Unbounded open set of $\mathbb{R}^n$ with finite measure

Question

I have been asked to find an unbounded open set of $\mathbb{R}^n$ with finite measure. I have thought about using a function such as $e^{-x^2}$ but I don't know very well how to start it. Any suggestions please?

Answer 1

Why not consider the set $$ G = \bigcup_{k=1}^{\infty} (2^k, 2^k+2^{-k}), $$ which has $m(G) = 1$.

Answer 2

Let $\{q_j\}_{j=1}^\infty$ be the rational points in $\mathbb{R}^n$ (ie each coordinate is a rational). Then the set $$ U=\cup_{j=1}^\infty B_{1/j^2}(q_j) $$ is open, dense, and has finite measure

Answer 3

Consider the graph of the function you suggested, from 1 to $\infty$. It's unbounded but it has finite 2-dimensional Lebesgue measure(area).