Why can we write any lottery as a convex combination of the degenerate lotteries?
Pick any set of non-negative $p_1,\dots,p_n$ such that $p_1+\cdots+p_n=1$. The convex combination of degenerate lotteries $L^1,\dots,L^n$ with the $p_i$'s can be written as 
\begin{align}
p_1L^1+\cdots+p_nL^n&amp;=p_1(1,0,\dots,0)+\cdots+p_n(0,\dots,0,1)\\
&amp;=\bigl(p_1(1)+p_2(0)+\cdots+p_n(0),\;\dots,\;p_1(0)+\cdots+p_{n-1}(0)+p_n(1)\bigr)\\
&amp;=(p_1,\dots,p_n).
\end{align}
This is just a lottery $L=(p_1,\dots,p_n)$.