Lebesgue measure of a set $G$

Question

I want to show that $\lambda(G)=\infty$ where $G=\left\{(x,y):1<x\ \text{and}\ 0<y<\frac{1}{x}\right\}$. I have tried to define each special polygon as:

$\lambda(P_{1})=\frac{1}{2}-\epsilon$

$\lambda(P_{2})=\frac{1}{3}-\epsilon$

and:

$\lambda(P_{n})=\frac{1}{n}-\epsilon$

Each $P_{n}$ is also a special rectangle then:

$\lambda(P)=\sum_{k=1}^{n} \lambda(P_{k})=\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-n\epsilon$, if $n\rightarrow\infty$ then $\lambda(G)=\infty$

Is it right?

Answer 1

I don't think you need any epsilons. You can bound the measure of G below by the harmonic series. $$P_j=\{(n,n+1) \times(0,\frac{1}{n+1}):n>1\} $$ $$P=\bigcup_{j=1}^\infty P_j \subset G$$ $$ \lambda(P)<\lambda(G)$$