Lebesgue measure proof for a set in R^2

[rustyjoker]

Homework Statement

Prove (using Lebesgue outer measure) that $$m_{2}^{*}(\{(x,y)\in\mathbb{R}^{2}\colon x>1,0<y<x^{-2}\})=m_{2}^{*}(A)<\infty$$

The Attempt at a Solution

I'm not sure if this is valid proof but I'd have done it like this:
$$\int_{1}^{k}\left|x^{-2}\right|\, dx <\infty \forall k\in\mathbb{N} \Rightarrow \int_{[1,\infty[}\left|x^{-2}\right|\, dx<\infty \Rightarrow A $$is measurable and$$ m_{2}^{*}(A)=\int_{1}^{\infty}x^{-2}\, dx<\infty.\qquad\square$$What else should I include to make sure the proof is valid?

 

[jbunniii]

I'm not sure if this is valid proof but I'd have done it like this:
$$\int_{1}^{k}\left|x^{-2}\right|\, dx <\infty \forall k\in\mathbb{N} \Rightarrow \int_{[1,\infty[}\left|x^{-2}\right|\, dx<\infty$$

No, this implication is not valid. If you replace $x^{-2}$ with $x^{-1}$, then your left hand inequality would still be true, but the right hand inequality would be false.

Similarly, the partial sums of an infinite series may all be finite, but that doesn't imply that the series converges. However, it is true that the series $\sum_{n=1}^{\infty}n^{-2}$ converges. Can you use this series to bound your integral?