Let
E⊆Rd. Let
{In}n∈N be a countable covering for
E of open real
d-intervals. Result
R2987: Set covering of Euclidean real scaled set shows than then the collection of scaled intervals
{λIn}n∈N is in turn a countable covering for
λE of open real
d-intervals. Results
now imply
μ∗(λE)≤μ∗(λ⋃n∈NIn)≤∑n∈Nμ∗(λIn)=∑n∈NVol(λIn)=|λ|d∑n∈NVol(In)
Since
{In}n∈N was an arbitrary covering of
E, we may apply
R1109: Antitonicity of infimum to extend the above inequality to the infimum element on the right hand side over all such coverings of
E to obtain
μ∗(λE)≤|λ|dinf
We may now proceed to apply this inequality in turn to the scaled set
\frac{1}{\lambda} E to obtain an inequality in the opposing direction
\begin{equation}
\mu^*(E) = \mu^* \Big( \lambda \Big( \frac{1}{\lambda} E \Big) \Big) = \mu^* \Big( \frac{1}{\lambda} (\lambda E) \Big) \leq \frac{1}{|\lambda|^d} \mu^*(\lambda E)
\end{equation}
Multiplying both sides by the nonzero quantity
|\lambda|^d yields now
|\lambda|^d \mu^*(E) \leq \mu^*(\lambda E). An inequality in both directions was established, whence
R1043: Equality from two inequalities for real numbers implies the equality
\mu^*(\lambda E) = |\lambda|^d \mu^*(E). This finishes the proof.
\square