User talk:Mindey/MathQuestions

Definition of Outer measure
Quotation: $$\mathfrak{m}( \cdot )$$ is outer measure for any subset $$A$$ of $$\mathbb{R}^n$$ $$\Leftrightarrow$$ $$\mathfrak{m}: \mathcal{P}(\mathbb{R}^n) \mapsto \mathrm{inf} \left\{ \sum_k \mathrm{Vol}(I_k) \ s.t. \ \forall k: A \in \bigcup_k I_k \right\}$$, where $$I_k = \{ x \in \mathbb{R}^n: a_i \leqslant x_i \leqslant b_i, \forall i = 1,2,..,n \}$$ and $$\mathrm{Vol}(I_k) := \prod_{i=1}^n (b_i - a_i)$$.

My question: is it that $$\mathfrak{m}(\cdot)$$ -- the volume of the smallest $$I_k$$ that covers $$A$$, or not always? When is it that it's not?