- #1

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*external measure*of the set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of ##A## by finite or countable families of ##n##-paralleliped ##P_k=\prod_{i=1}^n I_i##$, where ##I_i## is of the form ##[a_{i,k},b_{i,k}]## or ##(a_{i,k},b_{i,k})## or ##(a_{i,k},b_{i,k}]## or ##[a_{i,k},b_{i,k})##, with ##a_{i,k}\le b_{i,k}##, whose measure is defined as $$m(P_k):=\prod_{i=1}^n(b_{i,k}-a_{i,k}).$$

I am intuitively inclined to believe that, if ##T\in\text{End}(\mathbb{R}^3)## is a unitary transformation, then $$\mu^{\ast}(A)=\mu^{\ast}(T(A))$$but I have no idea how we can prove it.

Is that so and, if it is, how can it be proved?

I ##\infty##-ly thank any answerer!