$\partial([0,1]^2)$ as a cross product

Question

I am trying to write the boundary $\partial([0,1]^2)$ as the cross product of 4 sets representing each segment such as I end up with each segment looking like the cross product of closed intervals starting with $0$

i.e. $[0,x_1] \times [0,x_2]$

So I would have:

$$\partial([0,1]^2) = A_1 \cup A_2 \cup A_3 \cup A_4 $$

such that

$$A_1 = [0,1] \times [0,0] $$ $$A_2 = \{1\} \times [0,0] $$ $$A_3 = [0,1] \times \{1\} $$ $$A_4 = [0,0] \times [0,1] $$

I've been thinking about this for two days now and cannot figure it out. I get stuck at sets $A_2$ and $A_3$ I need to express the singleton $\{1\}$ as an interval in the form of $[0,$ "something"$]$ It must start with a zero in the x-axis because I need to use a measure defined as

$$\mu([0,x] \times [0,y]) = xy \ \ \ \forall (x,y) \in [0,1]^2 $$

and show that it gives $0$ everywhere on $\partial([0,1]^{2})$

Answer 1

As a complement of what I said in comments, what about using the properties of measure to show that such $\mu$ satisfies

• $\mu([0,x]\times[y,1]) = x(1-y)$
• $\mu([x,1]\times[0,y]) = (1-x)y$
• $\mu([x,1]\times[y,1]) = (1-x)(1-y)$

for $x,y\in[0,1]$ ?

Doing this, you can write $\{1\}$ as $[1,1]$ and the problem is done pretty straightforward.