# 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 $$L=(\omega_{1},...,\omega_{N};p_{1},...,p_{N})\in\mathcal{L}$$: $$$$\label{eq:vnm} U(L)=\sum_{i=1}^{N}p_{i}u_{i}.$$$$ My question: Is it true that the assignment of numbers $$(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.