chi_E$ is Riemann integrable on $\displaystyle [a,b]$.
That seems to be consistent with your notation. If so how can you use that?

If that is not the definition of content you have, what textbook are you following?
• Jul 30th 2010, 04:04 PM
Danneedshelp
Quote:

Originally Posted by TheEmptySet
If I am reading this correctly the set D has only a finite number of elements in it. So try a easier problem first. What if f is bounded and is only discontinuous at one point c in [a,b]. How could you create a partition that that would show f is integrable. If you can solve this problem it will generalize to any finite set of points of discontinuity.

Hint: squeeze the one point of discontinuity into a set of length $\displaystyle \frac{\epsilon}{2M}$ then the function is continuous on $\displaystyle [a,c-\frac{\epsilon}{4M}]$ and $\displaystyle [c+\frac{\epsilon}{4M},b]$

Well, I would want to construct a partition that minimizes the effect of the discontinuity at $\displaystyle c$ by embedding $\displaystyle c$ in a very small subinterval.

Quote:

Originally Posted by Plato
The concept of content is a bit dated.
If you are using the text material by T.H. Hildebrand then a subset $\displaystyle E\subseteq [a,b]$ has content if and only if its characteristic function $\displaystyle \chi_E$ is Riemann integrable on $\displaystyle [a,b]$.
That seems to be consistent with your notation. If so how can you use that?

If that is not the definition of content you have, what textbook are you following?

I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.
• Jul 30th 2010, 04:55 PM
Plato
Quote:

Originally Posted by Danneedshelp
Well, I would want to construct a partition that minimizes the effect of the discontinuity at $\displaystyle c$ by embedding $\displaystyle c$ in a very small subinterval.

I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.

I do not know Abbots' text book.
I your have reasonable university library access you may want to find Hildebrandt's book or a book by S K Breberian.
• Jul 30th 2010, 08:14 PM
Jose27
A set $\displaystyle X\subset \mathbb{R}$ has content zero iff for every $\displaystyle \varepsilon >0$ there exist $\displaystyle (I_k)_{k=1}^{n}$ intervals with $\displaystyle X \subset \cup I_k$ and $\displaystyle \sum m(I_k) < \varepsilon$.

So in your problem, given $\displaystyle \varepsilon >0$ just use an argument completely analogous to that where you have a finite number of discontinuities.