Prove that $\{(x,y)\in\mathbb{R}^2 | |x|+|y| = 1\}$ has Jordan content 0.

Question

Here is my working definition of what it means to have Jordan Content 0 (not that I am in a Real Analysis course, not an Algebra or Topology course).

A bounded subset $S$ of $\mathbb{R}^n$ is said to have Jordan content 0 provided that for each $\epsilon>0$, there is a finite collection $\mathcal{F}$ of generalized rectangles in $\mathbb{R}^n$ that cover $S$, the sum of whose volumes is less than $\epsilon$.

The textbook gives an example for if $S = \{(x,y) | 0\leq x \leq 1, y=x \}$, where it says that for every natural nuber $k$, if we let $\{I_j\}_{1\leq j \leq k}$ be the collection of $k$ generalized rectangles defined by $$ I_j = \left[ \frac{j-1}{k}, \frac{j}{k} \right] \times \left[ \frac{j-1}{k}, \frac{j}{k} \right],\ \ 1 \leq j \leq k$$ then this clearly covers $S$ and has volume $1/k$, which by the Archimedean property means that we can choose $k$ such that the volume is less than $\epsilon$.

I have no idea how to use this logic to solve the problem at hand without essentially solving it 4 times for 4 segments.

Answer 1

A good way that gives you a lot of such sets is to prove the following statements.

• The graph of a continuous function over a closed interval has Jordan content 0.
• The union of two Jordan content 0 sets has Jordan content 0.