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Suppose $E$ is a subset of $\mathbb{R^n}$ and exterior measure of $E$ is defined as $m_*(E)=\inf\sum_{i=1}^{\infty}|Q_i|$ where $E\subset\bigcup_{i=1}^{\infty}Q_i$ and $Q_i$ are closed cube. The question that i dont understand is that why it would not suffice to allow finite sums in defining outer measure?

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2 Answers 2

Consider $E = \mathbb{N}$ as a subset of $\mathbb{R}$. Then its exterior measure is clearly zero. But if you were to only cover it with finite cubes, you would have to cover at least two of the points in $E$ by one cube. That cube would have a size of at least 1.

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Consider the line $L=\{(x,y) \in \mathbb R^2 : x=y\}$ how would you cover $L$ by a finite number of cubes or rectangles so that the outer measure of the cubes was finite.

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