Lebesgue measure of an open unbounded set

Question

In his book Frank Jones defines the Lebesgue measure of an open set $U\subset\mathbb{R}^n$ by

$$\lambda (U):=\sup\{\lambda(P):P\subset U, P \text{ special polygon}\}$$

where a special polygon $P$ is a union of compact rectangles with disjoint nonempty interiors, with lebesgue measure $\lambda(P)$ defined as the sum of the areas of the compact rectangles.

Jones write: In case $U$ is not a bounded set, then we interpret the definition to produce $\lambda (U)=\infty$.

Question: Is it possible to actually prove from the definition that $\lambda (U)=\infty$ if $U$ is unbounded?

Answer 1

You have misquoted Frank Jones, which in PP.29-30 wrote (emphasis mine):

If $G \subset \Bbb R^n$ is an open set and if $G \ne \emptyset$, we define $$\lambda(G) = \sup\{\lambda(P) \mid P \subset G, P\text{ a special polygon}\}.$$ [...] consider the least upper bound of the set of numbers $\lambda(P)$ for $P \subset G$. In case this set is bounded, then $0 < \lambda(G) < \infty$. In case it is not a bounded set of numbers, then we interpret the definition to produce $\lambda(G) = \infty$. In particular, $\lambda(\Bbb R^n) = \infty$.

You thought "it" refers to $G$, in which case it would make no sense to say "bounded set of numbers", because elements of $G$ are not numbers.

In actuality, "it" refers to the set of numbers $\lambda(P)$ for $P \subset G$, the set which you are taking supremum of. This is the set of numbers that Frank Jones is referring to, and in the case that this set is not bounded, then surely the supremum is $\infty$.