# prove the unit interval has outer measure 1.

For a subset $$E \subset \mathbb{R}^d$$ define the outer measure

$$m_*(E)=\inf \{\Sigma \vert I_j \vert\}$$

Where the infimum is taken over all countable coverings of $$E$$ by closed intervals. So $$E \subset \bigcup_{j=1}^\infty I_j$$.

Show $$m_*([0,1])=1.$$

How do I go about this rigorously? What would my $$I_j$$ be?

How about for general $$[a,b]$$ how can I show its just $$b-a$$?

Assuming that $$d = 1$$ to keep things simple, we have by definition $$|[0,1]| = 1$$. (More generally, $$|[a,b]| = b - a$$, and for an elementary set which is a finite union of disjoint intervals we sum up their lengths.) This is the pre-measure that leads to the Lebesgue measure. Note that $$| \cdot |$$ can be shown to be countably subadditive for elementary sets.

Since $$[0, 1]$$ is itself a closed interval, we can take $$I_1 = [0,1]$$, $$I_j = \emptyset, \; j \geq 2$$ as a particular covering. Then the outer measure satisfies $$m^*([0,1]) \leq |[0,1]| = 1$$ (as it is an infimum).

Now for the other direction. Consider any countable covering of $$[0,1]$$ by closed intervals $$(I_j)_{j \geq 1}$$ (i.e. elementary sets with $$[0,1] \subseteq \bigcup_{j \geq 1} I_j$$). By countable subadditivity, $$\sum_{j \geq 1} |I_j| \geq |\bigcup_{j \geq 1} I_j|$$. By monotonicity, $$|\bigcup_{j \geq 1} I_j| \geq |[0,1]| = 1$$. Putting this together shows that $$m^*([0,1]) \geq 1$$ (again, because $$m^*$$ is an infimum).

Hence $$m^*([0,1]) = 1$$. For general $$[a,b]$$, which is an interval/elementary set, the proof is essentially identical. (More generally, this says that the Lebesgue outer measure $$m^*$$ agrees with the Lebesgue measure $$m$$ on elementary sets, which we call $$|\cdot|$$ here).

• I do not see how in the second paragraph you conclude the exterior measure is bounded above by 1. Commented Jan 15, 2021 at 21:46
• is it because exterior is defined as an infimum? Commented Jan 15, 2021 at 21:46
• Yup, in particular the lower bound part - we have exhibited a particular covering.
– JKL
Commented Jan 15, 2021 at 21:53