1. ## L-integrable?

Let $\displaystyle f_n(x)=\left\{\begin{array}{cc}1,&\mbox{ if for some integer i, x can be written } \frac{i}{2^n}\\0, & \mbox{ otherwise } \end{array}\right.$

(a) What is the limit of $\displaystyle f_n(x)?$
(b) Does $\displaystyle f_n(x)$ converge uniformly on $\displaystyle I = [0,1]?$
(c) Is $\displaystyle f_n$ Riemann integrable? Is it's limit?
(d) Show that $\displaystyle f$ is Lebesgue-integrable and evaluate the integral over $\displaystyle [0,1]$

I know this is a lengthy question, but I'm new to this Lebesgue stuff, so any miniscule bit of help is valued and appreciated.

2. Originally Posted by Anonymous1
Let $\displaystyle f_n(x)=\left\{\begin{array}{cc}1,&\mbox{ if for some integer i, x can be written } \frac{i}{2^n}\\0, & \mbox{ otherwise } \end{array}\right.$

(a) What is the limit of $\displaystyle f_n(x)?$
(b) Does $\displaystyle f_n(x)$ converge uniformly on $\displaystyle I = [0,1]?$
(c) Is $\displaystyle f_n$ Riemann integrable? Is it's limit?
(d) Show that $\displaystyle f$ is Lebesgue-integrable and evaluate the integral over $\displaystyle [0,1]$

I know this is a lengthy question, but I'm new to this Lebesgue stuff, so any miniscule bit of help is valued and appreciated.
a) Ever heard of the dyadic rationals? The limit will be $\displaystyle \mathbf{1}_{\mathbb{D}}$. Look it up if you have never seen it before.

b) You can figure this one out.

c) What do you get when you take the uniform partition and shrink it? In the case of the limit note that dyadic rationals are dense.

d) $\displaystyle \int \mathbf{1}_{\mathbb{D}} d\lambda=\lambda(\mathbb{D})$. Note the hint is in the name (dyadic rationals). Why is it measurable? What is its measure? (For both you can either use the cardinality or a superset of it).

If you have any problems with these, do post here.

3. (c) both are Riemann-integrable?

(d) I would assume the dyadic rationals are measurable since they are a subset of the rationals. I believe they have measure zero?

4. Originally Posted by Anonymous1
(c) both are Riemann-integrable?

(d) I would assume the dyadic rationals are measurable since they are a subset of the rationals. I believe they have measure zero?
c) The limit is not Riemann integrable, as any interval you have is going to contain a dyadic rational (they are dense), and also an irrational. Try to find the limsup and the liminf of the Riemann sum, and you will see they do not correspond.

d) That argument requires the sigma field to be complete, which in the case of the Lebesgue sigma field, it is. An other way to look at it is that each singleton set is closed, so the Borel sigma field contains any countable set (a countable set is a countable union of singleton sets).