Lebesgue measure of $\left\{ (x, y) \mid 1 < x, 0<y<\frac{1}{x} \right\}$

Question

I am following Lebesgue Integration on Euclidean Space (2001) by Frank Jones.

I am working through Chapter 2 where he develops the notion of the Lebesgue measure $\lambda (A)$ of a set $A \subset \mathbb{R}^n$. He begins by defining $\lambda$ for closed rectangles:

For $I = [a_1, b_1] \times \dotsb \times [a_n, b_n]$, define $\lambda (A) = (b_1 - a_1) \dotsb (b_n - a_n)$.

Then that of "special polygons," which are unions of closed rectangles:

For $\displaystyle P = \bigcup_{k=1}^N I_k$ where $I_k$'s are closed rectangles and non-overlapping, i.e. their interiors are pairwise disjoint, define $\displaystyle \lambda (P) = \sum_{k=1}^N \lambda (I_k)$.

Then that of open sets:

For an open $G \subset \mathbb{R}^n$, define $\lambda (G) = \sup\{\; \lambda (P) \mid P \subset G, P \text{ is a special polygon} \;\}$.

Given the above definitions, a question asks me to prove that the Lebesgue measure of $\displaystyle G = \left\{ (x,y) \mid 1<x, 0<y<\frac{1}{x} \right\}$ is $\infty$.

I am not allowed to use that $\displaystyle \int_1^\infty \frac{1}{x} \;\mathrm{d}x = \left[ \log x \vphantom{\frac11} \right]_1^\infty = \log \infty - \log 1 = \infty $.

I thought about constructing a sequence of special polygons that mimics Riemann integration, where I subdivide $G$ into closed rectangles whose widths decrease as we proceed in the sequence:

Define $\displaystyle P_k = \bigcup_{l=1}^\infty I_{k, l}$ where $I_{k, l} = \left[ 1+\frac{l-1}{k}, 1+\frac{l}{k} \right] \times \left[ 0, \frac{k}{k+l} \right]$. Then let $\displaystyle P = \lim_{k \rightarrow \infty} P_k$.

However I see two problems with this approach:

1. $\{ P_k \}$ is not strictly increasing or decreasing, so I don't have a nice representation of $\displaystyle \lim_{k \rightarrow \infty} P_k$ as $\displaystyle \bigcup_{k=1}^\infty P_k$ or $\displaystyle \bigcap_{k=1}^\infty P_k$, making it difficult to prove $P = G$.
2. Each $P_k$ consists of closed rectangles while $G$ is open. So $P_k \subset G$ is false for every $P_k$, meaning that I can't use something like $\displaystyle \lim_{k \rightarrow \infty} \lambda (P_k)$ to compute $\lambda (G)$.

What should I do instead?