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.



Inequality of Lebesgue integral with $L^p$-norm
It holds more generally replacing $L^p(\Omega)$ by a normed space $(B,\lVert \cdot \rVert\_B)$, assuming $M\colon t\mapsto X\_t(\cdot)$ goes from $[0,T]$ to $B$ is continuous. 


We have for each integer $N$ that 
$$\left\lVert\frac TN\sum\_{j=1}^NX\_{TkN^{-1}}(\cdot)\right\rVert\_B\leqslant\frac TN\sum\_{j=1}^N\left\lVert X\_{TkN^{-1}}(\cdot)\right\rVert\_B.$$
As $t\mapsto X\_t(\cdot)$ and $t\mapsto \lVert X\_t(\cdot)\rVert\_B$ are continuous, we conclude by a Riemann sum argument.


