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Let $(\mathbb{R}^n,\mathcal{L},\lambda)$ be a measure space, with $\lambda$ the Lebesgue measure. Is the following statement true? Let $E\subset\mathbb{R}^n$, be a $\lambda$-measurable set, such that $\lambda(E)<+\infty$, then $E$ is bounded.

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2 Answers 2

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No - for example, consider $E := \bigcup_{n=1}^\infty [n, n + 2^{-n}]$. (Or for a more extreme example, consider that $\mathbb{Q}$ has measure 0.)

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No. AT the point $(i, 0, 0, \cdots, 0)$, with $i \in \mathbb{N}$, center a ball of measure $2^{-i}$. The set of balls is unbounded but with measure $2$.

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