# All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral ?

1. Oct 18, 2012

### dumbQuestion

"All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral"?

I am trying to learn about the Lebesgue integral and Lebesgue measurability. None of my text books really cover it from the basics, but I found this document online which seems to be pretty through in explaining the motivation behind developing the Lebesgue integral http://web.media.mit.edu/~lifton/snippets/measure_theory.pdf [Broken]

However, there is a statement I am having a hard time grasping, on the bottom of the first page:

"Third, all Lp spaces except for L fail to be complete under the Riemann
integral"

Here is what I understand: when saying "Lp spaces" I'm assuming this means metric spaces, right? I know from functional analysis that a complete metric space is one where there are no "gaps", or formally, where every Cauchy sequence has a limit that's also in the space. (That's why, for example the rational numbers with the st. Euclidean metric (Lp with p=2) is not complete, because we have gaps at all the irrational places)

Here's what I don't understand: what does it mean to be complete under the Riemann integral? I don't understand what this means. I thought a metric space would be a set of numbers, with a metric defined on it, and it would be complete or incomplete just based on that information alone. Where does the Reimann integral come in to play?

Last edited by a moderator: May 6, 2017
2. Oct 18, 2012

### micromass

Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

No, the $L^p$-spaces are a special kind of Banach space. They are defined as (for $1\leq p <+\infty$):

$$L^p([a,b])=\left\{f:[a,b]\rightarrow \mathbb{R} ~\text{measurable}~\left|~ \int_a^b |f|^p < +\infty \right.\right\}$$

The norm on this space is

$$\|f\|_p=\sqrt[p]{\int_a^b |f|^p}$$

The thing is that the above integral must be the Lebesgue integral. If we just focus on the Riemann-integral, then we find out there are not enough Riemann integrable functions and the space will be incomplete.

3. Oct 18, 2012

### dumbQuestion

Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

oh ok, thank you very much for clearing up my confusion

4. Oct 21, 2012

### Bacle2

Re: "All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral

An example for L^1(ℝ) , which you can generalize:

Take an enumeration {an} of the Rationals, and Let XS be

the characteristic function of the set S .

Define:

f_1=χa1

...
fna1+.......xan

.....

So that your function fn is 1 at each of the rationals ≤ an, and is

0 everywhere else. Then each of the fn is R-integrable, but the

sequence converges to the characteristic function of the Rationals, which is not

R-integrable.

Notice the limit functions is Lebesgue integrable. As a nitpick, remember: Riemann, not Reimann; I don't mind so much, but your

prof. may cringe.

Last edited: Oct 21, 2012