In my book we defined $$(\mathbf{a}\otimes\mathbf{b})_{ij} = a_ib_j$$ then the book goes on and says the outer product is distributive. Then it does the following $$\mathbf{a}\otimes \mathbf{b} = (a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k})\otimes(b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k})= a_1b_1...

I have a set M of objects $m_{s}^r$ of domain size $s \in S=\{1 \ldots n\}$ containing $r \in R=\{1 \ldots n\}$ relevant objects. The highest ranked model will be $\alpha=max(\{f(\frac{r}{s}): m_{s}^r \in M \})$ \begin{equation} M=\{m_{s}^r | m_{1}^1 \ldots m_{n}^n \} \end{equation} \begin{e...

Let $A$ be a set and define $m^{**}(A) \in [0,\infty] $ by $m^{**} (A)= \inf\{m^{*}(\mathcal{O}) \ | \ A \subset \mathcal{O}, \mathcal{O} \ \text{open} \} $. And $m^{***}(A) = \sup \{m^*(F) \mid F \subset A, F$ is closed $\}$. Show how are $m^{**}(E)$ and $m^{***}(E)$ related to $m^{*}(E)$...