Let's consider a random variable $\Delta(t)$ and consider that one has to compute:
$$||\sup\limits_{t\in[0,1]}|\Delta(t)|||_{L^4}$$
It is computed like:
$$||\sup\limits_{t\in[0,1]}|\Delta(t)|||_{L^4}=\mathbb{E}(\sup\limits_{t\in[0,1]}|\Delta(t)|^4)^{\frac{1}{4}}\tag{1}$$
However, shouldn't it be like:
$$||\sup\limits_{t\in[0,1]}|\Delta(t)|||_{L^4}=\mathbb{E}(\color{red}{(}\sup\limits_{t\in[0,1]}|\Delta(t)|\color{red}{)}^4)^{\frac{1}{4}}\tag{2}$$
?
Could I consider $(1)$ the same as $(2)$? If so, why?