Example of decreasing sequence of sets with first set having infinite measure

real-analysis examples-counterexamples measure-theory lebesgue-measure

I was wondering if someone could please give me an example of a sequence of decreasing sets where the first set has infinite Lebesgue measure; i.e., $\{B_{n}\}_{n=1}^{\infty}$ such that $m(B_{1}) = \infty$ but $m(\cap_{n=1}^{\infty} B_{n}) \neq \lim_{n \to \infty}m(B_{n})$?

thank you.


Solution 1:

Hint: Consider $$B_n := \{x \in \mathbb{R}; |x| \geq n\}.$$

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