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Let $\mu^*:\mathcal P(\mathbb R^2)\longrightarrow \mathbb R$ defined by $$\mu^*(E)=\inf\left\{\sum_{i=1}^\infty m(T_i)\mid E\subset \bigcup_{i=1}^\infty T_i\right\}$$ where $T_i$ are triangles and $m$ is the Lebesgue measure.

1) Show that $\mu^*$ is an exterior measure.

2) Which measure is given by $\mu^*$ ?

My work

I did 1), and for 2), I'm sure it's Lebesgue measure, but how can I justify it ? Is the fact that a triangle is homeomorphic to a cube enough ?

José Carlos Santos
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MSE
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  • Other than the very elegant solution @Surb gave, You can show that it does not matter if we calculate it with triangles and with rectangles, and the one with rectangles gives the Lebesgue measure. – Ranc Jul 26 '17 at 14:26

1 Answers1

4

Big hint

Lebesgue measure is the only one invariant by translation.

Surb
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