I have a question to the following exercise: On $\mathbb{R}^n$ we look at the outer measure $\lambda_*(Q)$ = {${\inf{\sum{\vert Q\vert} | A \subseteq \bigcup_{j \in\mathbb{N} }Q_j}} $ } where the $Q_j$ on the right side are axis-parallel closed cubes.
Prove the following property: Let $E \subseteq \mathbb{R}^n$
$$\lambda_*(E) = \underset{E \subseteq O,O is open}{inf} \lambda_*(O).$$
How can I understand the notation on the right side of the second equation? And how can I write it better with MathJax?
The notation $$\inf_{\text{$E\subseteq O$, $O$ is open}}\lambda_*(O)$$ means $$\inf\big\{\lambda_*(O) : \text{$E\subseteq O$ with $O\subseteq\mathbb R^n$ open}\big\},$$ i.e., the infimum of $\lambda_*$ over all open subsets of $\mathbb R^n$ that include $E$. You could use
\\to stack the subscript: $$\inf_{~~~E\subseteq O\\\text{$O$ is open}}\lambda_*(O)$$ or you could use the fact that the set of all open subsets of $\mathbb R^n$, a subset of the powerset $\mathcal P(\mathbb R^n)$, is called topology and often denoted by $\mathcal T$ or $\mathcal O$: $$\inf_{E\subseteq O,~O\in\mathcal T}\lambda_*(O)$$