First Post

As part of an effort to continue to study of mathematics now that I am no longer taking classes I have decided to start blogging the mathematics I am studying. I intend for this blog both to provide a resource to others studying mathematics and to serve as a means for receiving feedback on my work. Any corrections, comments, or alternative proofs are welcome.

The general format of a post will a consist of brief introduction to a problem, including a discussion of some of the relevant definitions and theorems, the statement of the problem itself, and separate hint and solutions files.  The problems will mainly be based off standard problems from various math texts, but the introductions and solutions will be my own work unless stated otherwise.  My plan is to write-up two or three problems a week, but I am not going to commit to any deadlines just yet.

Any how without further ado, here is my first problem:

A set $X \subset \mathbb{R}$ is said to be of (Lebesgue) measure zero if for every $\epsilon >0$ there exists a countable set of open intervals $C = \{(a_i, b_i)\}_{i \in \mathbb{N}}$ such that $C$ covers $X$ (i.e $X \subseteq \bigcup_i (a_i, b_i)$) and $\sum_{i=1}^\infty
(b_i -a_i) < \epsilon$. The distance $b_i - a_i$ is referred to as the length of the interval $(a_i, b_i)$.

One can see right away that any finite set of points is measure zero. It isn't much more difficult to see that that the integers are measure zero, as is any countable set of points in $\mathbb{R}$. Even some uncountable sets, such as the Cantor set are of measure zero. This corresponds with our intuition about these sets not taking up space.

Exercise: Show that the interval $[0,1]$ is not measure zero.

Solution File

Questions or Comments are welcome.