Risky Assets over Bernoulli Utility
Regarding your first question, the space on which you can apply Lebesgue's theorem is $\mathbb{R}_{+}^{N}$. The relevant $\sigma$-algebra is the Borel $\sigma$-algebra and the integration corresponds to the Lebesgue measure. Formally, with your notation, the function $f_m$ is defined on vectors $x=(x_1,\cdots,x_N)$ by
\begin{equation*}
f_m(x_1,\cdots,x_N) = u(\sum_{n}{\alpha_n^m x_n})
\end{equation*}

Regarding your second question, it seems to me that constructing a counter-example would be difficult and tedious. Proofs of continuity are rarely done that way. You might ask your teacher what kind of argument he has in mind. A possibility that you might wish to explore is to go back to the definition of continuity. Basically, you want to show that:
\begin{equation*}
\forall \alpha, \forall \epsilon&gt;0, \exists \delta&gt;0 \text{ such that } \|\alpha-\beta\|&lt;\delta \Rightarrow \|U(\alpha)-U(\beta)\|&lt;\epsilon
\end{equation*}
You can try to replace $U(\alpha)$ and $U(\beta)$ by their expression and manipulate the resulting integral. But this essentially boils down to proving Lebesgue's convergence theorem again : this solution is therefore not going to be simpler, but it might help you to understand what is behind this result.