Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.

I would like to know if this property of Lebesgue integrals:

$|\int_0^TX_t(\omega)dt|\leq\int_0^T|X_t(\omega)|dt$

is also valid for $L^p$-norm, i.e. if this is true:

$||\int_0^TX_t(\omega)dt||_{L^p(\omega)}\leq\int_0^T||X_t(\omega)||_{L^p(\omega)}dt$

Indeed, I have just proved that:

$||\int_0^TX_t(\omega)dt||_{L^p(\omega)}=(\int_R (\int_0^TX_t(\omega)dt)^pd\omega)^\frac{1}{p}\leq(\int_0^T||X_t(\omega)||_{L^p(\omega)}^pdt)^\frac{1}{p}\cdot T^\frac{p-1}{p}$,

where I applied Hölder's inequality and Fubini's theorem.

Thank you for your attention.