How do I find the lebesgue measure of a interval $[n,n+\frac{1}{n^{2}}]$ when $n\in\mathbb{N}$? I have to use the following definition:
The set-function $\lambda^{n}$ on ($\mathbb{R}^{n}, \mathcal{B}(\mathbb{R}^{n})$) that assigns every half-open $[[a,b)) = [a_{1},b_{1}) \times \dots \times [a_{n},b_{n})\in\mathcal{J}$ the value: $ \lambda^{n}([[a,b))):=\prod_{j=1}^{n}(b_{j}-a_{j}) $ is called n-dimensional Lebesgue measure.
I have considered to write $[n,n+\frac{1}{n^{2}}]$ as a union of distinct half-open intervals, but with no luck. I have tried to write it:
$[n,n+\frac{1}{n^{2}}] = [n,n+\frac{1}{n^{2}}-1) \cup (n+\frac{1}{n^{2}}-1,n+\frac{1}{n^{2}}]$
But then I miss the point $n+\frac{1}{n^{2}}$?