Help understanding definition of expected utility function This is the definition of an expected utility function that I'm working with:

The utility function $U:\mathcal{L}\to\mathbb{R}$ has an expected utility form if there is an assignment of numbers $(u_{1},...,u_{N})$ to the $N$ outcomes $\{\omega_{1},...,\omega_{N}\}$, such that the following is satisfied for any simple lottery $L=(\omega_{1},...,\omega_{N};p_{1},...,p_{N})\in\mathcal{L}$:
\begin{equation}\label{eq:vnm}
    U(L)=\sum_{i=1}^{N}p_{i}u_{i}.
\end{equation}
My question: Is it true that the assignment of numbers $(u_{1},...,u_{N})$ are given by the Bernoulli function $u:\Omega\to\mathbb{R}$, where $\Omega$ denotes the sample space. And is it true that $U(L):\mathcal{L}\to\mathbb{R}$ is called the Von Neumann-Morgenstern expected utility function, where $\mathcal{L}$ denotes the set of simple lotteries?
That's true, but it should be constructed the other way round: The assignment of numbers $u_i$ to outcomes is not given by the Bernoulli utility function but defines it. The terminology of Bernoulli and Von Neumann-Morgenstern utility functions is common, but not universally used in the context of EUT.