Consider two distributions

$$F(x) =\begin{cases} 0 &\text{if} \ \ x < 0\\ \frac{1}{4} &\text{if} \ \ 0\leq x < 2\\ 1 &\text{if} \ \ 2\leq x \end{cases}$$ and $$G(x) =\begin{cases} 0 &\text{if} \ \ x < 1\\ 1 &\text{if} \ \ 1\leq x \\ \end{cases}$$ Show that the expected value of $x$ under $F$ is greater than the expected value of $x$ under $G$ but that $F$ does not first-order stochastically dominate $G$.

Attemped solution - It seems the expected value of both distributions will be infinity, so I do not know how to show the latter. For first-order stochastic dominance we have $$U(F) = \int_{-\infty}^{0}dx + \int_{0}^{2}\frac{1}{4}dx + \int_{2}^{\infty}dx$$ again it does not seem that we will get any meaningful evaluation.