In my problem I have a measure $ \eta $ at $ (0, \infty) \times (0, \infty) $ such that $ \eta (\text{d}s, \text{d}x) = \dfrac{C}{x^{\alpha + 1}} \text{d}s \text{d}x $ and they ask me for the values of $ \alpha $ such that $ \eta $ is $ \sigma-$ finite.
Context:
I have a reasoning that is wrong but I can not find the error. It is clear that: $$(0,\infty)\times (0,\infty)=\bigcup_{n=1}^{\infty} \left( \frac{1}{n},n \right)^{2}$$ Let us denote the interval $ (1/n, n) $ Let us denote the interval $ (1 / n, n) $ by $ A_{n}$. Then: $$\eta(A_{n}\times A_{n})=C\left(n-\frac{1}{n}\right)\int_{1/n}^{n}\frac{\,dx}{x^{\alpha+1}}<\infty$$
Therefore $ \eta $ is $ \sigma-$ finite, the value of $ \alpha $ does not matter. This reasoning is what leads me to think that I have some misconception in my solution. I would appreciate your help and in case I am wrong, I would appreciate any clues to find those $ \alpha $ values.